Download Target-oriented full waveform inversion of seismic data

Target-oriented full waveform inversion
of seismic data
- development and verification on field
data
X.R. Staal
M.Sc. Thesis
Supervisor: dr. ir. D.J. Verschuur
Mentors:
prof. dr. ir. A. Gisolf
Laboratory of Acoustical Imaging and Sound Control
Department of Imaging Science and Technology
Faculty of Applied Sciences
Delft University of Technology
Delft, October 2009
ii
Graduation committee:
prof. dr. ir. A. Gisolf
dr. ir. D.J. Verschuur
dr. K.W.A. van Dongen
dr. F. Bociort
ir. P. Spaans
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Contents
1 Introduction
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2 Target-oriented forward modeling
2.1 Linear modeling . . . . . . . . . .
2.2 Nonlinear modeling . . . . . . . .
2.3 Conversion to (x, t) domain . . .
2.4 Propagation to the surface . . . .
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3 Target-oriented data manipulation
3.1 Back propagation . . . . . . . . .
3.2 Redatuming . . . . . . . . . . . .
3.3 Traveltime operator estimation .
3.4 Amplitude operator estimation .
3.5 CMP domain . . . . . . . . . . .
3.6 Radon transform . . . . . . . . .
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4 Full
4.1
4.2
4.3
waveform inversion
Transmission effects . . . . . . . . .
Wavelet estimation . . . . . . . . .
Linear inversion . . . . . . . . . . .
4.3.1 Least-squares constraint . .
4.3.2 Multiplicative regularization
4.3.3 Final result . . . . . . . . .
4.4 Nonlinear inversion . . . . . . . . .
4.5 Full inversion scheme . . . . . . . .
5 Synthetic data results
5.1 Seismic data . . . . . . . . . . . . .
5.2 Linear inversion . . . . . . . . . . .
5.2.1 Least-squares . . . . . . . .
5.2.2 Multiplicative regularization
5.3 Nonlinear inversion . . . . . . . . .
5.4 Elastic inversion . . . . . . . . . . .
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5.5
Linear inversion with updated kinematics . . . . . . . . . . . . . . . . .
5.5.1 Least-squares . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2 Multiplicative regularization . . . . . . . . . . . . . . . . . . . .
6 Field data results
6.1 Seismic data . . . . . . . . . . . . . . . .
6.2 Linear inversion . . . . . . . . . . . . . .
6.2.1 Well matching . . . . . . . . . . .
6.2.2 Least-squares . . . . . . . . . . .
6.2.3 Multiplicative regularization . . .
6.3 Velocity extraction . . . . . . . . . . . .
6.4 Linear inversion with updated kinematics
6.4.1 Least-squares . . . . . . . . . . .
6.4.2 Multiplicative regularisation . . .
6.5 Comparison to in-house results . . . . .
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7 Conclusions and recommendations
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8 Acknowledgements
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A The radon transform
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B Genetic Algorithm
B.1 Chromosomes and populations
B.2 Initialization of chromosomes
B.3 Evaluation . . . . . . . . . . .
B.4 Selection . . . . . . . . . . . .
B.5 Reproduction of chromosomes
B.6 Adaptation . . . . . . . . . .
B.7 Final result . . . . . . . . . .
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Chapter 1
Introduction
The conventional way to gain structural information about the earth is by acquisition
and processing of seismic data. Seismic data is acquired by sending sound waves into
the earth (for example by means of an explosion) and recording the sound that is
reflected back towards the surface. The recordings (shot records) are made using an
array of receivers, resulting in a multi-angled measurement of the seismic response
of the subsurface. Figure 1.1 shows a simple model of the earth and a seismic data
modeled in the model.
The main objective of a seismic experiment is to image the subsurface. Imaging methods range from very simple ones (stacking) to quite advanced ones (pre-stack depth
migration). All methods require an analysis of the propagation speeds of sound waves
in the subsurface, such that corrections for the propagation effect can be made to the
measurements. After adding contributions from all shot records a final image of the
subsurface can be obtained. Imaging reveals the structural composition of the subsurface, the depth and shape of earth layers, and can be combined with geological
knowledge to get a good idea about the composition of the earth.
Figure 1.1: Seismic data. Left: a simple earth model. Right: a seismic measurement,
a single reflection event is highlighted in red.
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2
Chapter 1. Introduction
Imaging does a good job at using seismic data to learn about the subsurface, but
doesn’t use all the information that is contained in the seismic data. The amplitude
of reflected sound in the subsurface is a function of contrasts in material properties;
propagation speed of pressure waves (vp ), propagation speed of shear waves (vs ) and
density (ρ). AVO (Amplitude Versus Offset) analysis uses reflection amplitudes along a
single reflector, such as the red line in figure 1.1, to gain information on these material
properties; typically using statistical information about the effect on seismic reflection
amplitudes of transitions in sediments. With AVO analysis, reflection events that signal
a transition to oil-bearing sediments can be identified. For more information on AVO,
refer to Aki and Richards (1980), Castagna and Backus (1993).
Full waveform inversion as discussed in this thesis combines imaging and AVO analysis
into one process; a spatial map of material properties (vp , vs and ρ) is estimated.
Amplitude inversion problems are normally large problems, involving a large amount
of parameters (three parameters for every sample of the map) to be estimated from the
seismic data. With seismic shots typically spanning 2-5 kilometers laterally and in the
order of a kilometer in depth, the volumes to be inverted are large and very efficient
inversion schemes are needed. For more information on amplitude inversion, refer to
Aki and Richards (1980); an excellent overview of all the above subjects was written
by Yilmaz (2000).
Figure 1.2: An earth model with seismic wave paths. Left; surface data. Middle;
redatumed. Right; CMP domain.
DELPHI’s CFP technology offers a way to drastically reduce the inversion volume
with redatuming. Figure 1.2 schematically shows the scheme. On the left the surface
seismics require inversion of a large volume (in green), much larger than the zone of
interest (target zone, in red). In the middle the surface seismic data has been redatumed
to the target datum: using CFP technology, propagation effects in the seismic data
between the surface and the target datum are removed, greatly reducing the inversion
volume. On the right the redatumed seismic data has been resorted to the common
midpoint (CMP) domain, reducing the inversion to an approximately one-dimensional
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problem. The zone of interest can then be inverted piecewise with each sub-inversion
being one-dimensional. For more information on CFP technology, refer to Thorbecke
(1997).
Presented with many small inversion problems instead of one very large one, possibilities arise to improve on industry standard inversion techniques. This thesis presents
a first attempt at the target-oriented inversion scheme using redatuming, including
tests on a synthetic dataset and validation on a field dataset. Amplitude inversions
are performed using a standard linear algorithm, an improved linear algorithm and an
attempt was made at a fully nonlinear amplitude inversion.
Chapter 2
Target-oriented forward modeling
This chapter describes a way to numerically forward model a target-oriented seismic
record. The target-oriented seismic record contains only reflection information from a
(laterally invariant) part of an earth-model. First a one-dimensional target (in depth)
is chosen in a two-dimensional model of the earth (depth and one horizontal direction),
then the seismic response at the top of the target is numerically modeled as plane
waves under the assumption of a local laterally invariant medium. The modeled plane
wave response is transformed into a point source response and this process is repeated
for many points along the target horizon. Wave propagation effects are added to the
resultant data to simulate a measurement that was taken at the surface as would be
the case in an actual seismic measurement. For this project two separate numerical forward modeling algorithms were used, one is a linear algorithm that is computationally
efficient but not exact, the other one is a nonlinear algorithm that is computationally
expensive but exact.
2.1
Linear modeling
The linear forward modeling algorithm used for this project calculates a seismic response in the (sx , τ ) domain, where sx is the horizontal slowness and τ is the two-way
traveltime of a plane-wave through the medium. Slowness is defined as the inverse
of p-wave velocity ( v1p ) and the horizontal slowness is the horizontal component of
the slowness vector. The linear modeling algorithm thus represents the wavefield as
plane-waves traveling through the model.
In order to linearize the forward modeling of seismic data, medium properties are
separated into a slowly varying background model (vp0 ,vs0 ,ρ0 ) that will generate no
reflections but defines the kinematics of the wave propagation and a model (Δln(vp ),
Δln(vs ),Δln(ρ)) containing property contrasts across an interface that generate all the
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Chapter 2. Target-oriented forward modeling
reflection events.
In the algorithm the effect of a contrast in model parameters (vp , vs , ρ) is calculated
as the arrival of a pulse at the two-way traveltime (τtwt ) corresponding to the depth
of a contrast, convolved with the wavelet and multiplied with the reflection amplitude
corresponding to the amplitude of the contrast:
p(sx , τ ) = Rpp (sx )δ(τ − τtwt ) ∗ w(τ ),
(2.1)
which can be simplified to
p(sx , τ ) = Rpp (sx )w(τ − τtwt ).
(2.2)
In this notation the recorded pressure wave is a function of sx (horizontal slowness)
and τ (time) in the plane-wave domain. Rpp represents the linearized reflection coefficient adapted from Shuey (1985), a function of angle of incidence (calculated from
background pressure- and shear wave velocities) and property contrasts:
Rpp (sx ) = A(sx , vs0 )Δln(ρ) + B(sx , vp0 )Δln(vp0 ) + C(sx , vs0 )Δln(vs ),
in which:
2 2
sx )
A(sx , vs0) = 12 (1 − 4vs0
1
1
B(sx , vp0 ) = 2 1−v2 s2
p0 x
C(sx , vs0 ) =
(2.3)
(2.4)
2 2
−4vs0
sx .
τT W T represents the two-way traveltime of a plane-wave through the medium, down to
a reflector and back:
z 1 − s2x vp0 (z)2
.
(2.5)
τtwt (z, sx , vp0 ) = 2Δz
vp0 (z)
0
The linearized reflection coefficients allow a separation of variables into a matrix (M)
containing the functions τtwt , w, A, B and C as functions of (sx ,vp0 ,vs0 ,τ ,z) and a vector
(b) containing the contrasts between samples in the model as a function of (Δln(vp ),
Δln(vs ), Δln(ρ)). The final linear forward modeling algorithm is then given as:
p(sx , τ ) = M(sx , vp0 , vs0 , z)b(vp , vs , ρ).
(2.6)
Figure 2.1 shows a schematic interpretation is given on how the modeling scheme works.
For six contrasts between depth samples, partial responses are calculated in matrix
M, divided in different slowness values (sx1 and sx2 ). Each sub matrix incorporates
the wavelet at different two-way traveltimes τT W T , scaled respectively by A(sx , vs0 ),
B(sx , vp0 ) and C(sx , vs0 ). When these partial responses are multiplied with the property
contrasts given in vector b and added together, the full linear response of all the
contrasts is found. In this example, only the third contrast had a nonzero value, giving
a single wavelet response for each slowness value.
2.2. Nonlinear modeling
7
p
M
b
sx1
=
*
sx2
A*w(ʏͲʏTWT)
B*w(ʏͲʏTWT)
ȴln(ʌ)
1
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6
ȴln(vp)
1
.
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6
ȴln(vs)
1
.
.
6
C*w(ʏͲʏTWT)
Figure 2.1: Schematic representation of linear forward modeling. The modeled wavefield is a multiplication of matrix M, containing traveltime calculations and (in this
case) two incident angles, with vector b, containing reflecting contrasts at (in this
case) six depths.
2.2
Nonlinear modeling
The nonlinear forward modeling algorithm used in this project is the Kennett Invariant
Embedding Method (Kennett (1983)), an algorithm that calculates the full waveform
response of a 1-dimensional model in the (sx , ω) domain (horizontal slowness and radial
frequency). In this algorithm the total response of a horizontally layered medium is
built from bottom to top. To fully describe the response of a homogeneous layer with a
reflecting contrast at the top, as shown in Figure 2.2, reflection and propagation effects
need to be calculated.
P1+
S1+
P1-
S1-
vp1, vs1, ȡ1
d1
vp2, vs2, ȡ2
P2-
S2-
P2+
S2+
Figure 2.2: A single layer of thickness d1 with a reflecting contrast at the top.
There are four incoming wavefields, a down-going pressure wave P1+ , a down-going
shear wave S1+ , an up-going pressure wave P2− and an up-going shear wave S2− . The two
down-going waves will partly reflect immediately at the contrast and partly transmit
through the contrast and propagate through the layer. The two up-going wavefields
first propagate through the layer, then partly transmit through it and partly reflect,
propagating back through the layer. Resulting from all these interactions are P1− , S1− ,
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Chapter 2. Target-oriented forward modeling
P2+ and S2+ , calculated (in matrix notation) as:
⎤ ⎡
⎡
Q̂p
Q̂s
Rsp
Tpp
Tsp
Rpp
P1−
⎢ − ⎥ ⎢
Rss
Tps Q̂p
Tss Q̂s
⎢ S1 ⎥ ⎢ Rps
⎢ + ⎥=⎢
⎣ P2 ⎦ ⎣ Tpp Q̂p Tsp Q̂p Q̂p Rpp Q̂p Q̂p Rsp
Q̂s
+
S2
Tps Q̂s Tss Q̂s Q̂s Rps Q̂p Q̂s Rss Q̂s
⎤⎡
⎤
P1+
⎢
⎥⎢
⎥ ⎥
R T’ ⎢ S1+ ⎥
⎥⎢
⎥
⎥⎢
⎥=
⎢ − ⎥,
T R’ ⎣ P2 ⎦
⎦⎣
⎦
S2−
(2.7)
where R and T are reflection and transmission coefficients respectively and Q̂p and Q̂s
contain propagation effects for pressure- and shear waves respectively.
P1+
S1+
P2−
S2−
⎤
⎡
The reflection and transmission coefficients needed to evaluate these equations are
taken from the Zoeppritz equations. The propagation effects Q̂ over thickness d1 are
evaluated as:
√ 2 2
iω vd
1−v p
p
Q̂p (ω, d, vp, p) = e √ p
(2.8)
d
2 2
Q̂s (ω, d, vs, p) = eiω vs 1−vs p ,
representing a phase shift in the frequency domain.
P1+
S1+
P1-
S1-
vp1, vs1, ȡ1
d1
vp2, vs2, ȡ2
vp3, vs3, ȡ3
P2+
S2+
P2-
S2-
P3-
S3-
P3+
S3+
d2
Figure 2.3: Two layers of thickness d1 and d2 with reflecting contrast at the tops.
Having calculated the full response of one layer, it is possible to connect a second layer,
as depicted in figure 2.3. Then the total response, starting with the responses from
both single layers can be calculated as:
⎡
⎡
⎤
⎤
P1−
P1+
⎢
⎢ − ⎥ ⎥
R1 T’1 ⎢ S1+ ⎥
⎢ S1 ⎥
=
(2.9)
⎢ + ⎥
⎢
⎥,
T1 R’1 ⎣ P2− ⎦
⎣ P2 ⎦
S2+
S2−
and:
⎡
⎢
⎢
⎢
⎣
P2−
S2−
P3+
S3+
⎤
⎡
⎢
⎥ R2 T’2 ⎢
⎥
⎢
⎥=
T2 R’2 ⎣
⎦
P2+
S2+
P3−
S3−
⎤
⎥
⎥
⎥.
⎦
(2.10)
2.3. Conversion to (x, t) domain
9
With some rewriting, the combined response of both layers is found:
⎤
⎤
⎡
⎡
P1+
P1−
⎢
⎥
⎢ − ⎥ Rtot T’tot ⎢ S1+ ⎥
⎢ S1 ⎥
⎢ − ⎥,
⎢ + ⎥=
Ttot R’tot ⎣ P3 ⎦
⎣ P3 ⎦
+
S3
S3−
(2.11)
with indices:
Rtot = R1 + T’1 (I − R2 R’1 )−1 R2 T1 ,
(2.12)
T’tot = T’1 (I − R2 R’1 )−1 T’2 ,
(2.13)
Ttot = T2 (I − R’1 R2 )−1 T1 ,
(2.14)
R’tot = R’2 + T2 (I − R’1 R2 )−1 R’1 T’2 .
(2.15)
Using these equations, it is possible to add one layer at a time to the total response
matrix calculated so far, until the entire model is covered and the full seismic response is
obtained in the (sx , ω) domain. Using the wavelet as the input wavefield P1− and setting
all the other input wavefields to zero, the output P1+ then simulates a 1-dimensional
seismic shot record in the (sx , ω) domain. An inverse temporal Fourier transform finally
brings the result to the (sx , τ ) domain. More information on the Kennett Invariant
Embedding Method can be found in Kennett (1983).
2.3
Conversion to (x, t) domain
With the algorithms described in the two previous sections, a seismic measurement can
be numerically modeled in the (sx , τ ) domain with the source and receivers at the top
of the target. To obtain results that can be compared to actual seismic measurements,
these results should be converted to a result in the (x, t) domain and wave propagation
effects towards the surface should be added. The conversion of wavefields in the (sx , τ )
domain is governed by the inverse Radon transform given as:
p(x, t) = Ft−1 Fx−1 Fτ p̂(sx , τ )|sx = kx .
(2.16)
ω
For a derivation of this transform, please refer to appendix A.
2.4
Propagation to the surface
The final step to model target-oriented surface data is to propagate the wavefield
up to the surface, this process uses wavefield extrapolation as described by Berkhout
10
Chapter 2. Target-oriented forward modeling
(1984) and Thorbecke (1997). Wavefield extrapolation is closely related to the Huygens principle; all points on a wavefront act as secondary sources, thus propagating
the wave. Figure 2.4 shows this schematically; a source emits a spherical wavefront.
This wavefront acts as a string of new weaker (due to geometrical spreading) sources,
all emitting their own secondary wavefront (dotted lines). Together these secondary
wavefronts form the propagated primary wavefront.
Figure 2.4: Wave propagation by Huygens’s principle, every point on a wavefront acts
as a new source, creating a secondary wavefront.
When this idea is applied to seismic datasets, it is possible forward propagate seismic records, modeled at a target horizon in the subsurface, toward the surface using
knowledge about traveltime and amplitude decrease from the top of the target (xA , zA )
to receivers at the surface (x, 0), as shown in figure 2.5. With these traveltimes and
amplitudes, propagation operators can be constructed to insert into the wavefield propagation equations.
(x,0)
(xA,zA)
Figure 2.5: Receiver records, modeled at depth, can be forward propagated toward a
virtual receiver at the surface.
In a mathematical sense, one-way wave propagation can be described by the Rayleigh
integrals. In this project the Rayleigh II equation is used for two-dimensional inhomogeneous media:
P (xA , zA ; ω) δG
P (x, 0, ω) = 2ρ(x, 0)
dS,
(2.17)
ρ(xA , zA )
δz z=zA
S
with:
G(x, 0; xA , zA ; ω) = a(x, 0; xA , zA )e−iωtG (x,0;xA,zA ) ,
(2.18)
called a Green’s function, dependent on the amplitude decrease, a, of waves during the
propagation from (xA , zA ) to (x, 0) and travel-time, tG between (xA , zA ) and (x, 0).
2.4. Propagation to the surface
11
To evaluate the Rayleigh II integral correctly, knowledge of the density at depth
ρ(xA , zA ) as well as the density at the surface ρ(x, 0) is needed. Because this information is usually not available, the assumption is made that the earth is homogeneous
along layers, so ρ(x, 0) = ρ0 and ρ(xA , zA ) = ρA ; the Rayleigh II integral then simplifies
to:
δG
ρ0
P (xA , zA ; ω)
P (x, 0, ω) = 2
dS.
(2.19)
ρA S
δz z=zA
With the Rayleigh II integral, propagation effects can now be added to the data as
schematically shown in Figure 2.6. Starting with the modeled data in the (x, t) domain
(the left figure shows a subset of modeling results), propagation effects are first added
to the receivers to simulate a dataset with sources at depth and receivers at the surface
(middle figure). In this dataset propagation effects can be added to the sources to
simulate a dataset with a source and receivers at the surface, creating a target-oriented
seismic record (right figure). Figure 2.6 shows that many modeling results are needed
in order to fully simulate the surface seismics; the forward propagation steps are in
effect mixing the modeling results. The next chapter discusses the inverse step to
forward propagation; instead of forward propagating the modeled response at depth,
the surface seismics will be back propagated. A step that will in fact decouple the
contribution of each target location to the seismic data.
surface
target
datum
target
zone
Figure 2.6: Forward propagation of modeled data (left panel) toward the surface in two
steps. First the receivers are forward propagated (middle panel), followed by propagation
of the sources (right panel).
Chapter 3
Target-oriented data manipulation
The previous chapter shows two ways to model seismic data in a target oriented-fashion.
The modeling algorithms give as their output a plane wave (sx , τ ) response to a onedimensional earth-model at depth. This modeled wave field can be transformed to a
point source response in the (x, t) domain and forward propagated to the surface in
order to match real seismic data. The forward propagation requires modeling results
from many locations along the target datum, it uses many localized modeling results to
simulate surface data where the modeling results all contribute partly. In this chapter,
instead of transforming the modeled data to match seismic data, the seismic data is
transformed to match the modeled data. All propagation effects from the surface down
to the top of the target are removed from the seismic data (redatuming) and the data
is transformed into the plane wave domain. As an inverted process to the forward
modeling, the data is now taken from surface data to localized information along the
target datum.
3.1
Back propagation
Back propagation of seismic records is the central process in redatuming, it removes
propagation effects from seismic data. The equations governing back propagation are
the same as in forward propagation but reversed in time. Starting from the Rayleigh
II integral in equation 2.19, reversing time and swapping coordinates results in:
δG
P (x, 0; ω)
dS,
δz z=0
S
(3.1)
G(xA , zA ; x, 0; ω) = a(xA , zA ; x, 0)eiωtG (xA ,zA ;x,0) .
(3.2)
ρA
P (xA , zA ; ω) = 2
ρ0
with:
Again, knowledge is needed about the Green’s function G(xA , zA ; x, 0; ω), given in
13
14
Chapter 3. Target-oriented data manipulation
equation 3.2, to back propagate a receiver record; allowing for an unknown gain ρρA0
in the result. The Green’s function is fully determined by an amplitude operator
a(xA , zA ; x, 0) and a traveltime operator tG (xA , zA ; x, 0); it describes the amplitude
decrease that a wave will show while back propagating from the surface (x, 0) to the
evaluation point (xA , zA ) and gives the time that the wave will take to travel that
distance. Sections 3.3 and 3.4 present ways in which these operators can be extracted
from the data.
3.2
Redatuming
With the equations for back propagation, redatuming can be explained. Redatuming
is a two-step procedure that takes a series of seismic records with sources and receivers
at the surface and transforms them to a series of seismic records with sources and
receivers along a reflector at depth (datum).
The first step involves focusing of the receivers. Figure 3.1 shows the first step in
redatuming schematically. Starting with many single surface shot records, the receivers
are back propagated to a point (xA , zA ) on the datum. This step results in a virtual
record of a receiver at the datum, receiving data from many sources at the surface. By
applying reciprocity, this record should be equal to a record of receivers at the surface
recording the field created by a point source at the datum.
operators applied to many
surface shot records
sources at the surface,,
receivers at the target datum
reciprocity
receivers at the surface,
sources at the target datum
Figure 3.1: Step 1 of redatuming, back propagation of all shot records. This results (via
reciprocity) in the response of a source buried at (xA , zA ).
The second step is focusing of the sources. In figure 3.2, the second step of redatuming
is schematically shown. The virtual record of receivers at the surface measuring the
response to a source on a reflector is back propagated to many locations along the
reflector. The result of this back propagation step is a virtual record of a source and
3.3. Traveltime operator estimation
15
receivers at the surface,,
source at the target datum
multiple back
propagation steps
source and receivers at
the target datum
Figure 3.2: Step 2 of redatuming, back propagation of the buried source response creates
a receiver record of a source at rA and receivers along the datum.
receivers along the reflector. If this process is repeated for many virtual sources along
the datum, a complete (virtual) seismic survey is built with sources and receivers at
the datum. For more information on wavefield extrapolation and redatuming, refer to
Berkhout (1984) and Thorbecke (1997).
3.3
Traveltime operator estimation
For redatuming an amplitude operator a(xA , zA ; x, 0) and a traveltime operator tG are
needed, describing amplitude decreases and traveltimes between all surface locations
and all locations along the datum. This section shows the method used to find the
traveltime operator tG directly from the surface seismic data. In order to find traveltime
operators from the data, a global search algorithm was used. This method uses both a
genetic algorithm and a simplex algorithm to optimize a parameterized version of a set
of initial operators. Figure 3.3 illustrates the parameterization of traveltime operators
used in the method; the operators are defined by four parameters: the apex time t0 , the
average velocity v, skew speed w and fourth order moveout term α. In mathematical
notation:
x
x2
(tG + )2 = t20 + 2 + α4 x4 .
(3.3)
w
v
The algorithm now needs to optimize four parameters per operator. In order to reduce
the total number of parameters for optimization and to speed up the algorithm, a subset
of operators is used. Interpolation of the parameters is used to obtain the operators
needed to complete the optimization algorithm.
16
Chapter 3. Target-oriented data manipulation
x
v-w
t0
v+w
t
Į
Į
Figure 3.3: Parametrization of traveltime operators. The operators are defined by the
apex time t0 , second order moveout velocity v, skew velocity w and fourth order moveout
term α.
To quantify the performance of a set of traveltime operators, two-way times are calculated from the operators. Then the time samples corresponding to those two-way
times are selected from the seismic data and stacked together, resulting in a total energy which is at its maximum when the traveltime operators are optimal. In figure
3.4, construction of two-way times from the operators is illustrated. The procedure is
based on the minimum time assumption (Fermats principle). Each operator describes
the travel time from surface locations to one focal point, so adding the travel times
from two surface locations to one focal point results in a two-way time. In order to
find the two-way travel time for a source/receiver pair, the combination of operators
that describes this event in the smallest amount of time is evaluated
Figure 3.4: Construction of two-way times from traveltime operators. Three possible
paths between surface locations are shown, constructed from traveltime operators. The
shortest total time will be the used as the two-way time.
The full algorithm uses an initial set of traveltime operators, obtained from the seismic
data directly, picking the apex time t0 from a low offset stack and velocity v from the
second order moveout (NMO, Dix (1955)) velocities, a skew velocity set to zero and
fourth order moveout term as α = v12 (Sun et al. (1991)). The genetic algorithm is
used to globally optimize the parameters describing the set of traveltime operators (tG ),
finally a simplex algorithm is used to fine-tune the parameters. For a more detailed
description, refer to Verschuur and Marhfoul (2005).
3.4. Amplitude operator estimation
3.4
17
Amplitude operator estimation
When traveltime operators are obtained as in Section 3.3, amplitude operators are
needed in order to calculate the Greens functions (Section 3.1) required for a back
propagation step. For this project, the amplitude operators were calculated directly
from the traveltime operators with a homogeneous, two-dimensional subsurface assumption. In figure 3.5, a wave (forward) propagating through a homogeneous medium
is schematically shown. The wave has a circular wavefront that carries a total energy
|p(t)|2 , distributed over the extent of the wavefront.
|p(t)|2
L=0
vt2
vt1
vt3
|p(t)|2
L=ʌ*vt1
|p(t)|2
L=ʌ*vt2
|p(t)|2
L=ʌ*vt3
Figure 3.5: A wave, originating from a point source, propagating through a homogeneous medium. The energy content |p(t)|2 of the wave is distributed over the extent of
the wave front with length L.
With a wave propagating at a constant velocity, corresponding to a homogeneous
medium, the wavefront length grows at a rate of vp t and the total energy is distributed
over a larger line. For a single point on the wavefront then, the energy content |p(x, t)|2
decreases as 3.4 and the amplitude |p(x, t)| drops as equation 3.5.
|p(x, t)|2 =
|p(0, 0)|2
|p(0, 0)|2
∝
vp t
t
(3.4)
|p(0, 0)|
√
t
(3.5)
|p(x, t)| ∝
Finally, the amplitude operators are defined as
1
a(xA , zA ; x, 0) = √ ,
tG
(3.6)
describing the amplitudes correctly for the homogeneous case up to a constant |p(0, 0)|.
An improvement on this scheme for obtaining amplitude operators is being developed
by DELPHI at this moment, in which inhomogeneities in the propagation speeds are
correctly incorporated.
18
3.5
Chapter 3. Target-oriented data manipulation
CMP domain
In order to promote the assumption that the seismic datasets obtained by redatuming
have traveled through a locally one-dimensional medium, the dataset is resorted to
the CMP domain. Figure 3.6 schematically shows the process. In a shot record, the
response of all receivers to one source is collected. In a CMP gather, the data is
presented such that all source/receiver pairs in the record have the same midpoint; in
a one-dimensional medium this would mean that all the reflection energy in a CMP
gather has reflected at the same point on the reflector.
Figure 3.6: Schematic representation of a shot record (left) and a CMP gather (right).
3.6
Radon transform
Having transformed the seismic measurement in such a way that its sources and receivers are moved to the top of the target in the earth-model, the data needs to be
transformed from a point source response into a plane wave response in order to be
able to compare the measurement to the modeled data. The conversion from a point
source response measurement to a plane wave response measurement is governed by
the forward Radon transform (see appendix A):
p̂(sx , τ )|sx = kx = Fτ−1 Fx Ft (p(x, t)) .
ω
(3.7)
Chapter 4
Full waveform inversion
Chapter 2 discussed two ways to forward model a plane-wave response to a locally
horizontally layered medium. Next, these modeled responses along a target datum were
combined and forward propagated towards the surface in order to describe the measured
response in the field. Chapter 3 reversed the forward propagation step; instead of
forward propagating the modeled data, the measured field data is translated onto
the target datum via a redatuming process. This chapter reverses the target-oriented
forward modeling part in order to extract an earth-model at depth from the redatumed
seismic data. The first step (section 4.1) is to apply a transmission correction to o
correct for reflection events occurring in the measured seismics above the target. Next
a wavelet is extracted from the seismic data (section 4.2) to use as input to the forward
modeling algorithm and finally the earth model is extracted using the linear model
(section 4.3) or the nonlinear model (section 4.4).
4.1
Transmission effects
As described in section 3.1, redatuming corrects for propagation effects of the subsurface up to the target at depth. This process corrects for traveltimes and geometrical
spreading amplitude effects as described in sections 3.3-3.4. Redatuming does not,
however, account for the fact that the down-going waves lose energy with every reflection event; reflected energy is subtracted from the energy continuing downward. To
understand the dangers in neglecting these transmission effects, note that reflection
and transmission coefficients are a function of slowness as described in sections 2.1-2.2,
so the amplitude information in the redatumed seismic dataset will have a slowness
dependent imprint, which is exactly what inversion is sensitive to. To correct for these
transmission effects, a response to the well-logs is numerically modeled and compared
to the seismic measurement at the well location. The transmission imprint can be corrected by first evaluating the energy contained in the modeled data and seismic data
19
20
Chapter 4. Full waveform inversion
per slowness value, then applying a slowness dependent gain to the data accordingly:
τ
|p̂modeled (sx , τ )|
(4.1)
p̂corr (sx , τ ) = p̂seismic (sx , τ ) ττ =0
τ =0 |p̂seismic (sx , τ )|
4.2
Wavelet estimation
This section describes the wavelet estimation procedure. Based on a well log an impulse
response x(t) is forward modeled. By comparing the seismic data p(t) to this impulse
response, a wavelet w(t) can be estimated using a Wiener filtering technique (Robinson
and Treitel (1980)). The Wiener filter uses a convolution model, linking a true impulse
response x(t) to the observed response p(t) through the wavelet w(t):
p(t) = x(t) ∗ w(t),
(4.2)
which after applying a temporal Fourier transform, transforms into a straightforward
multiplication in the frequency domain:
P (ω) = X(ω)W (ω).
(4.3)
The wavelet can now be extracted using:
W (ω) =
P (ω)
.
X(ω)
(4.4)
However, because X(ω) is complex and may for some frequencies be (close to) zero,
two additional steps are needed. First the denominator is converted to a real, positive
number by multiplying both the numerator and denominator of the fraction with the
complex conjugate X H (ω), resulting in:
X H (ω)P (ω)
.
W (ω) =
|X(ω)|2
(4.5)
Now, a (small) real number 2 can be added to the denominator to remove the possibility of having a zero denominator, thus stabilizing the spectral division. The stabilized
spectral division equation is now:
W (ω) =
X H (ω)P (ω)
,
|X(ω)|2 + 2
(4.6)
the standard Wiener filter.
To further stabilize the extraction process, a constraint is placed on the wavelet extraction process such that the wavelet should have a limited length in the time domain.
4.3. Linear inversion
21
This constraint can be understood as follows; a signal that is limited in time (with
1
). To simulate a signal that is
tmax ) has a step in the frequency domain (Δf = Tmax
limited in time, equation 4.6 is smoothed in the frequency domain, effectively removing
fast variations and thus simulating a large frequency step. The final wavelet extraction
formula is given as:
H
X (ω)P (ω)
W (ω) = smooth
.
(4.7)
|X(ω)|2 + 2
4.3
Linear inversion
Extraction of an earth-model, using the linear forward modeling algorithm in the inversion process, is straightforward. Starting from the linear forward modeling formulae
in Section 2.1 a set of contrasts in b should be found, such that forward modeled
data optimally matches the measured data. The inversion results b contains contrast information (Δ(ln(vp )), Δ(ln(vs )), Δ(ln(ρ))), which can be combined with the
background properties (vp0 , vs0 , ρ0 ) that where taken from the well logs. In real-life
situations, measured data contains noise and does not match perfectly linear modeling
results; to stabilize the inversion results in the presence of these mismatches, additional
constraints are imposed on the solution.
4.3.1
Least-squares constraint
In the case of least-squares constrained linear inversion (to be called LS inversion),
a simple least-squares constraint is imposed on the solution. In stead of the purely
finding the best match of modeled data and forward model, the inversion now tries to
find the best match with the minimal amount of information in b, the function to be
minimized is:
f (b) =
(p − Mb)2 +
(b)2 .
(4.8)
In matrix notation:
f (b) = (p − Mb)T (p − Mb) + (Ib)T (Ib) .
To find the minimum, the derivative of f (b) to b is set to zero:
df
= −2MT (p − Mb) + 22 Ib = 0.
db
(4.9)
(4.10)
This finally leads to a very simple function for determining the solution b, given as:
4.11.
−1 T
M p.
(4.11)
b = MT M + 2 I
In the implementation of LS inversion the variable can be split into vp , vs and ρ ,
giving control on how influential the constraint is on the separate inversion results.
22
4.3.2
Chapter 4. Full waveform inversion
Multiplicative regularization
While the LS inversion leads to a very simple function for determining the solution,
the constraint put on the inversion result is not necessarily the most suitable. A more
suitable constraint would be one where realistic expectations are imposed on the result.
One such an expectation is that the earth is built up out of layers and a constraint
that approaches this is the constraint imposed by multiplicative regularization (MR
inversion). The function to be minimized is given as:
f (b) =
db 2
dz
(p − Mb)2 ∗ 2
db
+1
dz ref
+1
.
(4.12)
The constraint imposed on the solutions of the inversion is a measure of the average
2
variation (derivative) of the contrasts in depth, where the reference values db
dz ref
can be taken from the well-logs. The constraint is multiplied with the energy in the
residuals (p − Mb)2 , hence the name multiplicative regularization. The function
in equation 4.12 cannot be solved in a straightforward way as with LS inversion, but
instead a conjugate gradient scheme is used to solve this minimization problem. For
more information on multiplicative regularisation, refer to van den Berg (2003).
4.3.3
Final result
In the linear model, the seismic response of a medium is a linear function of the
medium properties; the seismic data only contains information in a limited frequency
band constrained by a wavelet. Due to the linear relationship between depth and time,
the limited frequency-band translates directly to a limited wave number band in depth
information that the seismics contain; in fact, this wave number limitation is what
allows a separation into a background model and a reflecting contrasts model. The
final procedure in the linear inversion process is to recombine the background model
and the inverted contrasts model, a summation of the two models:
ln(vp (z)) = ln(vp0 (z)) + zz =0 Δln(vp (z ))
z
ln(vs (z)) = ln(vs0 (z)) + z =0 Δln(vs (z ))
ln(ρ(z)) = ln(ρ0 (z)) + zz =0 Δln(ρ(z )),
and finally:
z
(4.13)
vp (z) = vp0 (z)e z =0 Δln(vp (z ))
z
vs (z) = vs0 (z)e z =0 Δln(vs (z ))
z
ρ(z) = ρ0 (z)e z =0 Δln(ρ(z )) .
(4.14)
4.4. Nonlinear inversion
4.4
23
Nonlinear inversion
To extract an earth-model from seismic data using the Kennett Invariant Embedding
Method a Genetic Algorithm (GA) was used. The GA is a global optimization scheme
that evaluates many permutations of a parameter set by comparing their responses
with the seismic data and combines the best solutions into a new generation. After a
number of iterations the solution found by the GA should be converging to the true
solution.
recorded
data
update
compare
GA
initial model
properties
in depth
modeled
data
Figure 4.1: The nonlinear inversion scheme: starting from an initial model a genetic
algorithm estimates properties in depth, forward models a seismic response, compares
it to the data and creates new models, based on minimization of a target function.
The nonlinear inversion method, like linear inversion, uses a target function that describes the data mismatch that should be minimized. As an additional penalty factor,
the number of contrasts in the model are counted and added to the target function.
This penalty factor will favor models with few contrasts.
The data match is calculated in two parts:
(pmeas (sx , τ ) − pmod (sx , τ ))2
sx ws (sx )
τ wτ (τ )
,
T1 =
pmeas (sx , τ )2
sx ws (sx )
t wτ (τ )
and:
w
(s
)
w
(τ
)
(env(pmeas (sx , τ )) − env(pmod (sx , τ )))2
sx s x
τ τ
.
env(pmeas (sx , τ ))2
sx ws (sx )
τ wτ (τ )
(4.15)
T2 =
(4.16)
The first part (T1 ) calculates the energy in the residuals, normalized to the energy in
the seismics and contains weighting functions in time wτ (τ ) and slowness ws (sx ). The
weighting functions allow the user to focus attention to a specific area in the seismics in
order to make sure that this area is most probable to be inverted properly. The second
part of the data matching function (T2 ) works exactly the same as the first, except for
24
Chapter 4. Full waveform inversion
the fact that it calculated the energy in the residuals of the envelope of the data. This
envelope is not sensitive to the phase of the seismic data and allows a more smooth
convergence of the solution to the correct timing and amplitudes of the modeled data
as explained in appendix B.
The additional constraint (T3 ) on the solutions is given as:
T3 =
n
α(Δvp (i) > thrvp ) + β(Δvs (i) > thrvs ) + γ(Δρ(i) > thrρ ),
(4.17)
i=2
with:
Δvp (i) = |vp (i) − vp (i − 1)|
Δvs (i) = |vs (i) − vs (i − 1)|
Δρ(i) = |ρ(i) − ρ(i − 1)| .
(4.18)
Every contrast larger than some threshold value is counted and weighted by α, β and
γ. This constraint drives the genetic algorithm to find the solution that can fit the
data well with the least amount of contrasts possible, corresponding to the assumption
that the earth is a layered medium containing only a few contrasts within the target
zone.
The final target function is given as:
Ttot (iter) =
T1 + wenv (iter)T2
+ T3
1 + wenv (iter)
(4.19)
The three contributions are added but the data matching functions are again weighted
with wenv (iter), where iter is the iteration number. The second data matching function
(4.16) was added to allow a smoother convergence toward the solution but as the
solutions converge on a good data match, it adds no more advantage to the scheme;
the weighting function thus reduces the contribution of T2 to zero over the course of
the iterations of the algorithm. The final result is evaluated purely by T1 , the weighted
data match. The other parts of the target functions are used during the iterations
to constrain the solution space and allow a smooth convergence, but are not used to
select the final best solution from the solution space after the final iteration. For more
details on the GA used in this project see appendix B, some other projects using a GA
for amplitude inversion are: Boschetti (1996) and Stoffa and Sen (1991).
4.5
Full inversion scheme
With all the theory being discussed, this section presents the target-oriented full waveform inversion scheme. Figure 4.2 shows a flow chart of the full target-oriented inversion scheme. The left top part represents the processing required to prepare the seismic
data: operators have to be found for backpropagation, then the dataset is redatumed
4.5. Full inversion scheme
25
and resorted into the CMP domain, based on forward modeled data from the well logs
transmission correction is applied to the seismic data. The right top part represents
the forward modeling needed to estimate the transmission correction amplitudes and
the broadband modeling needed for wavelet estimation. The forward modeling algorithm used in this part should always match the inversion method that will be used.
For linear inversion use linear forward modeling and for nonlinear inversion, use the
nonlinear forward modeling algorithm. Below, the inversion algorithms follow. From
the prepared seismic data and forward modeled data, a wavelet is estimated. The
wavelets are then used in the separate inversion algorithms; LS linear inversion, MR
linear inversion and the GA nonlinear inversion.
redatuming &
CMP
radon
well logs
transmission
correction
modeled narrow
band response
reference
f
data
modeled
d l d
impulse response
m
modeling
data m
manipulatio
on
surface seismic
data
inversion
wavelet
LS
linear inversion
MR
linear inversion
GA
nonlinear inversion
Figure 4.2: Flowchart showing all steps involved in the target-oriented full waveform
inversion scheme.
Chapter 5
Synthetic data results
This chapter shows results from the developmental stage of the target-oriented inversion
scheme. To test the proposed inversion scheme, an acoustic dataset was modeled based
on a two-dimensional vp and ρ model. The modeling was done using a finite difference
algorithm, in such a way that the data simulates a surface recorded seismic dataset.
An acoustic version of the target-oriented inversion scheme was applied to this dataset.
5.1
Seismic data
The acoustic seismic data was modeled using the velocity and density model shown in
figure 5.1. The model mostly consists of slightly curved reflectors which was done to
avoid the redatuming process to be an identical operation for every location. Within
the target zone both reflectors have a differen shape to make sure that the inversion
would not lead to an identical result for each location.
Figure 5.1: The velocity (left) and density (right) models used for modeling the seismic
data. In black the target datum is indicated, in red the target zone.
27
28
Chapter 5. Synthetic data results
Figure 5.2 shows a single shot record from the dataset as well as a zero offset gather.
The red arrows indicate the reflection event corresponding to the target datum. A zero
offset gather takes the trace corresponding to x = 0 (as in the figure) from each shot
record and pastes them together.
Figure 5.2: The modeled seismic data. On the left, a selected shot record. On the right,
a zero offset gather.
To perform the back propagation steps as shown in chapter 3, traveltime and amplitude
operators are needed. Because the velocity model is known, the traveltime operators
were obtained by solving the eikonal equation in the model. Amplitude operators
were found as in section 3.4, based on a homogeneous medium assumption. Using
the traveltime and amplitude operators the first back propagation step (focusing the
receivers) was done. Figure 5.3 shows one of the virtual receiver gathers from the
resulting dataset. Note that this is also called a CFP gather for focusing in detection.
Figure 5.3: The data, back propagated on the receiver side, i.e. a CFP gather for
focusing in detection.
5.1. Seismic data
29
Using the same traveltime and amplitude operators as above, the second back propagation step (focusing the sources) was done. Figure 5.4 shows a single virtual shot record
as well as a zero offset gather of the resulting dataset. This data represents virtual
sources and virtual receivers at the target datum. Note that t = 0 in the zero-offset
data now refers to the target datum level.
Figure 5.4: The fully redatumed dataset. On the left, a selected virtual shot record. On
the right a zero offset gather.
Finally, the Radon transform was applied to transform the dataset into the (sx , τ )
domain. Figure 5.5 shows the radon transformed data at a single location.
Figure 5.5: The radon transformed data at a single location.
Each Radon transformed gather is now assumed to contain plane wave responses on a
locally horizontally layered medium and, thus, can be inverted independently of each
other.
30
5.2
Chapter 5. Synthetic data results
Linear inversion
In this part, the redatumed and Radon transformed dataset is inverted linearly in order
to extract the medium properties (vp and ρ). As a reference for the inversion results,
figure 5.6 shows the part of the original model that is in the target zone, with the well
location shown in white. If these figures are exactly replicated by an inversion step the
perfect result has been found.
Figure 5.6: The original model used for modeling the seismic data, zoomed in on the
target zone. Inversion results should resemble this model. The white line indicates the
well location
As part of inversion, transmission correction and wavelet estimation based on the well
logs should be be performed. In this case, however, wavelet estimation is not necessary because the exact wavelet is known with which the seismic data was modeled.
Transmission correction, as described in section 4.1, was performed on the data. Transmission correction was based entirely on the well location.
5.2. Linear inversion
5.2.1
31
Least-squares
The first (linear) inversion algorithm to be applied is the least-squares inversion, which
requires some tweaking of the parameters (see equation 4.11 in order to constrain the
solution adequately; to set the parameters the well location is used. The parameters
are tweaked such that the inversion result matches the well-logs as closely as possible.
Figure 5.7 shows the inversion result at the well location with the optimal parameters.
On the left, bandlimited inversion results are plotted in blue and bandlimited well logs
are plotted in green. On the right, full inversion results (integrated with the background
properties) are plotted in blue, the background properties in red and the well logs in
green; this color scheme will be used throughout the results.
Figure 5.7: Least-squares inversion results at the well location. Left: bandlimited results. Right, the integrated results.
With these parameters all responses along the target zone were inverted, resulting in
figure 5.8.
32
Chapter 5. Synthetic data results
Figure 5.8: Least squares inversion results at all locations. On the left, the background
properties. On the right, the fully integrated results.
The LS inversion results are good withing it’s limitations; the shape of the medium is
reconstructed with rather accurate values. However, best visible at the well location,
contrasts are not put at the right depth and there is some overshoot on the second
reflector. A way to understand these errors is that the algorithm uses the background
vp0 properties as a link between the depth of a reflector and the timing of its response
and because the background vp0 only partially contains the true velocity behavior of
the medium this link is slightly incorrect; the inversion result then compensates for
these errors by placing contrasts at different depths and changing the contrasts. This
effect have been discussed by REF. Also visible are the effects of the bandlimitation
in the inversion result; away from the contrasts the inversion results quickly returns to
the background properties.
5.2. Linear inversion
5.2.2
33
Multiplicative regularization
The MR inversion was also tested on the acoustic dataset. One of its attractive features
is that the method requires no calibration at the well location. Figure 5.9 shows the
results at the well location.
Figure 5.9: MR inversion results at the well location. On the left, the bandlimited
results are shown. On the right, the full results (integrated with the background properties).
The results are quite similar to the LS inversion results in figure 5.7. Note that no
extensive testing of the regularisation parameter was required. The complete target
zone was inverted in a similar manner and figure 5.10 shows the results.
34
Chapter 5. Synthetic data results
Figure 5.10: MR inversion results at all locations. On the left, the background properties. On the right, the fully integrated results.
Again the contrasts are not put at the correct depth and there is an overshoot at
the second reflector, this is to be expected as the algorithm uses exactly the same
background property vp0 as the LS inversion. The overshoot at the second reflector is
more pronounced than in the LS inversion result, probably due to the fact that the
MR inversion regularizes itself while the LS inversion can be tuned to decrease the
overshoot. Overall both LS and MR inversion results match the original model quite
well and are comparable in quality for this dataset.
5.3. Nonlinear inversion
5.3
35
Nonlinear inversion
The nonlinear inversion was performed with the least-squares linear inversion as input.
Figure 5.11 shows the inversion result at the well location. Clearly the inversion results
(blue) are very close to the correct answer (green), showing that the inversion algorithm
is capable of finding an very close answer given a redatumed seismic dataset, corrected
for transmission imprint. Note that the blocky character of this result was for a large
part enforced by the constraint in equation 4.17.
Figure 5.11: Nonlinear inversion result at the well-location.
The inversion was repeated for all locations. Figure 5.12 shows the results. Away from
the well location, where the transmission correction might not be exact, the results
are slightly less accurate. To be complete, it should be noted that the results are very
close to the original model (figure 5.6) all the way down to the second reflector but less
so beyond. This is a result of the fact that there is no reflection energy to restrict the
model beyond the second reflector.
Figure 5.12: Nonlinear inversion results at all locations.
36
5.4
Chapter 5. Synthetic data results
Elastic inversion
The results from the nonlinear inversion of acoustic data are very promising, so an
attempt was made to invert an elastic dataset. The inversion now needs to solve for vp ,
vs and ρ. The elastic dataset was made with the Kennett Invariant Embedding Method
and no noise was added. The inversion could theoretically find the exact result with
seismic residuals equal to zero.
As an input to the Genetic Algorithm, a least-squares linear inversion was performed
on the dataset. The results from this inversion are shown in figure 5.13.
Figure 5.13: LS linear inversion on the elastic dataset. Left the bandlimited result,
right the integrated result.
Figure 5.14 shows the result found with the elastic nonlinear inversion. The result is
very close to the truth for the vp and ρ parameters, but not satisfactory for the vs
parameter. The residuals, shown on the right in the figure in the rightmost panel, are
very close to zero, which means that the algorithm can’t improve much on the result
as the target functions won’t decrease much in the process.
5.4. Elastic inversion
37
Figure 5.14: Elastic nonlinear inversion on the elastic dataset (left) and the residuals
(right). The three panels on the right hand side show the input data, the estimated data
and the residual.
Because the acoustic parameters were found to be close to the true answer, but the
vs wasn’t, an acoustic nonlinear inversion was performed on the same dataset. The
acoustic version of the Kennett Invariant Embedding Algorithm lends itself to much
faster calculation of the wavefield, so results will be found much faster. Figure 5.15
shows the results from this acoustic nonlinear inversion, where the vp estimation is still
very good but the ρ is estimated less accurately. Still the residuals, shown on the right
of the figure, are very small.
Figure 5.15: Acoustic nonlinear inversion on the elastic dataset (left) and the residuals
(right). The three panels on the right hand side show the input data, the estimated data
and the residual.
Just to check that the vs parameters do have an impact on the data, an acoustic dataset
was modeled on the same vp and ρ logs as the elastic dataset and both datasets were
compared. The elastic imprint can clearly be seen in figure 5.16. The figure shows
the elastic data (left), the acoustic data (middle) and the difference (right) between
the two. Apparently the Genetic Algorithm is not able to attribute this imprint to
contrasts in the vs parameter but instead works around the problem by tweaking the
38
Chapter 5. Synthetic data results
acoustic parameters.
Figure 5.16: The elastic imprint on the data. The elastic dataset (left) contains an
amplitude imprint when compared to the acoustic dataset (center), which shows op in
the residuals (right).
The above shows that the nonlinear inversion scheme as proposed in this project is
not sufficiently stable to perform an elastic inversion reliably. It is, however, good at
estimating a velocity model that can focus reflection energy to the correct depth samples, which is precisely what the linear inversion schemes lack. From this information a
new approach was formulated; use the acoustic genetic algorithm inversion algorithm
to extract a velocity (vp ) model from the data and use those velocities to improve the
linear inversions, the updated velocities are used to more accurately calculate τtwt in
equation 2.5.
5.5. Linear inversion with updated kinematics
5.5
39
Linear inversion with updated kinematics
In the previous section, it was proposed that vp properties found from an acoustic
nonlinear inversion of elastic data could be used to improve upon linear inversion
results. To evaluate this idea, a return was made to the (acoustic) synthetic dataset.
Using the velocities found from the nonlinear inversion, a second attempt was made to
extract medium properties using the linear inversion methods.
5.5.1
Least-squares
A second LS inversion was performed on the dataset, now using the updated kinematics.
First the parameters were calibrated at the well location and figure 5.17 shows result.
Figure 5.17: LS inversion results at the well location, using the updated kinematics.
On the left, the bandlimited results are shown. On the right, the full results (integrated
with the background properties).
With the calibrated parameters, the LS inversion was applied to all locations in the
target zone, figure 5.18 shows the results.
40
Chapter 5. Synthetic data results
Figure 5.18: LS inversion results at all locations. On the left, the background properties.
On the right, the fully integrated results.
The results immediately show an improvement in the depth of the inverted contrasts,
also the overshoot is mostly resolved. While the band gap of the inversion result
remains, the inversion results near each reflector are much improved as compared to
the LS inversion with only background kinematics (figure 5.8).
5.5. Linear inversion with updated kinematics
5.5.2
41
Multiplicative regularization
The MR inversion scheme was also performed with updated kinematics. Figure 5.19
shows the inversion result at the well location.
Figure 5.19: MR inversion results at the well location, in this case using the updated
kinematics. Left the bandlimited results, right the integrated results.
The MR inversion algorithm with updated kinematics was applied to all the locations
in the target zone, figure 5.20 shows the results. The left figures show the background
properties, the right figures show the fully integrated properties.
42
Chapter 5. Synthetic data results
Figure 5.20: MR inversion results at all locations. Left the background properties, right
the integrated results.
Again the gain as compared to the previous MR inversion result is clear: contrasts
are found at the correct depth and the large overshoot that was happening at the
second reflector is gone. As with the LS inversion, the effects of the band limitation
in the inversion are still visible with the result moving to the background properties
rapidly away from the reflectors. An interesting difference with the LS inversion that
is becoming apparent with this experiment is that the MR inversion seems to give a
more laterally consistent result; while the LS result contains jumps in properties from
one location to the next, the MR result behaves quite smoothly. This continuity is
most likely due to the parameter in the LS inversion that may be suitable for the
well location but not necessarily for other locations, while the MR inversion optimizes
itself for every location independently.
Chapter 6
Field data results
In the previous chapter the target oriented inversion scheme was applied to a synthetic
dataset. This helped to develop and improve an inversion scheme to a point where it
is very promising. In this chapter the inversion scheme is applied to a field dataset.
The data was provided by Saudi Aramco and concerns an OBC (Ocean Bottom Cable)
seismic measurement of an active oil reservoir. In the following sections all the steps
involved toward obtaining an inversion result from the seismic data are shown, starting with redatuming, then transmission correction and wavelet estimation and finally
the three inversion steps (1st linear inversion - extraction of velocities - 2nd linear
inversion).
43
44
6.1
Chapter 6. Field data results
Seismic data
The seismic dataset is an OBC measurement; figure 6.1 shows a single shot record. All
coordinates are removed in order to safeguard the depth and size of the reservoir. On
the left the data is shown as received from Saudi Aramco. The data has already been
pre-processed; some very strong low velocity events (mudroll) traveling through the
water and near bottom layers were mostly filtered out, leaving a region near zero-offset
where the amplitudes in the reflection events are quite low. This is a worry because
amplitude variations versus offset are exactly what inversion is sensitive to. Other
than filtering out the mudroll, some noise-removal techniques were applied that should
have no effect on the inversion at all. On the right the data is shown after near offset
interpolation.
t 0
t=0
Figure 6.1: A selected surface shot record. Left the pre-processed data, right the nearoffset interpolated data. The red arrows indicate the reflection event corresponding to
the target datum.
To get a feeling for the data and the earth in the target zone, a brute stack was made,
shown in figure 6.2. The seismic shot gathers were flattened using NMO correction
and stacked together to build an image where the lateral shape of the reflectors is
visible. Visible in the stack is the target datum, indicated by the red arrow, and the
target zone beneath it. There is quite some noise present and there are some variations
in amplitudes along the reflection events, indicating a variable data quality along the
target zone.
6.1. Seismic data
45
Figure 6.2: A brute stack of the seismic data. Left the shot record from figure 6.1.
Right the stacked data, revealing the shape of the reflectors.
Figure 6.3 shows the result of the first back propagation step. Traveltime operators
were estimated from the date as described in section 3.3 and amplitude operators were
calculated as in section 3.4. With these operators the Green’s functions were calculated
(equation 3.2) and with those, the data was backpropagated to the target datum. On
the left if figure 6.3, the raw back propagation result is shown, which contains some
unexpected events that behave like straight lines (plane waves) coinciding with the
asymptotes of the regular events. These events are due to the low offset amplitudes in
the surface shots being too low compared to the high offset ones; in back propagation
the high offsets (where the wavefield starts to more and more resemble plane waves)
contribute more than they should. On the right, the unwanted events are removed
by first flattening the events (NMO correction) and applying a dip filter (removing
non-flat events) and then undoing the flattening operation.
t=0
Figure 6.3: A shot, back propagated on the receiver side. Left the raw result, right the
result after applying a dip filter. Red arrows indicate the target datum.
46
Chapter 6. Field data results
Figure 6.4 shows a fully redatumed shot record. On the left, the result after applying
the second back propagation step. On the right, the result after removing some noise
by simply muting the data directly around the source position.
t=0
Figure 6.4: A fully redatumed shot. Left, the result after applying the second back
propagation step, right the result after noise removal. Red arrow indicate the target
datum.
Figure 6.5 shows the radon transformed data. Shown on the left is the direct output of
the radon transform. The data was smoothed along the reflection events in an attempt
to average out some noise and flattened along the first reflector (at zero time). In the
redatumed dataset, the ’direct wave’ (cross through zero time and zero offset in figure
6.4) should result in a flat reflection event at τ = 0 in the radon domain as shown by
Berkhout (1997); the flattening the data was needed probably due to small errors in
the traveltime operators used in the redatuming process.
ʏ=0
Figure 6.5: The radon transformed data, left the radon output, right the result after
lateral smoothing and residual flattening.
6.2. Linear inversion
6.2
47
Linear inversion
After the above processing steps, the data is ready for inversion. Before the inversion
algorithms can be applied, however, a transmission correction should be applied and a
wavelet should be extracted. The transmission correction and wavelet extraction was
done separately for the linear and nonlinear methods and again for the linear inversion
with updated kinematics; on the basis of the well logs data was forward modeled and
compared with the seismics, allowing both processes. With transmission correction
done and the wavelet extracted, the inversion algorithms can be applied; the following
subsections describe all the above steps for the field dataset.
6.2.1
Well matching
Figure 6.6 shows the final radon transformed data from section 6.1, at the well location
and imported into Matlab. The original surface data showed relatively strong amplitudes in the high offsets, translating into high amplitudes at large sx values. Although
these amplitude variations are most likely due to filtering of mudroll, the effects can
partly be compensated for in transmission correction.
Figure 6.6: Redatumed and radon transformed seismic data at the well location.
48
Chapter 6. Field data results
To apply a transmission correction modeled data is needed, based on the well logs; for
linear inversion the data is forward modeled using the linear method. The data was
modeled using a wavelet with corner frequencies of [0 10 45 60] Hz, the data obtained
with this wavelet approximates the seismics in frequency content and looks roughly
like the seismic data, the wavelengths being in the same range.
Figure 6.7: Linearly forward modeled data with a frequency content roughly like the
seismics. This data was used for transmission correction.
With the above datasets, a slowness dependent gain was calculated to correct for
transmission effects in the overburden. Applying the gain to the seismics yields the
data shown in figure 6.8.
Figure 6.8: Transmission-corrected seismic data. Slowness dependent amplitude behavior now more closely matches modeled data in figure 6.7.
The transmission-corrected dataset does not exactly match the forward modeled dataset.
The later reflection events seem to be much stronger in the seismics than in the modeled data, which could be attributed to filtering schemes used in the pre-processing
sequence that boost later events to make them more visible or, alternatively, to errors
in the well logs where particularly the vs measurements may sometimes be unreliable.
6.2. Linear inversion
49
Whatever the cause, the next step will be to extract a wavelet and apply the linear
inversion algorithms to see whether the high amplitudes have a deteriorating impact
on the results.
For wavelet extraction another dataset was modeled using the linear modeling method,
using a broadband wavelet [0 10 60 90] Hz. These frequency lie beyond the frequencies
found in the seismic data and allow extraction of a wavelet. Figure 6.9 shows the
modeling result.
Figure 6.9: linearly forward modeled data with a broadband frequency content, allowing
wavelet extraction.
Based on the broadband modeling result and the transmission-corrected seismic data,
the wavelet was estimated allowing one wavelet per slowness value, but with the restriction that they should not vary too much between consecutive slowness values. Figure
6.10 shows the extracted wavelets.
Figure 6.10: The extracted wavelets.
The extracted wavelets do vary over the slowness range, they show a phase change
of about π2 from the first to the last wavelet. This effect probably does not reflect
the actual situation in the seismic data where there most likely will not be such a
50
Chapter 6. Field data results
phase change over slowness values, but results from the mismatch between seismics
and broadband modeled data. The wavelet does give the optimal match between
seismics and data, thus they were used in the inversion process.
The forward modeled data, based on the well-log, now resembles the seismics in amplitude and frequency content. Figure 6.11 shows the seismic data on the left, the forward
modeled data in the middle and the residuals on the right.
Figure 6.11: Seismic data, forward modeled data and residuals.
The seismics and modeled data still do not match well and there is a lot of residual
energy, however, the strongest even at time t = 0 is matched quite well, the residuals
in that area are quite small.
6.2. Linear inversion
6.2.2
51
Least-squares
With the transmission corrected seismic data and the extracted wavelets, the inversion
algorithms are applied to the seismic gathers along the target zone. For LS inversion,
the well location is used to set damping parameters (vp , vs , ρ ) such that the inversion
result matches the well logs best. Figure 6.12 shows the calibrated results.
Figure 6.12: Least-squares results at the well location. On the left, the band limited
inversion result. On the right, the integrated inversion results.
The inversion results at the well do differ from the well-logs, but there are some indications of a good result. In the shallow parts the vp and vs results (in blue) are close to
the well logs (in green), with only a slight displacement in depth. In the deeper parts,
the ρ is very close to the well logs. The differences are telling as well; in the deeper
parts, contrasts in vp and vs are much larger than should be expected from the well
logs. This effect can be attributed to the strong reflection events in the seismic data as
discussed before. Again, either there has been some application of a time-dependent
gain in pre-processing or the well logs are unreliable in that specific area.
With the damping parameters found at the well-location, 300 gathers along the target
zone were inverted. Figure 6.13 shows the results. The results are satisying as they
show lateral consistency; exactly as one would expect there are identifiable layers in
the earth with more or less consistent material properties. Density (ρ) and p-wave
velocity (vp ) results seem the most consistent while the s-wave velocity (vs ) is a little
jittery from location to location.
52
Chapter 6. Field data results
The vs parameter is sensitive to data quality and as it turns out, the inversion result
shows exactly the regions where the data quality was the most affected by mud-roll
and noise (around locations 75-150). Also, at about location 160, there is a sudden
change in the results: the event at 225m depth is suddenly unclear, an event appears
at 175m depth and the event at 400m suddenly becomes much brighter.
Figure 6.13: LS inversion results for all locations
6.2. Linear inversion
53
The most interesting parameter to gain from inversion was in this case the ratio vvps ,
shown in figure 6.14. This ratio is sensitive to the porosity of a medium and indicates
regions where oil might be found. The inversion result shows regions of low vvps in
red, corresponding to regions that could be porous. The results are a bit inconsistent
laterally, again being very sensitive to data quality the vvps result reflects the data quality
varying from location to location.
Figure 6.14: LS inversion
vp
vs
results for all locations.
It is difficult to quantify the performance of the inversion algorithms on this dataset
as the true answer is unknown. In Section 6.5, the vvps amplitude inversion results are
compared to the in-house inversion results found by Saudi Aramco.
54
6.2.3
Chapter 6. Field data results
Multiplicative regularization
The MR Inversion was also run on the field dataset. This inversion algorithm does not
have parameters to be calibrated at the well location, still the inversion results at the
well location were calculated for comparison purposes; figure 6.15 shows the results.
Figure 6.15: MR inversion results at the well location. On the left, the bandlimited
inversion result. On the right, the inversion result integrated with the background properties.
It is clear that the difference in regularisation has an effect on the results. The data is
now explained with a minimum of contrasts, showing as a decrease in the number of
layers in the inversion result. Still the vp and vs results resemble the well logs in the
shallow parts and the ρ result resembles the well log in the deeper parts.
6.2. Linear inversion
55
Figure 6.16 shows the inversion results for all locations. Again the variations in data
quality are visible mostly in the vs results, but the data quality now has an effect on
the vp and ρ results as well. Again, around location 160, there is a sudden change in the
results, now more pronounced than in the LS inversion results. Finally, in comparison
to the LS inversion results, the MR results look a bit cleaner.
Figure 6.16: MR inversion results for all locations
56
Chapter 6. Field data results
As before, the most interesting result is the vp /vs ratio. Figure 6.17 shows this result.
The result is similar to the LS inversion but again look a bit cleaner.
Figure 6.17: MR inversion
vp
vs
results for all locations.
A final evaluation of these results will be done in Section 6.5, where the inversions are
compared to the Saudi Aramco in-house results.
6.3. Velocity extraction
6.3
57
Velocity extraction
This section shows the extraction of kinematics from the seismic data using the acoustic
nonlinear inversion algorithm. A subset of locations (locations 88 to 188) were treated,
incidentally the worst part of the data in terms of data quality (noise and mud-roll).
In the same way as before but now using the acoustic Kennett Invariant Embedding
Method, transmission correction was applied to the seismic data and a wavelet was
estimated for each slowness value, as shown in figure 6.18.
Figure 6.18: The extracted wavelets.
Figure 6.19 shows the estimated velocity profile at the well location, the estimated vp in
blue and the well log in green. The shallow part is excellently estimated, judging from
the match with the well log. The deeper part however does not match very well the well
log. The explanation for this behaviour can again be found in the unexpectedly high
amplitudes found in the later reflection events in the seismic data, the GA is trying
to match them by introducing contrasts in the velocity logs and thus compromises
between good kinematics and good amplitude matching. In spite of the errors made at
deeper locations, the results are encouraging as the deviations from the well logs are
at similar depths and have similar amplitudes as the linear inversion vp results.
58
Chapter 6. Field data results
Figure 6.19: Estimated velocities at the well location.
The process was repeated for a subset of all locations and figure 6.20 shows the results.
As with the linear inversion results, something is happening at cdp number 160. In
these results the effects are quite dramatic, the whole kinematics seem to change.
Again, this may well be caused by the fact that the GA does not simply look for
the correct kinematics, but also tries to correctly match the reflection amplitudes; at
around location 160, the reflection amplitudes seem to suddenly change and thus the
GA will give a different answer.
Figure 6.20: Estimated velocity profile from the nonlinear inversion process.
6.4. Linear inversion with updated kinematics
6.4
59
Linear inversion with updated kinematics
With the extracted vp described in the previous section, a second round of inversions
was performed. Before the LS and MR algorithms were applied, transmission correction
was applied to the seismic data and wavelets was estimated as before, now using the
updated kinematics to forward model the data from the well logs.
Figure 6.21: Estimated wavelets, using the updated kinematics.
6.4.1
Least-squares
Figure 6.22 shows LS inversion results at the well location. As can be expected, the
inversion result (blue) matches the well logs very well in the shallow parts; the depth of
contrasts in corresponds much more closely to the well than in the first LS result. In the
deeper parts specifically ρ performs a bit worse than before, which can be attributed
to the errors in the deeper arts of the estimated kinematics, as shown before.
Figure 6.22: LS inverted properties at the well location.
60
Chapter 6. Field data results
Figure 6.23 shows the LS inversion results with updated kinematics for all locations.
The jump in results at cdp 160 is clearly due to the jump in the kinematics as shown
in figure 6.20. The results look similar to the amplitude inversion results with the
original kinematics, some layers are slightly shifted; a direct comparison is made for
the vvps results in figure 6.24.
Figure 6.23: LS inversion results at all locations, using the updated kinematics.
6.4. Linear inversion with updated kinematics
61
Figure 6.24: LS inversion vvps result at all locations. Left the results using the original
kinematics, right the results using the updated kinematics.
In figure 6.23 the left panel shows the previous results zoomed in to cover the subset of
updated locations only, the right panel shows the inversion results using the updated
kinematics. Note that the porous layer at 200m depth in the original result has been
shifted downward to 250m and in the deeper areas the blurry results in the original
results seem to have been brought into focus, now showing some consistent layers. It
seems that overall improvements have been made, but the ’correct’ answer is unknown;
in section 6.5 the results will be compared to the Saudi Aramco in-house inversion
results.
62
6.4.2
Chapter 6. Field data results
Multiplicative regularisation
The updated kinematics were also used in the MR inversion algorithm and figure 6.25
shows inversion results at the well. Again the inversion show an improved match in the
shallow part, specifically contrasts in vp and vs are now placed at the correct depth.
Also as before, the ρ results seem to have suffered in the deeper areas, probably due
to the slight errors in the estimated kinematics.
Figure 6.25: MR inverted properties at the well location.
Figure 6.26 shows the vvps results. On the left the original results are displayed, on the
right the inversion results using the updated kinematics. The differences are again
marked, the porous layer at 200m depth in the original result has been shifted doward
to 250m and in the deeper areas and in the deeper areas the collection of porous
patches now seems to be a large porous area from 400m downward with two porous
regions above it at 350m. and 300m. Again it seems that overall improvements have
been made, in section 6.5 the results will be compared to the Saudi Aramco in-house
inversion results.
Figure 6.26: MR inversion
vp
vs
result at all locations.
6.4. Linear inversion with updated kinematics
63
Figure 6.27 shows the MR inversion results with updated kinematics for all locations.
The jumps in the results at location 160 is clearly due to the jump in the kinematics
as shown before. The results look similar to the amplitude inversion results with the
original kinematics, some layers are slightly shifted.
Figure 6.27: MR inversion results at all locations, using the updated kinematics.
64
6.5
Chapter 6. Field data results
Comparison to in-house results
This sections presents the final evaluation of the amplitude inversion results of the field
dataset. The vvps results are compared to the Saudi Aramco in-house results. Before
the results are compared, a comparison should be made of the methods in obtaining
the results.
The in-house inversion is done by a 3-D linear inversion method very similar to the LS
inversion used in this project. The main difference in the inversion results is that the
in-house results make use of the full 3-D seismic dataset while the results presented in
this thesis make use of only a small subset of the full dataset, being one 2-D receiver
line. The amount of data used in the inversions has a direct effect on data quality;
noise levels will have more impact when using only a subset of data and all processing
steps will suffer accordingly. In short, the results presented here really are a worst case
scenario.
Figure 6.28 shows the in-house results for the parameter vvps . The gamma ray measurements from the well logs are included in the figure. Gamma ray measurements can, like
vp
measurements, indicate porous rock formations. There may be differences in gamma
vs
ray logs and vvps measurements as they don’t actually measure the same properties but
the match is expected to be quite good. The in-house result will serve as a benchmark
for the inversion results obtained in this project.
Figure 6.28: In-house
vp
vs
result, with the gamma ray logs plotted in overlay.
6.5. Comparison to in-house results
65
Figure 6.29 shows the original LS inversion result, placed in depth and plotted with
the same colorscheme as the in-house results. Compared to the in-house result, the LS
result has much larger amplitudes (brighter colors) and seems to be less stable. However, the in-house result shows three layers that could be porous and the LS inversion
result shows the same three layers, albeit at slightly different depth. In the deeper
parts, the LS inversion results don’t have the lateral consisteny of the in-house results,
that’s most likely due to data quality issues.
Figure 6.29: LS inversion
vp
vs
result.
Figure 6.29 shows the LS inversion result with updated kinematics. The results only
cover a subset of the total target zone, so they are overlayed on the original LS inversion
results. There seems to be a bit of an improvement in the overal match with the inhouse results; the same porous layers are still there, but now the depths correspond
much better. These results should really be comparable to the in-house results as they
use closely related inversion algorithms. The in-house results does seem to be more
stable and matches the gamma ray measurement more closely; this shows the enormous
advantage in 3-D techniques compared to 2-D and the importance of using all available
data to improve signal to noise ratio.
66
Chapter 6. Field data results
Figure 6.30: LS inversion
vp
vs
result with updated kinematics.
Figure 6.31 shows the original MR inversion result. This result resembles the in-house
result much more closely than the LS inversion does. There still is a slight mismatch
in depth of the porous layers but the similarities are striking. The MR inversion result
does show the deepest porous layers as being more broken up than the in-house results,
but this most likely is due to the noise levels scrambling the results slightly.
Figure 6.31: MR inversion
vp
vs
result.
Figure 6.32 shows the MR inversion results with updated kinematics overlayed on the
original MR inversion results. Immediately it is clear that the porous layers now better
correspond in depth to the in-house result. Looking at the gamma ray measurement,
however, it seems that actually these MR inversion results match the gamma ray more
6.5. Comparison to in-house results
67
closely than the in-house results do. There is one porous layer predicted by the MR
inversion that is not visible in the gamma ray logs, but all the other areas actually
correspond to the gamma ray logs very well.
Figure 6.32: MR inversion
vp
vs
result with updated kinematics.
Based on the in-house results and the gamma ray logs, the MR inversion clearly is
an improvement on the LS inversion results and the extraction of kinematics helps
improve the results even further.
Chapter 7
Conclusions and recommendations
7.1
Conclusions
This project presents a first attempt at target-oriented full waveform inversion using
CFP technology. The localization of the inversion problems allowed the application of
more advanced inversion tools than are conventionally used, such as linear inversion
with a multiplicative regularisation and nonlinear inversion using a genetic algorithm.
The inversion scheme was applied to a synthetic dataset and a field dataset, each
offering valuable information on the feasibility of the scheme.
The synthetic experiments gave very promising results, they show that the proposed
target-oriented inversion scheme is feasible. With transmission correction, inversion
results can be very close to the true answer. Nonlinear elastic inversion using a GA is
not stable in the form presented here, but can provide a velocity map that describes the
kinematics in the target zone very accurately. With the kinematics from the nonlinear
inversion, the linear inversion schemes can be improved dramatically, because kinematics focus reflection information in depth. The difference between linear inversion
with least-squares regularisation and linear inversion with multiplicative regularisation
were not very pronounced in these experiments. On the basis of the synthetic experiments an updated target-oriented full waveform inversion scheme was proposed: after
redatuming and linear inversion, the genetic algorithm is used to extract a velocity
map from the data and then the data is inverted a second time using the updated
kinematics.
The field data tests show that the target-oriented inversion scheme can be succesfully
applied to field data. However, careful pre-processing of the data is pre-requisite.
In this case, the difference between the two linear inversion methods were clear, the
MR inversion gave results that fit more closely the layering of the target zone. With
multiplicative regularisation and kinematics extraction the results could compete easily
69
70
Chapter 7. Conclusions and recommendations
with the in-house result, despite the fact that only a 2-D subset of the full 3-D dataset
was used and the data quality caused some concern.
Finally, it can be concluded that target-oriented inversion using CFP technology shows
much promise. Nonlinear updates to linear inversion algorithms (kinematics and multiplicative regularisation) improve inversion results dramatically, showing that a fully
nonlinear target-oriented inversion scheme is the way forward.
7.2
Recommendations
First of all, the kinematic extraction procedure has to be improved. The kinematics
showed errors due to the compromises in matching reflection timing and matching
reflection amplitudes; the kinematics extraction step should be redesigned to be less
sensitive to amplitudes.
Next, apply the target-oriented inversion scheme as presented in this thesis to a new
dataset that has been pre-processed much more carefully and uses more of the available
seismic data. This should really show the potential of the scheme. Plans are already
in place to do this, with a dataset that will also be supplied by Saudi Aramco.
On the longer term, DELPHI is working towards a fully nonlinear target-oriented
inversion scheme in the CFP domain. Working in the CFP domain instead of the fully
redatumed domain is necessary when working with 3-D seismic datasets, where the
sparseness in source locations does not support the second backpropagation step. The
scheme will automatically incorporate all the advantages found in extracted kinematics
and the iterative MR inversion, but will also be able to fully use 3-D seismic datasets.
The nonlinear inversion scheme will be extended to inversion of 2-D and eventually 3-D
models of the earth instead of 1-D models, removing the need to make assumptions on
local invariance of the earth.
Chapter 8
Acknowledgements
Special thanks to Saudi Aramco and specifically the research division Expec Arc for
supplying the field data used in this project. Saudi Aramco has been a sponsor of
DELPHI for a long time and there have been many fruitful collaborations between the
two. This project will have a continuation, also in collaboration with Saudi Aramco,
we very much look forward to this.
Thanks again to Saudi Aramco for inviting us over to the kingdom of Saudi Arabia to
work on the field data in their premises. The stay has been very enjoyabe and we hope
to come again.
Personal thanks go out to:
Eric Verschuur, who seems to be able to explain everything in simple terms.
Dries Gisolf, especially for assisting me while working with field datasets.
Thierry Tonellot, my contact at Saudi Aramco, for helping me evaluate my results and
pushing forward this project.
71
Appendix A
The radon transform
The modeling algorithms used in both the linear and the nonlinear modeling algorithms
calculate a seismic response to models in the (sx , τ ) domain. The seismic measurement
to which the modeling results are compared are in the (x, t) domain. To convert data
from one domain to the other, the radon transform is used.
The first step in deriving the radon transform is to apply a spatial fourier transform
to data in the (x, t) domain:
p̂(kx , t) = Fx (p(x, t)) .
(A.1)
Then, apply a temporal fourier transform:
P̂ (kx , ω) = Ft Fx (p(x, t)) .
(A.2)
Now horizontal slowness is intruced to the left hand side of equation A.2 as sx =
P̂ (sx , ω)|sx= kx = Ft Fx (p(x, t)) .
kx
:
ω
(A.3)
ω
Next, P̂ (sx , ω) is written as the temporal fourier transform of p̂(sx , τ ):
Fτ p̂(sx , τ )|sx = kx = Ft Fx (p(x, t)) .
(A.4)
ω
Finally, the forward radon transform is obtained:
p̂(sx , τ )|sx = kx = Fτ−1 Fx Ft (p(x, t)) ,
ω
(A.5)
and the inverse radon transform is:
p(x, t) = Ft−1 Fx−1 Fτ p̂(sx , τ )|sx = kx .
ω
73
(A.6)
Appendix B
Genetic Algorithm
To extract an earth-model from seismic data using the Kennett Invariant Embedding
Method, a Genetic algorithm was used. A Genetic Algorithm is a search algorithm,
geared towards optimizing a target function. To achieve this goal, the GA creates many
parameter sets (chromosomes) divided into several groups (populations) and calculates
the target function for each parameter set. Next, parameter sets are selected based on
the score of their target functions and combined (mated) to form new parameter sets
in a semi-random way. Each generation of parameter sets should be an improvement
on the previous in terms of the target function, but is at least as good as the previous generation. Also included in a GA are random mutations on the chromosomes,
crossovers of successful chromosomes between populations and re-initializations of entire populations. For purposes of this project an adaptation step was included to the
scheme that changes chromosomes in a way that is expected to be beneficial to the
target function. Some suggested papers on the subject: Beligiannis et al. (2005) and
Whitley and Starkweather (1990)
The GA that was developed for this project was programmed that favors blocky results,
this was done to constrain the algorithm to solutions that fit a layered subsurface in
which every layer is homogeneous.
74
B.1. Chromosomes and populations
B.1
75
Chromosomes and populations
For the elastic inversion of seismic data, we are interested in three parameters: vp , vs ,
ρ. For each elastic parameter a log exists, giving the property value for every depth
step. Each chromosome in the GA consists of a log for vp , a log for vs and a log for ρ.
chromosome
ʌ
deepth
vs
deepth
deepth
vp
Figure B.1: Chromosomes in the Genetic Algorithm consist of logs of vp , vs and ρ
versus depth.
The chromosomes are divided into a number of populations, the chromosomes within
each population will be used to create a new generation in their respective population,
there is no crosstalk between the populations unless a crossover occurs in the mutation
step.
Chromosomes
Population1
Population2
Population3
Figure B.2: Chromosomes are divided into several populations.
76
B.2
Appendix B. Genetic Algorithm
Initialization of chromosomes
To start the GA, all the populations are filled with different chromosomes. Starting
with a single initial model, obtained by linear inversion or taken from well logs, many
blocky variations are created by converting the logs into homogeneous blocks of variable
thicknesses. If the initial model shows a lot of variation, there is a high chance of a
contrast in that region, and probability of a block to start there is also high. If the
initial model shows little variation, there is a low chance of a reflecting contrast in that
region and the probability of a block starting there is low. The property values within
such blocks are determined by taking the average value of the initial model in that area
and adding a random value, normal distributed in the range of the variations in the
initial model. Two examples of possible variations on the initial model are shown in
figures B.3 and B.4. At the end of this step, a set of chromosomes is obtained in each
population, containing the initial model and many blocky variations on that model.
chromosome
ʌ
deepth
vs
deepth
deepth
vp
Figure B.3: One possible blocky variation (red) of the chromosomes based on one initial
model (blue).
chromosome
ʌ
deepth
vs
deepth
deepth
vp
Figure B.4: A second possible blocky variation of the chromosomes based on one initial
model.
B.3. Evaluation
B.3
77
Evaluation
The goal of an amplitude inversion scheme is to match the measured seismic response
as closely as possible. To achieve this goal, the target function is defined as the energy
found in the residue, or the energy found in the difference between our measured seismic
response and the forward modeled response. This target function, given as:
w
(s
)
w
(t)
(pmeas (sx , t) − pmod (sx , t))2
sx s x
t t
T1 =
,
(B.1)
pmeas (sx , t)2
sx ws (sx )
t wt (t)
should be minimized to find the optimal match. Included in the target function are
weighting functions ws and wt , these can be used to emphasize a certain time-range
and specific sx -values in the calculation.
In the course of this project, minimization of the energy in the residuals was found to
be an insufficient requirement for the GA to converge to satisfactory solutions with a
low number of iterations. Instead, the GA needs to select the best solutions based on
both a less strict requirement on the residuals and some requirements on the shape of
the logs in a chromosome.
A less strict target function has been found by computing the energy in the residuals
of the envelope of the signals:
w
(s
)
w
(t)
(env(pmeas (sx , t)) − env(pmod (sx , t)))2
sx s x
t t
(B.2)
T1 =
env(pmeas (sx , t))2
sx ws (sx )
t wt (t)
Figure B.5 shows a schematic explanation why this is a less strict target function. If
the response is quite close to the measured field, but shifted in time with Δt, the pure
target function (T1 ) will have a very small minimum close to Δt = 0, but a maximum
at a small Δt. This maximum will stop the algorithm from smoothly converging to the
minimum. Now the T2 target function first computes the envelopes of both signals and
the results is that the minimum still is found at Δt = 0, but the maximum is gone,
enabling a much smoother convergence toward the minimum. Again included in the
target function are the weighting functions ws and wt . An important thing to note
is that the T2 target function does not take into account the polarity of the reflection
events in the signal, while T1 does.
ȴt
ȴt
ȴt
T2
Appendix B. Genetic Algorithm
T1
78
ȴt
Figure B.5: Target functions T1 (left) and T2 (right) as a function of the timeshift
between measured seismics and modeled seismics. T2 ensures a more gradual decline to
zero error.
The target function based on the shape of the logs, focuses on large contrasts. For
every log, the number of contrasts (between consecutive samples) larger than some
threshold value is counted:
T3 =
n
α(Δvp (i) > thrvp ) + β(Δvs (i) > thrvs ) + γ(Δρ(i) > thrρ ),
(B.3)
i=2
with:
Δvp (i) = |vp (i) − vp (i − 1)|
Δvs (i) = |vs (i) − vs (i − 1)|
Δρ(i) = |ρ(i) − ρ(i − 1)| .
(B.4)
Figure B.6 shows an example of the penalty this function imposes on property logs.
The goal of this target function is to give an advantage to logs that can generate a
reflection event with a single contrast instead of a series of contrasts; this represents
a property of the earth that is assumed, sparsely layered earth. Also included in the
target function are constants α, β and γ. These constants are included to balance the
impact of the different logs on the target function, but also to balance the impact of this
target function on the total target function; constraints on the log shape usually only
need to contribute a small part of the total target function to be maximally effective.
Finally, the total target function is given:
Ttot (iter) =
T1 + wenv (iter)T2
+ T3 .
1 + wenv (iter)
(B.5)
Again a weighting function (wenv ) is included, this time to reduce the impact of T2 as
the number of iterations in the GA increases. As the number of iterations increases,
the GA should be converging on a close match between modeled and measured seismic
response; in this regime, the stricter function T1 is to be preferred above T2 .
B.4. Selection
79
property
p
p y
property
p
p y
1
2
3
4
depth
depth
1
2
Figure B.6: Target function T3 counts the number of contrasts larger than a threshhold
value. The left property log is found preferable to the right one because it contains less
(large) contrasts
B.4
Selection
Based on the target function, a fitness function is defined. The goal of this fitness
function is to serve as a basis for selecting chromosomes that are suited to reproduce.
In this project, the fitness value was calculated as:
F =
max(Ttot )pop − Ttot
.
max(Ttot )pop − min(Ttot )pop
(B.6)
Within a population, the target value of each chromosome is rated on a scale between
the best and worst solution in that population.
Now every chromosome has a fitness value, those chromosomes with the highest fitness
have the lowest Ttot , thus are the most suitable for reproduction. The next step now is to
select wo chromosomes within a population for mating to produce the next generation
of chromosomes. To do this we sort the fitness values for all the chromosomes in a
population, add them together as a normalized cumulative sum and then choose a
number randomly between 0 and 1; the chromosome corresponding to this random
number then is chosen for reproduction. This process is schematically shown in figure
B.7.
chromosomes
1
2
3
4
fitness values
fitnessvalues
5
1
0.8
0.6
0.3
cumulative
normalized
0
0
0.4
0.7
0.9 1
randomnumber:0.3
Ͳ>chromosome1
> h
1
Figure B.7: Selection of chromosomes for reproduction. Based on it’s fitness value, a
chromosome is more or less likely to be used in the reproduction step
80
B.5
Appendix B. Genetic Algorithm
Reproduction of chromosomes
The final step in the GA is to take the best chromosomes in a generation, as discussed
above and create a new generation from them. The GA programmed in this project
uses four steps to create this new generation: combination, mutation, crossovers and
population restarts. Also, the best solutions from one generation are copied directly
into the next generation to ensure that ensuing generation will never perform worse
than the current one.
Combination
After the best solutions are copied into the new generation, the rest of the available
chromosomes in each population are filled by combination of chromosomes selected by
the procedure discussed above. For this project a combination method was used that
first generates a number of sample-numbers, and then mixes two selected chromosomes
by swapping the chromosome properties between those sample-numbers. This process
is schematically shown in figure B.8, where three depths are chosen and two selected
logs are mixed in-between those depths.
property
p
p y
property
p
p y
depth
depth
property
p
p y
Figure B.8: Combination of two chromosomes. The property logs are mixed in-between
randomly chosen depth samples.
Mutation
After combining the chromosomes, a mutation step is taken. This step introduces new
features into the populations that would not appear by combination of chromosomes
alone. In this project a form of mutation was chosen that adds block to the chromosomes as shown in figure B.9. The chance of such a mutation being added to a
chromosome is a function of the mutation rate (mr), which in turn is a function of the
standard deviation (std) of properties as measured in a whole population; this relation
is described as:
mrpop =
i
max
αmr ∗ std(vp (i)) + βmr ∗ std(vs (i)) + γmr ∗ stdpop (ρ(i)).
(B.7)
i=0
As diversity within a population increases, the mutation rate for that population decreases; the motivation for this is that when there is a lot of diversity, the selection
B.6. Adaptation
81
criterion will work well to improve the total target function over generations. When
there is little diversity, combination will not work well and some external change is
needed such as mutation.
property
p
p y
property
p
p y
depth
depth
property
p
p y
Figure B.9: Mutation of a chromosomes. Blocks of random length are added to property
logs at random depth levels.
Crossover
After mutation, there is a chance that a crossover will occur, this means that the
best solution of one population is copied into another population. This might help
a population escape a local minimum or might help improve diversity and thus the
overall convergence of the GA.
Population restarts
The final form of reproduction is the restart of an entire population, where there is
again only a slight chance that it will occur. If a population restart happens, the global
best chromosome at that moment is taken and a whole population is formed around
it. This is done in a way similar to the initialization of the first populations; first
the large contrasts in the chromosome are found, these contrasts are shifted in depth
by a random number of samples and inbetween an average value of the property is
assigned plus a normal distributed random value. So again, this step will create blocky
variations of a chromosome that is known to perform well.
A population restart can help to bring a whole population closer to the best solution
at that moment, thus enabling it to focus on that solution and improving upon it very
quickly.
B.6
Adaptation
The final step in an iteration of the GA is adaptation of the chromosomes. This step entails making changes to chromosomes that should be expected to improve their seismic
matches, it can be seen as a more extensive mutation round. During this step, first a
copy is made of every chromosome in a population; on these copies the adaptations are
made. After the evaluation of all chromosomes, original and adapted, creating the next
generation restores the original size of the populations. Adaptations that are made to
82
Appendix B. Genetic Algorithm
the chromosomes are fourfold: shifting contrasts, introducing or removing contrasts,
updating kinematics of chromosomes and creating a best-of-chromosome within each
population. For each adaptation a chance exists that it is applied to a chromosome, in
this way a few chromosomes are not adapted at all, but most are adapted in one way
or more ways.
Shifting contrasts
The first adaptation is to look for large contrasts in the logs and the shift them over
one sample, either downward or upward. This process is schematically shown in figure
B.10.
property
p
p y
depth
property
p
p y
depth
property
p
p y
Figure B.10: Shifting contrasts. In this step, large contrasts can be shifted upward (left)
or downward (right) by one depth-sample.
Introducing or removing contrasts
The second adaptation is to compare large contrasts in the different logs (vp , vs , ρ)
within one chromosome. If there is a contrast at a certain depth in one log (say vp ),
usually there should be a contrast in the other logs (vs and ρ) as well. If a contrast
exists in one log but not in another, the adaptation step will either add a contrast or
remove a contrast to match the different logs in behavior. The size of the introduced
contrast is taken as a percentage of the total variation in the logs. This updating step
is shown in figures B.11 and B.12.
depth
depth
property2
p
p y
depth
property2
p
p y
property1
p
p y
Figure B.11: Adding a contrasts. Two property logs within one chromosome are compared. If there is a large contrast in property 1, but not in property 2, a new contrast
is introduced to property 2.
B.7. Final result
83
depth
depth
property1
p
p y
depth
property2
p
p y
property1
p
p y
Figure B.12: Removing a contrast. Two property logs within one chromosome are
compared. If there is a large contrast in property 1, but not in property 2, the contrast
in property 1 is removed.
Updating kinematics
The third adaptation is to correct the vp logs in order to match the measured seismics
better. This step first looks for the large contrasts in a chromosome, then calculates
the traveltime (ttravel ) corresponding to the depth of this contrast at a set sx value.
Next a seismic response is forward modeled for this set sx value and the resulting data
is correlated with the measured seismics within a time-window around ttravel . From
this correlation, a time shift can be found that would optimally match the response
to the seismic; this time shift is then translated into an update of the velocities in
the chromosome. Due to the focus on one sx value used in this updating step, it can
be helpful to give this sx value more weight in the target function (Ttot ) in order to
maximize the effect of the update.
Best-of-chromosome
The final adaptation step is to evaluate a sample-by-sample data-match of the chromosomes within a population. Again, the step focuses on large contrasts, calculates travel
times and calculates the seismic data match (T1 , equation B.1) within a time-window
around those travel times. The resulting target function is assigned to all samples
in-between two consecutive contrasts and finally the best combination of the chromosomes in a population is formed by taking the samples with the lowest errors assigned
to them.
B.7
Final result
The final inversion result is obtained when the GA is stopped, which can be due to
two reasons: a maximum number of iterations is made or the target function is not
decreasing anymore. Usually the result with the best score on the target function is
chosen as the final result, for this project however, the final result was chosen purely
on the best match between forward modeled and measured seismic response according
to T1 (equation B.1). All the other parts of the target function are only used for
84
Appendix B. Genetic Algorithm
reproduction in the iterations. In addition to the best result taken from the GA,
there is also an opportunity to look at all the other available chromosomes with their
respective seismic matches. From this information, it is possible to take some kind of
statistical look at the results, for instance showing results within 5 percent of the best
available solution.
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