Piecewise smooth models for electromagnetic inverse ... - CiteSeerX

Piecewise smooth models for electromagnetic inverse problems Hugo Hidalgo,

Jos´e Luis Marroqu´in, and Enrique G´omez-Trevi˜ no

Abstract— This paper presents a new method for constructing onedimensional electrical conductivity models of the Earth from surface electromagnetic measurements. The construction of these models is a nonlinear inverse problem that can be approached by linearization techniques combined with iterative methods and Tikhonov’s regularization. The standard application of these techniques usually leads to smooth models that represent a continuous variation of conductivity with depth. In this work we describe how these methods can be modified to incorporate what is known in Computer Vision as the line process (LP) decoupling technique, which has the ability to include discontinuities in the models. This results in piecewise smooth models which are often more adequate for representing stratified media. We have implemented a relaxation technique to construct these types of models and present numerical experiments and an application to field data. These examples illustrate the performance of the combined LP and Tikhonov’s regularization method. Keywords— Magnetotelluric, electromagnetic soundings, regularization, inversion.

I. Introduction

E

LECTROMAGNETIC sounding methods have been applied for a long time to investigate the interior of the Earth from depths of a few meters to hundreds of kilometers. The magnetotelluric (MT) sounding method is one of the most popular today, and it is aimed at estimating the electrical conductivity distribution at depth associated with subsurface geological structures. It has been successfully applied in shallow exploration problems and to infer the Earth’s deep electrical conductivity distribution, temperature regimes, and geological structure. The method uses measurements of natural electric and magnetic fields recorded at the surface of the Earth. If it is assumed that the underground structure is one-dimensional (1D), the inverse problem consist of recovering the vertical conductivity distribution from the ratio of surface electric and magnetic field measurements taken at different frequencies. One of the main characteristics of the MT inverse problem is its non-linearity. Several methods have been developed to cope with this problem [1]: linearization, asymptotic behavior, and exact methods. Constable et al. [2] consider smooth models using a procedure called Occam’s inversion. The method consists of regularization applied to the linearized equations of the 1D MT problem. We propose a modified algorithm to obtain piecewise smooth models, This work was supported by Consejo Nacional de Ciencia y Tecnologia, Mexico. Hugo Hidalgo and Enrique G´ omez-Trevi˜ no are with CICESE, Ciencias de la Tierra Km. 107 Carr. Tijuana-Eda. Ensenada, Baja California 22860 M´ exico. (E-mail: [email protected] [email protected]). Jos´ e Luis Marroqu´in is with CIMAT, Callejon Xalisco S.N. Valenciana, Guanajuato, Gto., 36240 M´ exico. (E-mail: [email protected]).

as opposed to continuous smooth models obtained by the direct application of Occam’s inversion. A fundamental characteristic of Occam’s models is that the resulting subsurface conductivity distribution is a continuous function. This is very convenient from a mathematical point of view [3] [4]. However, other authors [5] [6] [7] [8] [9] represent the Earth as composed of several layers over a homogeneous half-space, i.e. a piecewise continuous model. This is a more realistic model, particularly for the case of stratified media such as is common in sedimentary basins. Our main objective here is to develop algorithms that incorporate discontinuities in what otherwise is a smooth model. In order to develop the discontinuous models, we apply a method known in Computer Vision as Line Process (LP) [10] [11] [12]. LP models assume piecewise smoothness, which is obtained by switching off the smoothness constraint at points where the magnitude of the signal derivative exceeds a certain threshold. As in Occam, we work with a model that approximates a continuous function and define LP in a lattice. For every interface between two model segments there exists a binary line variable li . When li = 1 (inactive) the model smoothness is not affected, but if li = 0 (active), a discontinuity has been detected and it is inserted between segments mi−1 and mi . The algorithms are based on the regularizing Tikhonov [13] functional, by adding the binary line variables, and also restricting the number of discontinuities in the model. Results are presented for the MT problem, but can also be applied to other types of electromagnetic soundings. II. The forward MT problem. In the MT problem, natural electromagnetic fields are used to investigate the electrical conductivity structure of the Earth. The simplest model that can adequately describe the subsurface does not vary laterally. The resistivity structure of the model consists of horizontal layers, each with different resistivity and thickness, overlaying a uniform half-space. This is called a 1D geoelectrical model of the Earth. Any cross-section of such an Earth in any location and in any direction will appear identical. This model consists of M layers, where the ith(i = 1, ...M − 1) layer is characterized by two parameters: the resistivity ρ i and thickness τi . The M − 1 layers of the model overlay a homogeneous geological half-space with resistivity ρ M . The resistivity is given in Ohm-m and the thickness in meters. The impedance for√the homogeneous half-space of resistivity ρM is ZM = iωµ0 ρM , where ω = 2π/T is angular frequency, T is period of the electromagnetic field, µ 0 √ is the magnetic permeability of free space in H/m, and i = −1. Beginning at the top of the homogeneous half-space, which

is overlain by a stack of M − 1 laterally homogeneous layers, one can progressively evaluate Zi , at the top of layer i, using the recursive relationship [14] [15] Zi = γ i ρi

γi ρi (1 − ui ) + (1 + ui )Zi+1 , γi ρi (1 + ui ) + (1 − ui )Zi+1

iteration step. Assuming a small step ∆ = mk+1 − mk and neglecting , the algorithm for mk given an initial model m0 is given as:

(1)

where i = M − 1, M − 2, ...M . The variable γi is the wave number given by p γi = iωµ0 /ρi

and the propagation operator ui is given by

ˆk , mk+1 = [λSt S + (WJk )t WJk ]−1 (WJk )t Wd

(5)

where ˆ k = d − F[mk ] + Jk mk . d

The measurements of electric and magnetic fields at the surface are commonly reported in terms of ρa and the phase 1) of Z1 , φ = arctan( Im(Z Re(Z1 ) ), at a given frequency ω.

This algorithm, named Occam’s inversion by Constable et al, generates smooth models, as shown in Fig. 1. The results were obtained after eight iterations using the synthetic data presented in Fig. 2. We assumed a 1 percent standard deviation for both amplitude and phase data. The final model fits the data to a RM S = q PN (di −Fi (m))2 1 = 1. That is, within 1 percent, and j=1 N σi2 with λ = 236. The discretization of the depth axis is such that none of the discrete values coincide with the boundaries in the true model. We use this same discretization in the examples that follow.

III. The smoothest model.

IV. Piecewise smooth models.

ui = e−2γi τi . The impedance at the surface is converted to apparent resistivity, defined by ρa = |Z1 |2 /(ωµ0 ).

(2)

Following the notation of Constable et al [2], the electromagnetic inverse problem can be approached by minimizing the following functional: U = kWd − WF[m]k2 + λkSmk2 ,

(3)

where m ∈ M, the space of models, F[m] = {f1 (m), ..., fN (m)} is the non-linear forward mapping {F[m] : M → D}, D is the space of realizations of the data. In the MT problem, due to the large variation in parameters and data, it is better to work with the logarithm of components, then {m}i = log ρi , is the logarithm of resistivity of the ith layer, f2j−1 [m] = log ρa (ωj ) is the logarithm of apparent resistivity, f2j [m] = φ(ωj ) is the phase of Z1 at frequency ωj . N is the total number of observations. d represents the measurements made at a number of frequencies. Here, as in [2] we assume a multilayer model in which the thickness of every layer is known and constant. W is a diagonal matrix that incorporates the variances in the measurements, assuming Gaussian errors, and with no cross-correlation among them, W = diag{ σ11 , σ12 , · · · , σ1N }, where σi represents the standard deviation of the ith measurement. k · k denotes the Euclidean norm. S is the firstorder finite-difference operator such that kSmk 2 is the disR dm crete equivalent of ( dz )2 dz, in matrix form:   0 0   −1 1 . (4) S=   −1 1 0 −1 1 Constable et al [2] linearize equation (3) by considering that F[m] can be approximated in a small vicinity of m by F[mk + ∆] = F[mk ] + Jk ∆ +  where Jk is the Jacobian ∂fi evaluated at vector mk , at the kth matrix {J}ij = ∂m j

A. Global LP algorithm LP is incorporated in equation (3) by reinstating the formulation with the inclusion of binary variables l i , in between each term of the roughness penalizing term, and adding also a penalizer for this new li as follows: t

ULP = kWd − WF[m]k2 + λ(Sm) D(Sm) + θ(M − trD), (6) where D = diag[0, l2 , l3 , · · · , lM ], M is the number of layers, and θ is the parameter that permits to penalize the activation of lines. Minimization with respect to m, i.e. for D fixed, produces the modified Occam algorithm: Ak mk+1 = bk ,

(7)

where Ak = [λSt DS + (WJk )t WJk ], and

ˆk. bk = (WJk )t Wd

The algorithm start with k = 0, m = 0, and obtains new models by solving equation (7). Minimization with respect to li is carried out after the accomplishment of some iterations of equation (7), and it is realized by lowering the threshold (θ) term as follows. Consider the terms that incorporate li in functional ULP : (mi − mi−1 )2 li + θ(1 − li ). When (mi − mi−1 )2 < θ then li = 1, else li = 0. The threshold is decreased until the first discontinuity is detected, then it remains constant, while another iteration of (7) is performed, then θ is lowered again, and another iteration of (7) is performed, the cycle continues until RMS is

less than or equal to its desired value. It can be observed that the regularization term is now   l2 −l2 0  −l2 (l2 + l3 ) −l3      t −l3 (l3 + l4 ) −l4 S DS =  .   . .   . −lM

0

lM

Note that, at the beginning, when all li = 1, the regularization term becomes the Occam’s algorithm roughness penalizing term. One advantage of the LP method is that it can include penalizers for other conditions, for example, not allowing that two consecutive lines be active simultaneously. This would represent a very discontinuous model, not so frequent in real situations. In the case of two dimensions, more general conditions can be included [11]. The main problem with a global decoupled algorithm is that the insertion of discontinuities deteriorates the condition of the matrix A. In the limit, as more and more layers become discontinuous, D approaches zero, and the regularizer vanishes gradually, leading to an ill-posed problem. Another inconvenient is the great amount of space needed to solve the system of equations, particularly for a two-dimensional case. The performance of this method is illustrated in Fig. 3, using the same data as before. Again, the final model fits the data to a RMS = 1. The corresponding comparison between the response of the model and the data is almost identical to that shown in Fig. 2 for the case of the Occam model. V. Local LP algorithms. A. Non-linear algorithm. Local approximation modeling can be easily described by writing equation (6) in a element wise form:

ULP

=

N X (di − fi (m))2

σi2

i=1

+

M X (1 − lj ). θ

+λ

M X (mj − mj−1 )2 lj

k−1 mkr (lr + lr+1 ) − mkr−1 lr − mr+1 lr+1 = g k (mkr ),

where g k (mkr ) =

−p

= + −

X (di − fi (m)) ∂fi σi2 ∂mr λmr (lr + lr+1 ) λ(mr−1 lr + mr+1 lr+1 ).

i=2

(2 − li − li−1 ),

B. Linearized local algorithm Considering the linear approximation of the functional used in the development of the Occam algorithm, a linear version of the local algorithm can be developed. Rewriting equation (5) in a component-wise form and setting the partial derivative with respect to mr equal to zero, the following Gauss-Seidel type algorithm is obtained: mkr = where

k mkr−1 lr + mk−1 r+1 lr+1 + h (m) , k P ∂f (lr + lr+1 + λ1 i ( ∂mir )2 /σi2 )

hk (m) =

−

A local relaxation algorithm can be obtained by setting the last equation equal to zero and solving for mr , in terms of the current values of mk . The Gauss-Seidel type algorithm that results consists in finding mk as follows. Start with k = 0, m = 0, and solve for every element mr (r = 1, ...M ) in

M X

with p some suitable constant to penalize the simultaneous activation of li and li−1 . A model with RM S = 1.0 produced by this algorithm is shown in Fig. 4, using λ = 250.

In order to minimize with respect to the rth element of m we obtain partial derivatives: ∂ULP ∂mr

1 X (di − fi (mk )) ∂fi k ) . ( λ σi2 ∂mr

Due to the nonlinear character of g k (mkr ) we have to apply a zero finding technique to solve the equation for every element of the model, which is a slow process. The minimization with respect to l is also realized after some iterations of the algorithm with a high threshold, then, when the model variation kmk − mk−1 k is small, the threshold is lowered until a line becomes active. When this occurs, θ is kept constant and several iterations are carried out until the model variation is small again, proceeding to lower the threshold again. The algorithm stops when the RMS misfit is less than or equal to its desired value. We have noted that this algorithm does not have the problem of instability when a discontinuity is detected. The problem may happen when two contiguous discontinuities are present, but this is discouraged by inserting a proper cost function to ULP [11]:

j=2

j=2

(8)

(9)

1 X (dˆi − s1 − s2 ) ∂fi , λ i σi2 ∂mr

s1 = and s2 =

r−1 X ∂fi k m ), ( ∂mj j j=1

M X ∂fi k−1 m . ∂mj j j=r+1

In this algorithm, every step is faster to compute, because hk (m) is a linear combination of the other elements of the model, computed on the previous and present step. Optimization over lr is realized as in the previous algorithm.

Fig. 5 shows the model obtained by this algorithm with λ = 250. The main advantages of this algorithm over the global version is that it is more efficient in the use of memory, and it is also more stable because there is an implicit regularization in the local relaxation technique. Furthermore, it is a simple matter to include local restrictions in order to allow the accommodation of a-priori information. In Occam, parameter λ is obtained by using crossvalidation. Here we obtain an empirical value by running several tests, observing that lower values increase convergence, but with some inaccuracies in the location of discontinuities. On the other hand, large values of λ increase the accuracy of the model but also slow down the convergence rate. As noticed, this algorithm runs a lot faster than its non-linear version, based on equation (8). The non-linear algorithm requires the solution of equation (8) for every element in the model. This means that f (m) and its partial derivatives must be evaluated several times in the solution process. On the other hand, in the case of the linear algorithm, the functional f (m) and its derivatives need to be evaluated only once to obtain a new model. C. Further improvements An over-relaxed local linear algorithm is obtained by the inclusion of a damping parameter α to equation (9):

mk+1 =α r

k hk (m) + mk+1 r−1 lr + mr+1 lr+1 + (1 − α)mkr . (10) P ∂f lr + lr+1 + λ1 i ( ∂mir )2 /σi2

The best value of α is related to the eigenvalues of the product A0 A1 · · · Ak [16], which are expensive to compute, but after several test runs, an empirical value of 1.4 was obtained. In Fig. 6, a comparison is shown of the local linear algorithm with λ = 250, α = 1.0 and the same algorithm but with α = 1.4. An increase in the speed of convergence can be observed in this figure, and also a considerable decrease in RM S error after the inclusion of every discontinuity, as marked with special symbols. On the other hand, it can be observed that the smoothing parameter λ affects all elements of the model with the same weight, regardless of depth. It is possible by varying λ with depth to increase or decrease the amount of smoothing at any desired section of the model. This would allow for the generation of different but equally valid models from the same data set. For instance, we may choose λ to decay logarithmically with depth. Fig. 7 shows the models developed by the linear algorithm with constant λ = 250, and with decreasing λ from 10, 000 to 10, α = 1.0 in both cases. It is observed an improvement in the accuracy of the model at larger depths, but at the expense of loosing precision at layers near the surface. Fig. 8 shows the resulting model of the local linear algorithm with constant λ = 200., and λ decaying from 500 to 10, α = 1.4 using the COPROD data set [2], and also the resulting model using directly the Occam technique. The

models are compared to that obtained by Jones and Hutton [17], who obtained their model by trial and error. The four models fit the data to the same level RM S = 1. It can be observed that the models developed by the local linear algorithms cannot be retrieved directly from Occam’s model, as could be suspected from the form that LP models are generated. Notice that the models differ particularly at shallow depths. This is because the data set does not define sufficiently the short period asymptote, and because the data available at the shorter periods have relatively large errors. Fig. 9(a-b) shows the degree of fitness to the data for the Occam and the local algorithms. VI. Conclusions Relaxation techniques were applied to the solution of electromagnetic inverse problems by allowing the generation of piecewise smooth models. The algorithms were tested with data generated with a known model. The models generated are close to the original, and the discontinuities detected are also very close to those of the test model. The algorithms were also tested with real data, developing layered Earth models equivalent to those obtained by other methods. The advantage of the present method is that it does not require to specify a priori the number of layers or the location of the discontinuities. This is done automatically using a combination of the Tikhonov regularization method and the line process decoupling technique. The recommended procedure is to employ local linear algorithms, either with constant or decaying λ, and using α = 1.4. Acknowledgments The authors would like to thank S. C. Constable for providing the computer code for Occam algorithm. References [1] Whittal, K. P. & Oldenburg, D. W., Inversion of magnetotelluric data for a one-dimensional conductivity. Geophysics Mono. Ser., Society of Exploration Geophysicists, 1992. [2] S. C. Constable, R. L. Parker, C. G. Constable ”Occam’s inversion: a practical algorithm for generating smooth models from electromagnetic sounding data,” Geophysics, vol. 52 no. 3, pp. 289-300, March 1987. [3] E. G´ omez-Trevi˜ no, ”Nonlinear integral equations for electromagnetic inverse problems,” Geophysics, vol. 52 no. 9 pp. 1297-1302, Sept. 1987. [4] H. Hidalgo, E. G´ omez-Trevi˜ no, ”Application of constructive learning algorithms to the inverse problem.” IEEE Tran. Geosci. Remote Sensing, vol. 34 no. 4, pp. 874-885, July 1996. [5] J.R. Inman ”Resistivity inversion with ridge regression,” Geophysics, vol. 40 no. 5 pp. 798-817, May 1975. [6] F. Esparza, E. G´ omez-Trevi˜ no, ”One-dimensional inversion of resistivity and induced polarization data for the least number of layers,” Geophysics, (in press), 1997. [7] F. J. Esparza, E. G´ omez-Trevi˜ no, ”Inversion of Magnetotelluric soundings using a new integral form of the induction equation,” Geophys. J. Int., vol. 127 no. 2 pp. 452-460, Nov. 1996. [8] S. E. Dosso, D. W. Oldenburg, ”Linear and non-linear appraisal using extremal models of bounded variation” Geophys. J. Int., vol. 99 no. 3 pp. 483-495, Nov. 1989. [9] D. W. Oldenburg, R. G. Ellis, ”Efficient inversion of magnetotelluric data in two dimensions” Physics of the Earth and Planetary Interiors, vol 81, nos. 1-4 pp. 177-200, Dec. 1993. [10] S.Z. Li, ”On discontinuity-adaptive smoothness priors in computer vision,” IEEE Tran. Pattern Anal. Mach. Intell., vol. 17 no. 6 pp. 576-586, June 1995.

[11] J. L. Marroqu´in, Probabilistic Solution of inverse problems, Ph.D. thesis, MIT AI Lab, 1985. [12] J. Marroquin, S. Mitter, T. Poggio, ”Probabilistic solution of illposed problems in computational vision,” Journal of the American Statistical Association, vol. 82, no. 397, March 1987. [13] A. Tikhonov, Vasily Y. Arsenin, Solutions of Ill-Posed Problems. Washington, D.C.: V.H. Winston & Sons, 1977. [14] M. Dobrin, Introduction to Geophysical Prospecting. New York: Mc Graw-Hill, 1976. [15] H. P. Patra and K. Mallick, Geosounding Principles, 2 TimeVarying Geoelectric Soundings. Amsterdam: Elsevier, 1980. [16] W. Ames, Numerical Methods for Partial Differential Equations San Diego: Academic press, 1977. [17] A.G. Jones, R. Hutton, ”A multi-station magnetotelluric study in southern Scotland I. Fieldwork, data analysis and results.” Geophys. J. Roy. Astr. Soc., vol. 56, pp. 329-349, 1979.

Hugo Hidalgo received the B.Sc. in electronics engineering from Instituto Tecnol´ ogico de Chihuahua, M´ exico, in 1980 and the M.Sc. in electronics and communications from Centro de Investigaci´ on Cient´fica y Educaci´ on Superior de Ensenada, (CICESE), in 1985. Currently, he is a computer science Doctoral candidate at Centro de Investigaci´ on en Matem´ aticas (CIMAT) in Guanajuato, M´ exico. He has been a faculty member of CICESE on Computer Science subjects and in the University of Baja California. His interests are in inverse problems, computer vision, and neural networks.

Enrique G´ omez-Trevi˜ no received the B.Sc. in physics from the Univ. A. de Nuevo Le´ on, M´ exico, in 1974 and the M.Sc. (1977) and Ph.D. (1981) both in geophysics from the University of Toronto. He did postdoctoral research at the University of Toronto during 1981-82, was a member of the faculty of the Univ. A. de Nuevo Le´ on in 1982-83, and since 1984 has been a member of the faculty at CICESE. His interests are in modeling and inversion of potential and electromagnetic data. He is a member of SEG, EAEG, UGM, and AGU.

Jos´ e Luis Marroqu´in received the B. S. degree in chemical engineering in 1968 from the National University of Mexico, and the M.Sc. and Ph.D. degrees in systems science from the Massachusetts Institute of Technology, Cambridge, MA. He has worked for PEMEX, a mexican petroleum company, in geophysical data processing. Currently, he is head of the Computer Science Department at the Center for Research in Mathematics, Guanajuato, M´ exico, and is conducting research related to automatic learning and computational vision. Dr. Marroqu´in is a Fellow of the National Research System of the Mexican Goverment.