Inverse problems in geophysics

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Inverse problems in geophysics
Advanced Inverse Theory - February 14

• Daniël Loeve Tonie Davidsen

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Introduction

• Aim Reconstruct the model from a set of
  measurements
• General problem of inversion is
• Not ideal situation
• Unstable
• Degrees of freedom
• Inversion consist of two steps
• Estimation problem
• Appraisal problem

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Introduction

• Estimated model differs from true model
• Non-uniqueness
• Data error
• Model error
• Appraisal problem only solved for linear problems

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Finite linear inverse problem

• A-g generalized inverse of A
• RA-gA

Ideal case Estimated model equal to true model
Ridentity matrix
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Least square estimation

• Number of independent data gt number of unknowns
• Solution given by minimizing the following cost
  function
• This is minimized by the following model
  estimate

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Minimum norm estimation

• Number of independent data lt number of unknowns
• Solution given by minimizing the following cost
  function
• This is minimized by the following model
  estimate

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Mixed determined problems

• Least square estimation ATA regular
• Minimum norm estimation AAT regular
• Commonly the model parameters suffer from both
• Some are over-determined
• Others are under-determined
• ATA and AAT cannot always be inverted
• problem ill-posed or ill-conditioned

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Damped least square solution

• Levenberg 1944 introduced damped least squares
  solution
• Solution is found by minimizing the following
  cost function
• With the solution
• Cost function express the goal
• Finding a model that fits the data
• Model size is not too large
• The trade-off parameter g controls the emphasis
  on these conflicting requirements

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Consistency problems

• When scaling/transforming a problem, the solution
  differs!!
• Look for generel regularization
• Damped least squares
• With regularization
• Giving
• Which minimizes

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How to choose the weight matrices

• Bayesian approach
• Statistical point of view on a-priori information
• The weight matrices become
• With the a-priori covariance matrices for data
  and model as
• Another approach
• Define the misfit function to favor certain
  models (smooth, small, )

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Transformation of weight matrices

• Cost function of generalized least squares
• We look at the last part, using mSm
• We look at the first part, using dQd and AQA

In Bayes
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Solving the system of linear equations

• Least square solution
• Rewritten
• This represents the normal equations
• Estimation part inverting B not needed
• Appraisal part it is needed
• Problem B may be extremely large and close to
  singular (even after regularization g)
• Solution Singular Value Decomposition (SVD),
  this technique is excluding singular values

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Singular value decomposition

• A matrix can always be rewritten as BUSVT
• Where
• U are eigenvectors of dataspace
• V are eigenvectors of modelspace
• S is a diagonal matrix of eigenvalues
• U and V are partitioned in sub-matrixes one
  corresponding to real eigenvalues the other to
  singular eigenvalues
• Final result of the inversion is

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Iterative least squares

• Problem Least squares solution include BTB which
  may be too large to handle!
• Solution Iterative process of least squares
• Nth iteration
• Update the iteration to the better
• Model update
• Least squares Still BTB!
• Replace by P-1
• P can be set to cI because is the direction of
  descent of the cost function, with c a constant.