The application of TSVD in MT inversion with improved MSE as determination

of the regularization level

SUN Ya, LIU Jian-xin, TONG Xiao-zhong ,GUO Zhen-wei School of Info-physics and Geomatics Engineering, Central South University, Changsha, China

Sunya0624@163.com

Abstract-The total mean-square error (MSE) of the estimated

model, defined as the sum of the standard model variance and

the bias variance, is used to define the truncation level of the

singular-value decomposition to give a reasonable balance

between model resolution and model variance. This balance is

determined largely by the data and no further assumptions are

necessary except that the bias terms are estimated sufficiently

well. This principle has been tested on the 1D magnetotelluric

inverse problem with special emphasis on high-frequency

magnetotelluric (MT) data. Simulations clearly demonstrate that

the method provides a good balance between resolution and

variance. Starting from a homogeneous half-space, the best

solution is sought for a fixed set of singular values. The model

variance is estimated from the sum of the inverse eigenvalues

squared, up to a certain threshold, and the bias variance is

estimated from the model projections on the remaining

eigenvectors. By varying the threshold, the minimum of the MSE

is found for an increasing number of fixed singular values until

the number of active singular values becomes greater than or

equal to the estimated number.

Keywords- inverse problem; TSVD; mean-square error;

magnetotelluric model

I. INTRODUCTION

In many environmental/near-surface applications, the geology can be assumed to be pseudo-1D and the use of 1D model for representing the geology is often adequate. Robust methods for the automatic inversion of magnetotelluric data using plane-layered models can provide a first image of the electrical structure of a general 3D earth (Szarka and Menvielle 1997). This is particularly important for environmental/near surface studies in which a vast number of

radio magnetotelluric (MT) or controlled-source MT (CSMT) stations are measured along profiles or on dense grids. For traditional interpretation of MT soundings, the most commonly used algorithms are those of Constable, Parker and Constable (1987) and Smith and Booker (1988) using a smooth layered model and the Fischer scheme (Fischer et al. 1981) attempting to construct models with a small number of layers. In order to be sure that the data have been adequately fitted, it is common practice to fit the data using the D+ representation (Parker and Whaler 1981; Parker and Booker 1996), which will provide a better fit than any layered earth model.

Smooth layered inversion has become very popular, one reason being that it works automatically, another that the inverse problem is regularized in a systematic way. However, it is not easy to know beforehand which regularization parameters to use for a particular case or which functional or combination of functional to select from among the flattest model, the smoothest model and the nearest model [1-5]. In the following, a short description of the 1D model is presented together with an introduction to the principles of linear minimum MSE inversion. The non-linear approach taken here is illustrated by a detailed description of the inversion of a single data set. A suite of synthetic examples is inverted with and without parameter bounds. Finally, a set of MT data along a profile crossing a pollution plume spreading from a waste dumpsite in the Netherlands is inverted with the new technique.

II. TRUNCATED SVD AND LINEAR INVERSE PROBLEMS

Let the model be described by M model parameters

978-1-4244-4507-3/09/$25.00 ©2009 IEEE

1 1( ,...., ) ( ,...., )T TM Mm m m x x= = and N data

points 1( ,...., )TNd d d= . The linear forward problem can

then be formulated as d= Gm, where the data kernel G is independent of m. The inverse problem can be formulated as finding a model such that the prediction error energy E. and make the E minimize

2E d Gm= − (1)

Applying singular value decomposition to the matrix A, we have

1

nT T T

i i ii

G U V u vλ=

= Λ =∑

1 2 1 1( , , , ); ( , , ); ( , , ),m m m n n n nU u u u V v v diag λ λ× ×= = Λ=

2 2 21 2 0nλ λ λ≥ ≥ ≥ ≥ > .

So we can obtain the solution of the style (1)

1ˆest Tp p p pm m V U d−= = Λ (2)

In practice, the definition of null space is somewhat arbitrary because of numerical difficulties in the accurate calculation of the smallest eigenvalues. However, as we shall see, this is not so important because the choice of truncation level p is normally made so that p is smaller than the rank of G. If p is chosen to be too large, small data errors may give rise to large

oscillations in ˆ pm . Conversely, if p is chosen to be too small,

the solution will be very stable and smooth, but it may become heavily biased. So the MSE is introduced [6].

III. THE METHOD OF MINIMUM MSE

Smoothing ill-posed problems is highly desirable for obtaining maximum information from measurements, since the LS solution is too uncertain and may even result in incorrect signs of x. Regularization by TSVD could offer improvement over the LS solution if the truncated parameter p is properly chosen. The conventional principal component method fails to establish a connection between the goodness of an estimator and the pre-selected significance level [7-9].

How to choose the estimator become very important, the MSE is used to estimate the regularization parameter.

So the solution (2) can be written as

1 1

ˆ ˆ ˆTp pi

p i i i p pi ii

um v a v V aλ= =

= = =∑ ∑ (3)

Where ˆ ˆ ˆ /

ˆ0 otherwise

i i ii

a a ca

λ σ⎧ ≥⎪= ⎨⎪⎩

Note that the covariance matrix of αˆ p is of diagonal form and is given by

ˆ ˆ ˆ ˆ ˆcov[ ] [( [ ])( [ ]) ]

ˆ ˆ ˆ ˆ = [{ } [{ })]({ } [{ }) ]

( [ ])( [ ]) =

cov[ ]u u =

Tp p p p p

Tp p p p

TTji

i j

T Ti j i ij j

i j

a E a E a a E a

E a E a a E a

u d E du d E dE

u d u I

λ λ

δλλ λ

= − −

− −

⎡ ⎤⎧ ⎫−⎧ ⎫− ⎪ ⎪⎢ ⎥⎨ ⎬⎨ ⎬

⎪ ⎪⎢ ⎥⎩ ⎭⎩ ⎭⎣ ⎦⎧ ⎫⎪ ⎪=⎨ ⎬⎪ ⎪⎩ ⎭

2{ }ii j

diag λλ

−⎧ ⎫⎪ ⎪=⎨ ⎬⎪ ⎪⎩ ⎭

(4)

Where E [. . .] is the expectation operator. Then

2ˆcov[ ] Tp p p pm V V−= Λ (5)

Thus bias of the estimator is defined as:

ˆ ˆ( ) [ ]p pbias m E m m≡ − (6)

Where ˆ[ ]pE m is the expectation of the random vector

variable ˆ pm and m which is the true but unknown model.

Denoting the null space of model space by 0V , we can express

the true model m as:

0 0 0ˆ ˆp p pm m m V a V a= + = + (7)

Then the expectation is given

by ˆ ˆ[ ] [ ] Tp p p p pE m V E a V V m= = using (7), and the bias is

given by

0 0ˆ[ ] Tp p pbias m V V m m V V m≡ − = − (8)

Thus, the mean squared error of the TSVD estimator ˆ pm is

defined by

{ }{ }2

0 0

2 2

1 1

ˆ ˆ ˆ ˆ( ) cov( ) ( ) ( )

=

= 1/

T

p p p p

T T Tp p p

p M

i ii i p

MSE m tr m bias m bias m

tr V V m VV m

bλ

−

= = +

⎡ ⎤= + ⎣ ⎦

Λ +

+∑ ∑

(9)

Where ,Ti i ib v m v= is the ith column vector of 0V .

We are now in a position to define our new TSVD estimator by mean squared error, which is the solution of the following minimization problem:

2 2

1 1

ˆmin : 1/p t

i ii i p

bλ= = +

+∑ ∑ (10)

ˆ pm is some proper estimator of x.

IV. Numerical Simulation

The follow is the inversion of a 4-layers magnetotelluric model with the parameters

( 1 2 3

4

100 , 10 , 100 ,30

m m mm

ρ ρ ρρ

= Ω = Ω = Ω= Ω

,

mhmhmh 1500,1300,700 321 === ).Gaussian noise

was added to the data. Data errors are assumed to be 1% on the impedance and about half a degree on the phase.

5 10 15 20 25 30 35 40 45 50

0

5

10

15

20

25

30

columns

row

s

G matrix plot

matrix element nomalized vaule

Figure1. The operator matrix G

100

105

1010

100

101

102

103

104

trade-off u

mis

fit

misfit-tradeoff relationship

misfit-tradeoff curl

Figure2. The misfit-tradeoff relationship

0 20 40 60 80 100 1200

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

parameter result

z de

pth

directness inversion result

recoverd model

true model value

Figure3.directness inversion result and the true model

0 20 40 60 80 100 120

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

parameter result

z de

pth

result for misfit at the p (deleted number)

tsvd with MSE result

true model value

Figure4. The tsvd with MSE result and the true model

Comparing with figure3 and figure4, we can see the figure4’s inversion result which is inversed by TSVD with the MSE as determination of the regularization level is better fitting than the directive inversion. The present technique relies more directly on the data themselves. The models are in general smooth versions of ethereal structure but, due to the sharp cut-off of eigenvalues, some oscillations of the solutions may be introduced as seen in the figure4 example presented here. Similarly to regularize solutions with smoothness constraints, there is a tendency for resistive layers buried at depth to be smeared out. The reason is shown on the figure2. The curve of the figure2 is shown that the forward operator matrix is singular.

V. CONCLUSIONS

An attempt has been made to use the MSE concept for defining the truncation level of the SVD for linear inverse problems, on the 1D non-linear magnetotelluric inverse problems. Truncated SVD is shown to be extremely effective in finding solutions that give the best fit to the data even when data are noisy. Moreover, the MSE concept when applied locally in the vicinity of a given solution gives a reasonable estimate of the best truncation level when the part of the model not spanned by the active model eigenvectors is used to estimate the bias term in the MSE. The incorporation of rigid bounds on the solution can remove some of the smooth character of the solution.

At the same time it is dangerous to apply constraints that are too tight, because they may act then as barriers and prevent the solution from reaching a global minimum or they may slow down the optimization process to unacceptable levels. The depth of penetration of a given data set can be estimated efficiently using the present technique by noting at what depth level the resistivity remains constant and close to that of the starting model. Different starting models may produce different estimates for the depth of penetration if care is not exercised in specifying the resistivity correctly.

References

[1]. Uchida T. 1993. Smooth 2-D inversion of magnetotelluric data based

on statistical Criterion ABIC. Journal of Geomagnetism and Geoelectricity

45, 841–858.

[2]. Li Y. and Oldenburg D.W. 1999. 3-D inversion of DC resistivity data

using an L-curve criterion. 69th SEG meeting, Houston, USA, Expanded

Abstracts, 251–254.

[3]. Li Y. and Oldenburg D.W. 2003. Fast inversion of large-scale

magnetic data using wavelet transforms and a logarithmic barrier method.

Geophysical Journal International 152, 251–265.

[4]. Farquharson C.G. and Oldenburg D.W. 2000. Automatic estimation of

the trade-off parameter in nonlinear inverse problems using the GCV and

L-curve criteria. 70th SEG meeting, Calgary, Canada, Expanded Abstracts,

265–268.

[5]. Haber E. and Oldenburg D.W. 2000. A GCV based method for

nonlinear ill-posed problems. Computational Geosciences 4, 41–63.

[6]. MenkeW. 1989. Geophysical Data Analysis: Discrete Inverse Theory.

Academic Press, Inc.

[7]. Peiliang Xu. 1994. Truncated SVD methods for discrete linear

ill-posed problems. Geophys. J. Int. (1998) 135, 505–514.

[8]. Laust B. Pedersen. Determination of the regularization level of

truncated singular-value decomposition inversion: The case of 1D inversion

of MT data. Geophysical Prospecting, 2004, 52, 261–270.

[9]. Thomas Kalscheuer and Laust B. Pedersen. A non-linear truncated

SVD variance and resolution analysis of two-dimensional magnetotelluric

models. Geophys. J. Int. (2007) 169, 435–447