Download Ray tracing in geophysics

Ray tracing in geophysics
Seminar
Author: Klemen Kunstelj
Adviser: dr. Andrej Gosar and dr. Jure Bajc
18.12.2002
Abstract
Alp 2002 is an international seismological project. The main issue is to
construct the structure of the Lithosphere in the area of Suth-Eastern Alps.
The experimental part took place in July 2002 and the analysis of these
results is planned to be completed aproximately in two years. In the seminar
some basic modelling methods that are used in the analysis of the Alp2002
data are presented.
Figure 1: Network of geophones with seismograms in Slovenia during Alp2002
1
Figure 2: Seismograms of an explosion in Vojnik during Alp2002 gathered
on the network of seismographs in Slovenia
CONTENTS
2
Contents
1 Introduction
3
2 Ray
2.1
2.2
2.3
4
4
4
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theory and seismic waves
Wave equations for P- and S-waves . . . . . . . . . . . . . . .
Ray equation . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ray propagation in media of constant velocity . . . . . . . . .
3 Ray tracing with RAYAMP
4 Ray tracing with JIVE3D
4.1 Model parametrization . . . . . . . . .
4.2 The forward problem . . . . . . . . . .
4.2.1 Two-point ray tracing . . . . .
4.2.2 Ray perturbation theory (RPT)
4.2.3 Frechet derivatives . . . . . . .
4.3 The inverse problem . . . . . . . . . .
5 Conclusion
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1 Introduction
1
3
Introduction
The main issue of Alp 2002 is to discover in detail the tectonics and geodynamics at the junction of Europian, Adriatic (fragment of African tectonic
plate) and Tisza plates (contact zone between South-Eastern Alps, Dinarides
and Panonian basin) by using combined seismic refraction/wide-angle reflection method. For this two techniques the maximal distance between the
source (explosion) and the targets (geophone with seismograph) is around
few hundred km.
The velocity of seismic waves is in general increasing with depth. Waves
are refracted at discontinuities and travel in deeper regions almost horizontally (refraction) or they are reflected under blunt angle (wide-angle reflection). Because seismic waves travel long distances, projects as Alp 2002 give
us information about the lithosphere.
The Earth Lithosphere is in general characterized by three discontinuities:
• Moho discontinuity between Earth’s crust and mantle. It is named
after Croatian seismologist Mohorovičič, who discovered it by analysing
the earthquake in 1909 in the region of Kolpa. Moho is characterized
by an increase of P-wave velocity (compresional waves or prima waves)
from 6.5–7.2 km/s in the crust to 7.8–8.5 km/s in the mantle, and an
increase of S-wave velocity (shear waves) from 3.7–3.8 km/s to 4.8 km/s
.Moho is situated 25–40 km below continents, 5–8 km below the ocean
floor, and 50–60 km below certain mountains regions.
• Conrad discontinuity between upper part (granite) and lower part
(bazalt) of the crust, is located at depth 17–20 km and is characterized
by an increase of P-wave velocity from 5.8–6.2 km/s to 6.5 km/s. It is
not observed everywhere.
• Third discontinuity which is often seen on a seismic data is a junction
between the sediments and magmathic base.
In the framework of the Alp2002 project a network of 12 profiles of total
length 4100 km was spreading over seven countries (Austria, Chech republic, Hungary, Croatia, Slovenia, Italy and Germany). 1055 portable seismographs were deployed and 31 strong (300 kg) explosions fired. In Slovenia
127 seismographs were placed along five profiles of total length 575 km and
two explosions were fired; one near Vojnik and the other near Gradin. The
collected data will allow construction of the three-dimensional model of the
lithosphere.
2 Ray theory and seismic waves
2
2.1
4
Ray theory and seismic waves
Wave equations for P- and S-waves
2
Displacements u(x, t) in Navier’s equation ρ ∂∂tu2 = (λ + µ)∇(∇ · u) + µ∇2 u
form a vector field in an elastic medium. We use Helmholtz theorem in order
to represent u in terms of a curl-free scalar potential φ and divergence-free
vector potential ψ
u = ∇φ + ∇ × ψ.
(1)
The Navier equation may than be re-written as
∂2φ
∂2ψ
∇ (λ + 2µ)∇ φ − ρ 2 + ∇ × µ∇2 ψ − ρ 2 = 0,
∂t
∂t
"
#
"
#
2
(2)
where ρ, λ and µ are taken to be costant troughout the medium and the
change in the gravitational field strength are ignored. Equation (2) is setisfied, if
s
1 ∂2φ
λ + 2µ
2
∇ φ − 2 2 = 0, α =
(3)
α ∂t
ρ
and
1 ∂2ψ
∇ ψ − 2 2 = 0,
β ∂t
2
s
β=
µ
.
ρ
(4)
Equations (3) and (4) are two independent wave equations that describe
compressional waves traveling at the P -wave velocity α and shear waves
which travel at the S-wave velocity β. They represent a good aproximation to the equation of motion for heterogeneous media provided that the
elastic propertis of the medium do not change significantly over a single
wavelength. If the elastic properties do change rapidly, both wave equations
become coupled and P - and S-waves may no longer be considered to propagate independently. In the layer-interface formalism both wave equations
are normally treated as independent within each layer, but converted waves
may be produced at intefaces, where there is a sudden change in the elastic
properties of the medium.
2.2
Ray equation
The procedure of execution the ray equation is similar in geophysics and
optics. The solution of (3) for harmonic waves of frequency ω with constant
amplitude and zero initial phase can be written (for P -waves)
φ(x, t) = φ0 (x)e−iωt
(5)
2.2 Ray equation
5
without the loss of generality. If φ0 (x) is defined as
φ0 (x) = A(x)eik0 S(x) ,
(6)
where k0 = ω/α0 and α0 is a reference wave speed, equation (5) will form
a solution to the wave equation provided that certain constraints on A(x)
and S(x) (called the eikonal) describe the spatial variation of amplitude and
phase within a general wave field.
Let us now introduce the concept of a ray, which describes the path of a
wave packet through the isotropic medium, being at all times perpendicular
to the wavefront. It is clear that the unit vector
α
p̂ = ∇S
(7)
α0
is always perpendicular to the wavefront in the direction of the increasing
phase. Like the travel time T (x), which is defined as a time taken for wavefront to travel from a reference point x0 to the arbitrary point x, the eikonal
is defined relative to the phase at the reference point S(x) = α0 T (x) and the
eikonal equation can be re-written in terms of the travel time and the wave
speed v(x),
α2
1
(∇S)2 = 02 =⇒ (∇T )2 =
.
(8)
α
v(x)2
The ray path may be described by the function x(s) where s is the curvilinear
distance from the reference point along the ray path. Let us also define a
second function p(s) for which p = ∇T . This is called the slowness vector,
because its magnitude at a point x(s) is equal to the reciprocal of the velocity
at that point. Applying the condition that the ray path is orthogonal to the
wavefronts yields
dx
= v∇T = vp.
(9)
ds
By differentiating the eikonal equation (8) with respect to s and combining
the result with (9) we obtain the ray equation
d
ds
1 dx
v ds
!
=∇
1
.
v
(10)
Once the ray trajectory has been obtained, travel times may be calculated
by integrating
T =
Z
raypath
|∇T |ds =
Z
raypath
!
1
ds.
v(x(s))
(11)
So the travel time from A to B is equal to the travel time from B to A. This
principle of reciprocity may be used to improve the efficiency of ray-tracing
for certain problem types.
2.3 Ray propagation in media of constant velocity
2.3
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Ray propagation in media of constant velocity
In this case we will take a layer of thickness H and velocity v 0 over a halfspace of velocity v. This situation allows the study of ray trajectories and
travel times due to the presence of a surface of contact between the layer
and a half-space. The thickness of the layer is the parameter that scales a
spatial dimension of the model. For this reason, the application of ray theory
is valid, if the wave lengths are much smaller than the thickness of the layer.
This example represents, in a simplified way, the situation of the Earth’s
crust over the upper mantle for small distances, for which the flat-Earth
approximation is valid. We consider the case v 0 < v and the focus F at the
Figure 3: Seismic waves
surface (h = 0)(figure 3). There are three types of rays travelling from F to
P, which is a distance x from the focus F: (a) direct rays; (b) rays reflected
on the surface between the layer and the half-space (FCP); and (c) criticaly
refracted rays or head waves, i.e. rays with a critical angle of incidence ic ,
propagate a certain distance horizontally through the half-space and come
back to the free surface with the same angle of incidence ic (FBDP).
In the next sections of this seminar I will present some possible approaches
of modelling the seismic data.
3
Ray tracing with RAYAMP
Rayamp is a ray tracing modelling program that is used to obtain 2D models
without inversion stage. To define the velocity structure within a 2D model,
there are two types of boudaries, model and divider boundaries. A model
boundary is a straight line of arbitrary dip. It has assigned a costant velocity
along its length and a nonzero velocity gradient normal to its length. A
divider boundary is assigned a velocity zero and it separates two regions
3 Ray tracing with RAYAMP
7
Figure 4: Profil from Rijeka to Vojnik
with different velocity and velocity gradient. Blocks may thus be defined, in
which the velocity, magnitude and direction of velocity gradient are arbitrary.
The ray path within a given block is a circular arc (because of constant
velocity gradient), for which the travel time and the distance traveled may
be calculated using very simple analytical expresions.
The source may be located along any model boundary, targets are usually
located on the surface of the model. If the incidence to a boundary is at the
Figure 5: Profil from Vojnik to Ivanič grad
critical angle, the head waves may occur. Beyond the point of intersection of
4 Ray tracing with JIVE3D
8
the critical ray with the boundary, the head waves are simulated by shooting
critically refracted rays off the boundary at regular intervals along its length.
If the incident angle is more than the critical, then the ray reflects from the
boundary. For the precritical and multiple reflections, only the boundaries at
which reflection is studied need to be specified. At all other boundaries the
behaviour of a ray is controlled by the angle of incidence at the boundary.
Thus, if no precritical or multiple reflections are studied, then a single
specification of the range of take-off angles gives all wide-angle reflections,
turning rays and head waves. The corresponding travel-time curve is divided
into branches such that the distance along each branch increases or decreases
monotonically with distance. The family of rays associated with each traveltime branch is labeled with a unique identification number, which is used in
the synthetic seismogram routine for purposes of interpolation within a given
ray family.
4
Ray tracing with JIVE3D
Modelling with Jive3D can be divided into forward modelling and inversion
stjpg. At each iteration of the algorithm, a set of synthetic travel-time data
are produced from a working velocity model, and the Frechet derivatives
(subsection 4.2.3), which link small changes in model parameters to small
changes in travel-time data are calculated. These synthetic data and Frechet
derivatives are then passed to the inversion stage, which compares the synthetic data with the given real data and calculates a new model based on a set
of linear aproximations until the model converges to a point that optimizes
the specified norm for smoothness and best fit.
4.1
Model parametrization
An important feature of any tomographic inversion method is its approach
to model parametrization. We require a parametrization that is able to
describe models in the conventional layer-interface formalism. Each layer and
interface is then parametrized by a grid of velocity and depth nodes, allowing
the inclusion of reflected and refracted travel times. To define a 3D field of
seismic velocity and an interface in a 3D velocity model, we will use qudratic
B-splines in three-dimensions B23D (x, x0 ) = β2 (x1 , x01 )β2 (x2 , x02 )β2 (x3 , x03 ) and
cubic B-splines in two-dimensions B32D (x, x0 ) = β3 (x1 , x01 )β3 (x2 , x02 ). The β
functions are basis functions for interpolation, also called B-spline functions.
4.2 The forward problem
9
Velocity field is then given by
v(x) =
27
X
B23D (x, xi )vi ,
(12)
i=1
where vi are the velocity B-spline coefficients that have influence at position
x; xi are the node positions corresponding to those coefficients. Similarly,
an interface in a 3D velocity model (modelled as a 2D surface), constructed
using cubic B-splines is given by
z(x) =
16
X
B32D (x, xi )zi ,
(13)
i=1
where zi are depth B-spline coefficients that have influence at position x;
xi are equivalent node positions. The interfaces represent discontinuities in
seismic velocity at which reflections and refractions may occur.
4.2
4.2.1
The forward problem
Two-point ray tracing
In the section 2.2 it was stated that the ray equation may be used to determine the trajectory of a ray through a velocity model given the appropriate
boundary conditions. These conditons normally take one of the following
forms:
1. Source position and direction of propagation specified - find the position
at which the ray returns to the surface (if at all) and the travel time;
2. Source position and direction specified - find the position within the
model at which a given travel time is specified;
3. Start and end points specified - find all rays which join the two together
and their travel times.
In order to ensure that all two-point solutions may be found for any
possible ray phase (including reflections and refractions through multiple
layers), the shooting method is used in JIVE3D. This approach is relatively
easy to implement for the models using the ray perturbation theory (RPT).
It may be applied in a consistent manner to wide-angle reflection, wide-angle
refraction and normal incidence reflection phases. All possible ray paths from
source to receiver are being explored.
4.2 The forward problem
4.2.2
10
Ray perturbation theory (RPT)
RPT is a powerful set of tehniques allowing ray paths, travel times, amplitudes, paraxial rays and even waveforms to be approximated rapidly by
applying perturbations to models, for which a previous solution has been
calculated. Two mathematical formulations of RPT have emerged: Hamiltonian formulation and Lagrangian formulation. In both, a similar
procedure is adopted. A ray is traced through a reference medium cell by
cell for which an analytical solution exists and the quantities sought by the
procedure (travel time, amplitude, final position and slowness vector) are
calculated for reference ray (figure 6). One or more perturbations to the
system are then applied and their effect on the ray is estimsted to first order,
producing the perturbed ray. When ray-tracing through isotropic media, the
Hamiltonian may be defined as follows
1
H(x, p, τ ) = (p2 − u2 (x)),
2
where u(x) is slowness function, or the reciprocal of the seismic
x and p are the position and the slowness vector perspectively,
parameter τ is a measure of the extent of propagation along the
may be defined by its differential relation to the travel time dT
Hamilton’s equations yield a representation of the ray equations,
dx
= ∇p H,
dτ
(14)
velocity;
and the
ray, and
= u2 dτ .
dp
= −∇x H.
dτ
(15)
In order to apply RPT, the function u2 is written in the form u2 = u2ref +∆u2
where u2ref is a reference function for which rays may be traced analytically
and ∆u2 is perturbation that will be applied. This perturbation of u2 results
in a perturbation of the Hamiltonian ∆H = − 12 ∆u2 . The perturbations to
the position and slowness now read
∆x(τ ) = ∆x(τ0 ) + (τ − τ0 )∆p(τ0 ) +
1Z τ
(τ − τ 0 )∇x (∆u2 (x))dτ 0 ,
2 τ0
1Z τ
∆p(τ ) = ∆p(τ0 ) +
∇x (∆u2 (x))dτ 0 .
2 τ0
The travel time as a function of τ is given by
T (τ ) =
Z
τ
u2 (x)dτ 0
(16)
(17)
(18)
τ0
and the perturbation in travel time is Taylor expansion up to first order
∆T (τ ) =
Z
τ
τ0
2
0
0
∆u (x(τ ))dτ +
Z
τ
τ0
Γ0 ∆x(τ 0 )dτ 0 ,
(19)
4.2 The forward problem
11
where Γ0 is the gradient of the slowness squared function (Γ0 = ∇u2 (x0 )).
These results are all expressed as polynomials in τ therefore the integrals
in (16), (17),and (19) are straightforward. The equations above provide a
mechanism for tracing a ray through the medium described in subsection 4.1
using quadratic B-splines.
4.2.3
Frechet derivatives
Once the final position, final slowness and travel time have been determined
for a ray within a single cell in the grid, the Frechet derivatives relating the
travel time to each of the model parameters defining the velocity within this
cell must be obtained (see figure 6). Frechet derivatives for ray solutions to
Figure 6: A represantation of cell ray-tracing
the two-point problem describe a change in ray path as well as travel time
in response to small changes in the velocity or slowness squared parameter
(see figure 7). Frechet derivative of the travel time with respect to the j-th
model is given by
Z s2
∂T
∂ Z
∂u(x)
=
ds
u(x)ds =
∂m j ∂m j x0 (s)
∂mj
s1
(20)
where (s1 , s2 ) is the range along the ray over which the parameter mj influences the travel time. Making use of the identities udτ = ds and d(u2 ) =
2udu, we obtain
∂T
1 Z τ2 ∂u2 (x)
=
dτ
(21)
∂mj
2 τ1 ∂mj
4.2 The forward problem
12
Figure 7: Ray geometry for Frechet derivatives calculated under (a) initial
value boundary condition and (b) two-point boundary conditions. Shaded
area is part of velocity field that is perturbed. The new ray path is shown
as a dotted line.
which can be rewritten, by substituting x(τ ) = xref (τ )+∆x(τ ), as a reference
and a perturbation term,
∂T
=
∂m j
where
∂Tref
∂mj
!
and
∂T
∆
∂mj
1 Z τref
=
2 τ0
!
∂Tref
∂mj
!
∂T
+∆
∂mj
!
,
(22)
∂u2 (x0 ) ∂Γ0
+
· (xref − x0 ) dτ
∂mj
∂mj
!
1 Z τf inal
=
2 τ0
∂Γ0
∂∆u2 (x0 )
· ∆x +
dτ.
∂mj
∂mj
(23)
!
(24)
The 3D quadratic B-spline used to produce the slowness squared field in a
cell is
2
u (x) =
27
X
B23D (x, xj )mj .
(25)
j=1
Since Γ0 = ∇u2 (x0 ), we have
∂u2 (x0 )
= B23D (x, xj ),
∂mj
∂Γ0
= ∇B23D (x, xj ).
∂mj
(26)
4.3 The inverse problem
13
So far, these expressions have been derived for the case, in which the model
parameters describing the seismic velocity field contain B-spline coefficients
in units of u2 (represented as mu2 ). When solving the inverse problem using
regularized inversion (subsection 4.3), the model parameters must be con√
verted to units of velocity using the relation mv = 1/ mu2 and the Frechet
derivatives
∂T
∂T
2
=−
(27)
2
∂mv
(mv ) ∂mu2
Beginning with the differential expression for the change in travel time δT =
pδx along a short segment of a ray δx, for which the slowness vector is
p, a simple expression for the change in travel time for a ray crossing an
interface due to perturbation in that interface may be obtained from Snell
law; δT = δp3 δz, where δp3 is the change in the vertical component of the
slowness vector at the reflection/refraction and δz is the change in the interface depth. The interface is defined by equation (13), so an expression for
Frechet derivative may be obtained as
∂T
∂z
= δp3
= δp3 B32D (x, xj )
∂mj
∂mj
(28)
The results derived above allow a ray to be traced through a single cell in a
model parametrized as described in subsection 4.1, producing final position
and slowness vectors, travel times, and Frechet derivatives.
4.3
The inverse problem
In the linearized inversion approach, the inverse problem may be formulated
as: we have a model, described by the vector m, the components of which are
the interface depths and velocity parameters included in the inversion. From
this model, the forward modelling code provides a set of synthetic travel time
data, contained in the vector t and a matrix A of Frechet derivatives, which
measure the sensitivity of the model and synthetic travel times. We also have
the set of real travel-time data, treal , the uncertainties in those data, σ, and
information on the geometric arrangement of model parameters. In order
to measure the fit of the current model to the data, we define the traveltime residual vector r = treal − t, which is a subject of the optimization by
least-squares formulation.
For example I would like to present the evolution of salt dome inversion
from the starting to the final model. We can notice from pictures bellow that
χ2 is decreasing step by step, so we are getting more and more realistic two
dimensional model of investigating area.
5 Conclusion
14
Figure 8: From starting to the final model
5
Conclusion
We conclude by comparing both modelling techiques. Rayamp is only used
for 2D forward modelling, which means, that it can not produce inversions to
the starting model. If more realistic models are to be obtained with Rayamp,
one would have to include some sort of alghoritm for automatic search of the
best set of parameters, which are defining the model, such as; different model
boundaries, velocity field and smoothing criteria. The procedure of Jive3D
is more practical, it also includes inversion and is therefore able to constuct
the final model as an evolution of the starting model.
REFERENCES
15
References
[1] Agustin Udias: Principles of Seismology (Cambridge University Press
1999);
[2] James William Douglas Hobro: Three-dimensional tomographic inversion of combined reflection and refraction seismic travel-time data
(Chuchill College Cambridge 1999);
[3] Andrej Gosar: Exploration of the South-Eastern Alps lithosphere with
3D refraction seismic (Project Alp2002) data acqusition in Slovenia (Geologija);
[4] G. D. Spence, K. P. Whittall, R. M. Clowes: Practical Synthetic Seismograms for latterally varying media calculated by asymptotic ray theory
(BSSA 1984; 1209-1223);