Download Geophysical Inversion

Lectures 12 & 13:
Geophysical Inverse Theory I,
II
H. SAIBI & SASAKI
Kyushu University
Dec. 24, 2015 & Jan. 7, 2016
1
Geophysical inverse problems
Inferring seismic properties of the Earth’s interior
from surface observations
2
Inverse problems are everywhere
When data only indirectly constrain quantities of interest
3
Thinking backwards
Most people, if you describe a train of events to
them will tell you what the result will be. There are
few people, however that if you told them a result,
would be able to evolve from their own inner
consciousness what the steps were that led to that
result. This power is what I mean when I talk of
reasoning backward.
Sherlock Holmes,
A Study in Scarlet,
Sir Arthur Conan Doyle (1887)
4
Reversing a forward problem
5
Inverse problems=quest for information
What is that ?
What can we tell about
Who/whatever made it ?
Collect
data:
Measure size, depth
properties of the ground
Can we expect to reconstruct the
whatever made it from the evidence ?
Use our prior knowledge:
Who lives around here ?
Make guesses ?
6
Anatomy of an inverse problem
Hunting for gold at the beach with a gravimeter
X
X
X
X
Gravimeter
?
7
Forward modelling example: Treasure Hunt
We have observed some values:
10, 23, 35, 45, 56  gals
X
How can we relate the observed gravity
values to the subsurface properties?
Gravimeter
X
X
X
X
?
We know how to do the forward problem:
G (r ' )
( r )  
dV '
r  r'
This equation relates the (observed) gravitational potential to the
subsurface density.
-> given a density model we can predict the gravity field at the surface!
8
Treasure Hunt: Trial and error
What else do we know?
Density sand: 2.2 g/cm3
Density gold: 19.3 g/cm3
X
X
X
X
X
Do we know these values exactly?
Gravimeter
?
Where is the box with gold?
One approach is trial and (t)error forward modelling
Use the forward solution to calculate many models for a rectangular box
situated somewhere in the ground and compare the theoretical (synthetic)
data to the observations.
9
Treasure Hunt: model space
But ...
... we have to define plausible models
for the beach. We have to somehow
describe the model geometrically.
X
We introduce simplifying approximations
X
Gravimeter
X
X
X
?
- divide the subsurface into rectangles with variable density
- Let us assume a flat surface
x
x
x
surface
x
x
sand
gold
10
Treasure Hunt: Non-uniqueness
Could we compute all possible models
and compare the synthetic data with the
observations?
X
- at every rectangle two possibilities
(sand or gold)
- 250 ~ 1015 possible models
X
X
X
X
Gravimeter
Too many models!
- We have 1015 possible models but only 5 observations!
- It is likely that two or more models will fit the data (maybe exactly)
Non-uniqueness is likely
11
Treasure hunt: a priori information
Is there anything we know about
the treasure?
How large is the box?
Is it still intact?
Has it possibly disintegrated?
What was the shape of the box?
X
X
X
X
X
Gravimeter
This is called a priori (or prior) information.
It will allow us to define plausible, possible, and unlikely models:
plausible
possible
unlikely
12
Treasure hunt: data uncertainties
Things to consider in formulating the inverse problem
Do we have errors in the data ?
Did the instruments work correctly ?
Do we have to correct for anything?
(e.g. topography, tides, ...)
X
X
X
X
X
Gravimeter
Are we using the right theory ?
Is a 2-D approximation adequate ?
Are there other materials present other than gold and sand ?
Are there adjacent masses which could influence observations ?
Answering these questions often requires introducing
more simplifying assumptions and guesses.
All inferences are dependent on these assumptions.
(GIGO)
13
What we have learned from one example
Inverse problems = inference about physical
systems from data
X
X
X
X
X
Gravimeter
- Data usually contain errors (data uncertainties)
- Physical theories require approximations
- Infinitely many models will fit the data (non-uniqueness)
- Our physical theory may be inaccurate (theoretical
uncertainties)
- Our forward problem may be highly nonlinear
- We always have a finite amount of data
Detailed questions are:
How accurate are our data?
How well can we solve the forward problem?
What independent information do we have on the
model space (a priori information) ?
14
Estimation and Appraisal
15
Objective of geophysical survey
to obtain information about geologic structure or
buried object (such as the existence of the object
and its size and shape) by generating the
distribution of physical properties
Data
(Earth response)
Distribution of
physical property
16
Forward & inverse problems
Forward
modeling
Distribution of
physical property
(Earth model)
Earth response
(Data)
Inversion
17
Forward problem
Earth model
“Convolved” (or
averaging) process
Earth response
Governing
equation
We have a unique solution!
m
F (m)
d
e.g., finite difference modeling
18
Governing equation
in forward problem
Density
Laplace’s equation
Gravity
Magnetic
susceptibility
Laplace’s equation
Magnetic field
Electrical
resistivity
Poisson’s equation
Electric potential
Electrical
resistivity
Maxwell equation
(diffusion or wave)
EM field
-----
19
Inverse problem
Earth model
“Deconvolved”
process
Earth response
Governing
equation
We may have many (non-unique) solutions!
m1
m2
-
1
F (m)
d
20
Why is the inverse solution non-unique?
(Why is the inverse problem ill-posed?)
• Data are incomplete; data include
measurement errors and various errors.
• Data are insufficient; data coverage is
limited in space (and in frequency).
21
How to tackle this problem of
non-uniqueness
Incorporate additional ( a priori)
information into the inverse problem
Approach to solving the inverse
problem as an least-squares problem
with constraints (or optimization
problem)
22
Regression analysis
as an inverse problem
In this section, we regard regression analysis as a kind
of the inverse problem. Learning regression analysis
help to understand the basis of inversion.
23
Linear regression
y
y  a  bx
Assume that the observations can
be expressed as
y  a  bx
(xi , yi)
x
where y is an observation, x is
independent variables, a and b
are unknown parameters.
Determine a and b when we are
given N data sets
(x1 , y2), - - -, (xN , yN) .
24
Formulation of regression analysis (1)
y1  a  bx1
y2  a  bx2
Observation equation
- - - - - - -
y N  a  bxN
One can solve the above observation equation
in a least-squares sense.
N
   ( y i a  bxi ) 2  min
i 1
25
Formulation of regression analysis (2)
Minimization of



a
with respect to a and b requires that
 2(y  a  bx ) 0
i
i

  2( yi  a  bxi ) xi  0
b
26
Formulation of regression analysis (3)
y
i
 na  b xi
x y
i
i
Normal equation
 a xi  b xi
2
Solving the simultaneous equations, we obtain
x  y x x y

a
n x   x 
2
i
i
i
i
2
2
i
i
i
b
n xi yi   xi  yi
n xi   xi 
2
2
27
Regression from perspective of inversion
Model parameter:
a and b
Model function (response): y  a  bx
Data (observations):
y1 , y2 , - - -, yN
Independent variable such as
measuring location: x1 , x2 , - - -,xN
28
How to formulate least-squares
problem
In this section, we don’t consider non-uniqueness
problem and thus incorporation of constraints. We
just focus on how to find a model that matches
the data in a least-squares sense. So the
problem so far may be called unconstrained
least-squares problem. We assume that we can
do forward modeling.
29
Setting of inverse problem
Model parameters
Model response
Data (observations)
m  m1 , m2 , m3 ,  , mM 
F1 (m)
-----
FN (m)
d  d1 , d 2 , d3 ,  , d N 
m1 m2 m3
2D discretization
Note that model response is nonlinear function with respect to model
parameters.
Formulation of unconstrained
nonlinear least-squares problem
F1 (m)  n1  d1
F2 (m)  n2  d2
d (m)   d j  F j (m)
------
 d  F(m)  min
FN (m)  nN  d N
Gaussian noise
2
N
j 1
2
d
: objective function
(data misfit)
2
 : L2 (least-squares) norm
Observation equations
31
Linearized objective function
(0)
m
Initial model estimate:
Applying Taylor expansion to model response gives
M
 Fj
i 1
 mi
F j (m)  F j (m )  
(0)
mi
Then we have
F j


d   d j  
mi 
j 1 
i 1  mi

N
M
2
mi : Model parameter change (unknowns)
( 0)
d j  d j  Fj (m ) : Difference between data
and model response
32
Linearized objective function
in matrix form
d  d  Am  min
2
m : parameter change vector (unknowns)
d
A
: data residual vector
: N x M Jacobian matrix
(matrix of partial derivative
with respect to model parameters)
33
Minimization of objective function
  Am  d  min
2
Minimization of

with respect to m requires that

 0 (i  1,2,    M )
 mi
Then we have the normal equation,
AT A m  AT d
T : transpose
Note that the normal equation is M x M linear system of equations
34
There is an alternative way to obtaining the leastsquares solution by forming the normal equation. It
is through forming the over-determined system
equations. In general, this approach is more
accurate particularly when the number of the
unknowns is large.
35
Minimization of objective function is
equivalent to obtaining least-squares
solution from over-determined system of
equations
A m d
=
A m  d
Over-determined: the number of unknowns is
smaller than the number of (observation) equations
36
Exercise 1
When the initial estimate m0 is
given, show the linearized data
objective function in both
summation and matrix forms. In
the matrix form, show all the
elements.
Assume that we have
m  m1 , m2 , m3 
d  d1 , d 2 , d3 , d 4 
F1 (m) F2 (m) F3 (m) F4 (m)
Use the notation x   xi
2
2
i
We define the data objective
function as
d   d j  F j (m)
4
2
j 1
37
Exercise 2
Based on the result of Exercise 1, form the
over-determined system of equations.
38
How to solve least-squares
problems
In this section, we just focus on obtaining the leastsquares solution when we are given an overdetermined system of equations
Ax  y
A
x y
=
39
Classic approach to least-squares problem
Ax  y
A x y
  Ax  y  min
=
Minimization of
AT A
AT y
=
M x M matrix
2

with respect to x requires that

 0 (i  1,2,    M )
 xi
Then we have the normal equation,
A Ax  A y
T
T
40
Why is it called normal ?
A Ax  A y
T
A
T
r  Ax  y
Error vector
A r0
T
a1 a2
ai r  0
T
i=1, 2, - - -, M
Inner product
The error vector is normal (or orthogonal) to
the column vectors of A
41
Another approach to least-squares problem:
least-squares solvers
A
x y
=
Ax  y
• Modified Gram Schmidt
method (QR
decomposition )
• Householder transformation
method
• Singular value
decomposition (SVD)
method
42
Least-squares solver: QR decomposition
A
x y
=
Ax  y
A can be decomposed using
modified Gram Schmidt method
into
A=QR
Q: N x M orthogonal matrix
R: M x M upper triangular matrix
Rx  Q y
T
43
How to incorporate constraints (or a
priori information) into a least-squares
problem
In this section, we consider the most important aspect of
inversion, that is, how to pose some soft restrictions on
the least-squares solution so that the solution has the
likely features of the reality.
44
Prerequisites for solving inverse
problems
1) Forward problem can be solved.
(Model response can be calculated)
2) Jacobian matrix (or sensitivity matrix) can be
obtained
3) Incorporate a priori information (or the likely or
expected earth structure)
45
Sidetrack: Occam’s inversion
To find the smoothest model so that its features
depart from the simplest case only as far as is
necessary to fit the data (Constable et al., 1987)
Occam’s razor: the explanation of any
phenomenon should make as few assumptions
as possible, eliminating those that make no
difference in the observable predictions of the
explanatory hypothesis or theory (Wikipedia).
46
Constrained least-squares problem
Find a model that fit the data as well as
incorporates additional information
Design an objective function that include
a norm (measure) for model structure
besides the data misfit
  d   m  min
m : model objective function (constraints)

: tradeoff (regularization) parameter
47
An example of model objective function
(simplest one)
m  m  m ref
2
  d    m
 Am  d   m  m ref
2
Data misfit
2
 min
Size of model structure
(Constraint)
48
Selection of model objective function
  d   m
 Am  d   ?
2
Data misfit
2
 min
Model objective
function
49
Various desirable characteristics of structure
and their representations (1D)
• be as smooth as
possible
• be as flat as possible
• be as close to a
reference model as
possible
• be as blocky as
possible
 m
m 
z2
2
2
m
m 
z
2
m  m  m ref
m
m 
z
2
 m


 c 
 z

2
50
Linear expression for
m1
m2
m3
.
.
1D discretization
m
z
2
0 
1  1
 1 1




C
1 1





 0
1  1
M 1
 m Cm   (mi mi 1 )
2
2
i 1
51
 m
2
z
2
Linear expression for
m1
m2
m3
.
.
 m Cm
2
0 
 1 1
 1 2  1

0


 1 2 1 0 
C




 0  1 2  1


 1 1 
 0
2
M 1
 (m1  m2 )   (mi 1  2mi  mi 1 ) 2  (mM 1  mM ) 2
2
i 2
52
Classic approach to least-squares problem
  Am  d   C m  min
2
2
Minimization of  with respect to m requires that

 0 (i  1,2,    M )
 mi
Then we have the normal equation,
A A   C Cm  A d   C C m
T
T
T
mm
( 0)
T
 m
( 0)
53
Another approach: least-squares solver
  Am  d   C m  min
2
2
m  m( 0)  m
m
A
=
 A 
 d


 m  
(0) 
  C
  C m 
d
 C   Cm
( 0)
The constraints can be regarded as a part of observation
equations.
54
Choosing the tradeoff parameter
small

large

the data fit gets better
(likely to overfit)
the data fit gets poorer
(likely to underfit)
• Use a fixed value based on experience.
• Search an optimum value at each iteration so that the
data misfit is minimized.
• Use a more sophisticated search algorithm like L-curve
and ABIC methods.
55
Exercise 3
Assume that the model is 1D. Show the model
objective function (using the first derivatives) in
both summation and matrix forms.
m1
m2
m3
56
Exercise 4
The (total) objective function is given below.
Form the corresponding over-determined
system of equations using the answers to
Exercise 2 and 3.
  d  Am   C m
2
where
2
m  m  m
0
57
Practical tips for evaluating the
inversion results
You may encounter the results of geophysical inversion
as a client of geophysical service companies or you
may have to use inversion codes to interpret
geophysical data. This section provides practical tips for
those cases.
58
In the case of being shown
the inversion results
Pay attentions to :
1)Data misfit; how well the data fit the
model responses.
If the data fit is poor, there are two
possibilities: the data quality is poor or
inversion itself is poor.
2)What kind of a priori information is used.
59
In case of using inversion codes
Test the code on the synthetic data for
which you have the correct ‘answer’.
The simplest way to do this is use the data
from homogeneous half-space. For example,
for dc resistivity data, you provide 100 ohm-m
as the apparent resistivirty data. Start the
inversion with the initial model of, say 500
ohm-m and see if the solution converges to the
true value of 100 ohm-m.
60
Recap: Characterising inverse problems
 Inverse problems can be continuous or discrete
 Continuous problems are often discretized by choosing a
set of basis functions and projecting the continuous
function on them.
 The forward problem is to take a model and predict
observables that are compared to actual data. Contains
the Physics of the problem. This often involves a
mathematical model which is an approximation to the real
physics.
 The inverse problem is to take the data and constrain the
model in some way.
 We may want to build a model or we may wish to ask a
less precise question of the data !
61
Three classical questions
(from Backus and Gilbert, 1968)
The problem with constructing a solution
 The existence problem
Does any model fit the data ?
 The uniqueness problem
Is there a unique model that fits the data ?
 The stability problem
Can small changes in the data produce large
changes in the solution ?
(Ill-posedness)
Backus and Gilbert (1970)
Uniqueness in the inversion of inaccurate gross earth data.
Phil. Trans. Royal Soc. A, 266, 123-192, 1970.
62
References
Oldenburg, D.W., and Y. Li, 2005, Inversion for applied
geophysics: A tutorial, in D.K. Butler, ed., Near surface
geophysics: The society of Exploration Geophysicists,
89-150.
Constable, S.C., R.L. Parker, and C.G. Constable,
1987, Occam’s Inversion: A practical algorithm for
generating smooth models from electromagnetic
sounding data, Geophysics, 52, 289-300.
Lines, L.R., and S. Treitel, 1984, A review on leastsquares inversion and its application to geophysical
problems, Geophysical Prospecting, 32, 159-186.
63
Reference works
Understanding inverse theory
Ann. Rev. Earth Planet. Sci., 5, 35-64, Parker (1977).
Interpretation of inaccurate, insufficient and inconsistent data
Geophys. J. Roy. astr. Soc., 28, 97-109, Jackson (1972).
Monte Carlo sampling of solutions to inverse problems
J. Geophys. Res., 100, 12,431–12,447,
Mosegaard and Tarantola, (1995)
Monte Carlo methods in geophysical inverse problems,
Rev. of Geophys., 40, 3.1-3.29,
Sambridge and Mosegaard (2002)
64
Tikhonov Regularization
65
Lanczos bidiagonalization of gravity
66
3D Inversion of gravity data
67
Cuckoo Optimization of gravity
68
3D inversion of magnetic data
69
Books
See also Menke ‘Geophysical data
analysis: discrete inverse theory’
(Academic Press, 1989)
70
Books
Chapter 7 on inverse problems
Introductory Chapter
on inverse
problems
Useful Bayesian
tutorial
(First 5 chapters)
71