Dummies guide to robust inversion

(This is part of a series of documents I created during the PhD, with no intention to submit or reproduce them anywhere else. If you find it helpful, that's great. The formatting was butchered in the upload so there's a link to original document here: https://drive.google.com/open?id=1HJthXFj5Vta8cNSOtWFSJSN84yXQX012)

Introduction to Norms

A ‘norm’ is an expression of the size, or length, of a vector. These are typically expressed with a single bar ( ) for vectors, or a double-bar ( ) for matrices.

They are used in conjunction with misfit to restrict the rate-of-change of model values during successive iterations of a deterministic numerical inversion process.

Geophysical inversion presents an under-determined, or ill-posed, linear problem. The under-determined problem is easily surmised as a system where more variables than data are present. A mathematically infinite number of solutions exist to solve this problem, and so requirements are imposed on the range of possible solutions. Geological scenarios can be simplified into relatively smooth changes and typically a ‘minimum structure’ constraint is imposed.

Minimum structure implies that the difference between spatially nearby values should not vary wildly over a small area. This can occur when a mathematically ‘perfect’ fit is allowed, however, this seldom represents reality. Theoretically, with an infinite number of noiseless points, the perfect model can be constructed, but this is neither possible nor plausible for geophysical measurements.

In order to impose the structure requirement into the solution, the ‘norm’ is introduced to regulate changes in the model space. A norm which minimises structures is called the  norm, or the ‘Euclidean’ norm, and is simply the square root of the sum of the vector components. The  norm is defined as the ‘p’ vector norm (below) where ‘p’ is equal to 2.

Here ‘ ’ represents our vector, ‘j’ is the component of the vector (i.e. the index position), and ‘p’ is the ‘flavour’ or ‘order’ of the norm.

The value of ‘p’ determines the flavour of the norm:             (Aka “Taxicab geometry”) ·                     (Aka “Euclidean Norm”) ·      =            (Aka “Maximum Norm”) ·

Introduction to Misfit

‘Misfit’ traditionally refers to the difference between some measured data and related predicted data. This is minimised to produce a model which is closest to the measured data.

i.e.

Here  is defined as “the standard deviation of Gaussian distributed errors associated with the datum ”. Statistically, the difference between the predicted and observed datums

Relevance to Inversion Theory

Symbols:

Symbol	Description
	Data misfit vector containing the difference between logarithms of measured and calculated apparent resistivity values
	Change in model parameters of the i-th iteration
	Model parameter vector for previous iteration
	Jacobian matrix of partial derivatives
	Roughness filter
	Damping factor
	Weighting matrices for data misfit roughness
	Weighting matrices for model roughness
	Weighting applied to smoothness filter in x, y, z directions
	Smoothing matrices/filters in x, y, z directions
	Vector of model parameters, (1 to n)

Definitions:

-norm

·         Minimise the sum of the absolute values of the data misfit (Loke)

-norm

·         Minimise the sum of the square of the values of the spatial changes in the model resistivity and data misfit (Loke)

Loke’s Course notes Eq. 1.23 (Optimisation Method)

Loke (Course notes):

Mathematical Explanation

Default Smoothness constrained inversion formulation

Robust Inversion (aka L1-norm, Blocky, Piece-wise constant)

Sole mention of cut-off factor and usage:

“There is a cut-off factor which controls the degree in which this robust model constrain is used. If a large value is used, for example 1.0, the result is essentially that of the conventional smoothness-constrained least-squares inversion method. If a very small value is used, for example 0.001, the result is close to the true L1-norm inversion method.” (Page 71, October 2015 Res2DInvx64.pdf Manual)

Loke (2003, Comparison of Blocky and smooth norms)

I.e. Introduces the weighting matrices  and  

Thus far, nothing suggests the L1 or L2 require a particular ‘value’ to operate with.

The Huber norm (aka Huber misfit function) uses a ‘Huber threshold’ ( ) to determine the norm.

As does the Ekblom norm:

From Farquarson 2003, “As  becomes small, this measure tends towards the  norm. For large values, the measure behaves like a scaled sum-of-squares measure”

In addition, the Minimum Support Functional (Zhdanov 1999) also contains a value for epsilon:

However, this is likely not used in Res2Dinv.

Resources

“Inversion theory:  Model norms, data misfit and non-uniqueness”

 https://www.eoas.ubc.ca/ubcgif/iag/tutorials/invn-theoryintro/

“IRLS and the Huber solver, an overview”

http://sepwww.stanford.edu/public/docs/sep103/antoine2/paper_html/node2.html

“Robust inversion using the Huber norm”

http://sepwww.stanford.edu/public/docs/sep121/paper_html/node10.html

Books

Snieder, R., & Trampert, J. (1999). Inverse problems in geophysics. In Wavefield inversion (pp. 119-190). Springer Vienna. http://inside.mines.edu/~rsnieder/snieder_trampert_00.pdf

Scales, J. A., & Smith, M. L. (1994). Introductory Geophysical Inverse Theory 9DraftE. http://terra.rice.edu/department/faculty/zelt/esci441/scales/scales.pdf

Menke, W. (2012). Geophysical data analysis: Discrete inverse theory (Vol. 45). Academic press.

Parker, R. L. (1994). Geophysical inverse theory. Princeton university press.

Zhdanov, M. S. (2002). Geophysical Inverse Theory and Regularization Problems (Vol. 36). Elsevier.

Tarantola, A. (2005). Inverse problem theory and methods for model parameter estimation. siam.

Meju, M. A. (1994), Geophysical Data Analysis: Understanding Inverse Problem Theory and Practice. Vol. 6: Society of Exploration Geophysicists Tulsa,, OK.

Menke, W. (2012). Geophysical data analysis: Discrete inverse theory (Vol. 45). Academic press.