Malcolm Sambridge - Research School of Earth...
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Inverse Problems and Seismic tomography PHYS3070 Malcolm Sambridge Research School of Earth Sciences Australian National University [email protected]
Transcript of Malcolm Sambridge - Research School of Earth...
Inverse Problems and Seismic tomography
PHYS3070
Malcolm SambridgeResearch School of Earth Sciences
Australian National [email protected]
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Geophysical inverse problems
Inferring seismic properties of the Earth’s interior from surface observations
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Inverse problems are everywhere
When data only indirectly constrain quantities of interest
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Reversing a forward problem
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Forward and inverse problems
Given a model m the forward problem is to predict the data that it would produce d
Given data d the inverse problem is to find the model m that produced it.
The forward operator might be linear or nonlinear, mmight be a finite set of unknowns or a complete function.
Terminology can be a problem. Applied mathematicians often call the equation above a mathematical model and m as its parameters, while other scientists call G the forward operator and m the model.
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Linearized inverse problems
Nonlinear inverse problem
Choose a reference model mo and perform a Taylor expansion of g(m)
Linearized inverse problem
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Example: Travel time tomography
Seismic travel times are observed at the surface, and we want to learn about the Earth’s structure at depth. Travel times are related to the wave speeds of rocks through the expression
t =ZR
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v(x)dl =
ZRs(x)dl
The raypath, R also depends on the velocity structure, v(x). R can be found using ray tracing methods.
Is this a continuous or discrete inverse problem ?
Is it linear or nonlinear ?
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Travel time tomography example
We can linearize the problem about a reference model so(x) or vo(x). We get either...
δt =ZRo
δs(x)dl δt =ZRo− 1v2o
δv(x)dlor
How do elements of the matrix relate to the rays ?
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Travel time tomography example
The element of the matrix is the integral of the j-th basis function along the i-th ray. Hence for our chosen basis functions it is the length of the i-th ray in the j-th block.
δti = Gi,jδmj
δd = Gδm
G =
⎡⎢⎢⎢⎣l1,1 l1,2 · · · , l1,Ml2,1 l2,2 · · · , l2,M... ... . . . ...lN,2 lN,2 · · · , lN,M
⎤⎥⎥⎥⎦
δmj = sj − so,jδdj = toi − tci(so)
li,j =
Travel time residual for i-th path
Slowness perturbation in j-th cell
Length of i-th ray in j-th cell
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Travel time tomography example
One ray and two blocks
δti = Gi,jδmj
Non-uniqueness
δt1 = l1,1 × δs1 + l1,2 × δs2
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Travel time tomography example
Many rays and two blocks
δti = Gi,jδmj
Uniqueness ?
δti = li,1 ∗ δs1 + li,2 ∗ δs2 (i = 1, N)
NO !
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Travel time tomography example
δt1 = l1,1 ∗ δs1 + l1,2 ∗ δs2
δt2 = l2,1 ∗ δs1 + l2,2 ∗ δs2
δd = Gδm
Can we resolve both slowness perturbations ?
G has a zero determinant and hence problem is under-determined
Same argument applies to all rays that enter and exit through the same pair of sides.
Zero eigenvalues => Linear dependence between equations => no unique solution. An infinite number of solutions exist !
l1,1
l1,2=l2,1
l2,2⇒ |G| = 0
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Travel time tomography example
Two rays and two blocks
δti = Gi,jδmj
Uniqueness ?
δti = li,1 ∗ δs1 + li,2 ∗ δs2 (i = 1,2)
YES
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Travel time tomography example
Two rays and two blocks
δti = Gi,jδmj
Over-determined Linear Least squares problem
δti = li,1 ∗ δs1 + li,2 ∗ δs2 (i = 1, N)
Model variance is low but cell size is large
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Travel time tomography example
Many rays and many blocks
δti = Gi,jδmj
Simultaneously over and under-determined Linear Least squares problem
Mix-determined problem
Model variance is higher but cell size is smaller
Model variance and resolution trade off
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Linear discrete inverse problem
To find the best fit model we can minimize the prediction error of the solution
But the data contain errors. Let’s assume these are independent and normally distributed, then we weight each residual inversely by the standard deviation of the corresponding (known) error distribution.
We can obtain a least squares solution by minimizing the weighted prediction error of the solution.
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Linear discrete inverse problem
We seek the model vector m which minimizes
Note that this is a quadratic function of the model vector.
Solution: Differentiate with respect to m and solve for the model vector which gives a zero gradient in
A solution to the normal equations:
This gives…
This is the least-squares solution.
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This is an ill-posed or under-determined problem
with no unique solution
Discrete ill-posed problems
What happens if the normal equations have no solution ?
Recall that the inverse of a matrix is proportional to the reciprocal of the determinant
The determinant is the product of the The determinant is the product of the eigenvalueseigenvalues. Hence the inverse . Hence the inverse
does not exist if any of the does not exist if any of the eigenvalueseigenvalues ofof are zero are zero
We have seen examples of this in the tomography problem
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Parametrizing a continuous function is a choice, which affects the nature of the inverse problem.
In a linear problem, if the number of data is less than the number of unknowns then the problem will be under-determined.
If the number of data is more than the number of unknowns the system may not be over-determined. The number of linearly independent data is what matters. This is the true number of pieces of information.
Linear discrete problems can be simultaneously over and under-determined. This is a mix-determined problem.
The is a trade-off between the variance (of the solution) and the resolution (of the parametrization).
Recap:
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Example: tomography
Idealized Idealized tomographictomographic experimentexperiment
δd = Gδm
G =
⎡⎢⎢⎢⎣G1,1 G1,2 G1,3 G1,4... ... ... ...... ... ... ...... ... ... ...
⎤⎥⎥⎥⎦
What are the entries of G ?What are the entries of G ?
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Example: tomography
Using rays 1Using rays 1--44
G =
⎡⎢⎢⎢⎢⎣1 0 1 00 1 0 1
0√2√2 0√
2 0 0√2
⎤⎥⎥⎥⎥⎦
GTG =
⎡⎢⎢⎢⎣3 0 1 20 3 2 11 2 3 02 1 0 3
⎤⎥⎥⎥⎦
This has singular 0, 2, 4, 6.This has singular 0, 2, 4, 6.
Vp =
⎡⎢⎢⎢⎣0.5 −0.5 −0.50.5 0.5 0.50.5 0.5 −0.50.5 −0.5 0.5
⎤⎥⎥⎥⎦ Vo =
⎡⎢⎢⎢⎣0.50.5
−0.5−0.5
⎤⎥⎥⎥⎦
δd = Gδm
What type of change does the null space vector correspond to ?What type of change does the null space vector correspond to ?
Gvo = 0
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Worked example: Eigenvectors
Vp =
⎡⎢⎢⎢⎣0.5 −0.5 −0.50.5 0.5 0.50.5 0.5 −0.50.5 −0.5 0.5
⎤⎥⎥⎥⎦
Vo =
⎡⎢⎢⎢⎣0.50.5
−0.5−0.5
⎤⎥⎥⎥⎦
S12=6 S2
2=4
S32=6
The end
...for now
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