A hybrid genetic-linear algorithm for 2D inversion of sets of vertical electrical sounding

A hybrid genetic-linear algorithm for 2D inversion of sets of vertical electrical sounding

Niraldo R. Ferreira^I; Milton José Porsani^II; Saulo Pomponet de Oliveira^III

^IEscola Politécnica - Universidade Federal da Bahia -Rua Prof. Aristides Novis, 02 - Federação - CEP: 40210-730 Salvador- BA - E-mail: [e-mail]

^IICentro de Pesquisa em Geofísica e Geologia - Instituto de Geociências - Universidade Federal da Bahia - Rua Caetano Moura, 123 sala 312- C - Campus Universitário de Ondina - Salvador - BA - CEP: 40170-115 - Telefax: (71) 203-8551 - E-mail: [e-mail]

^IIICentro de Pesquisa em Geofísica e Geologia - Instituto de Geociências - Universidade Federal da Bahia - Rua Caetano Moura, 123 sala 312- C - Campus Universitário de Ondina - Salvador - BA - CEP: 40170-115 - Telefax: (71) 203-8551 - E-mail: [e-mail]

ABSTRACT

The inversion of vertical electrical sounding (VES) is normally performed considering a stratified medium formed by homogeneous, isotropic and horizontal layers. The simplicity of this geophysical model makes the inversion simple and computationally fast, and together with the main characteristics of the electroresistivity method, it was greatly responsible to make VES one of the most popular geophysical method for groundwater exploration and engineering geophysics. However, even in a sedimentary basin where the geology is more conform, the assumption of horizontal and homogeneous layers is not necessarily valid, limiting the reliability of the inversion results.

In this paper we present a fast and robust 2D resistivity modeling and inversion algorithm for the interpretation of sets of VES. We consider three inversion algorithms: the Gauss-Newton method of linearized inversion (LI), the genetic algorithm (GA), and a hybrid approach (GA-LI) that uses LI to improve the best model at the end of each step of the GA. The medium parametrization consists of the partition of the domain into fixed homogeneous rectangular blocks such that their resistivities are the only free parameters. The apparent resistivity is evaluated by an iterative scheme that is derived from a finite-difference discretization of the potential differential equation. We enhance the convergence rate of the scheme by adopting an incomplete Cholesky preconditioner.

Numerical results using synthetic and real 2D apparent resistivity data formed by sets of VES for the Schlumberger configuration illustrate the performance of the hybrid GA-LI algorithm. The VES field data were acquired near Conceição do Coité, state of Bahia, Brazil. We compare the performance of the LI, GA and GA-LI algorithms.

Keywords: Incomplete Cholesky, 2D resistivity modeling, geophysical inversion, genetic algorithms, linearized inversion, hybrid optimization.

RESUMO

A inversão de uma sondagem elétrica vertical (SEV) normalmente assume que o meio é estratifcado e formado por camadas horizontais homogêneas e isotrópicas. A simplicidade deste modelo geofísico torna a inversão simples e com reduzido custo computacional. Esta simplicidade, junto às principais qualidades do método de eletroresistividade, foi responsável por tornar a SEV um dos métodos geofísicos mais populares nos trabalhos de exploração de águas subterrâneas e geofísica aplicada à engenharia. Porém, mesmo em bacias sedimentares, onde a geologia é mais conforme, a hipótese de camadas planas e homogêneas não é válida, o que limita a confiabilidade dos resultados da inversão.

Apresentamos neste artigo um algoritmo rápido e robusto de modelagem e inversão eletroresistiva para a interpretação de conjuntos de SEVs. Consideramos três algoritmos de inversão: o método de inversão linearizada de Gauss-Newton (LI), o algorítmo genético (GA), e uma abordagem híbrida (GA-LI) que usa a inversão linearizada para aprimorar o melhor modelo obtido ao final de cada geração do algoritmo genético. A parametrização do meio consiste na partição do dommínio em blocos retangulares e homogêneos, de modo que a resistividade de cada bloco é um parâmetro do modelo. A resistividade aparente é calculada com um método iterativo baseado numa aproximação por diferenças finitas da equação do potencial elétrico. Um precondicionamento do tipo Cholesky incompleto é utilizado para acelerar a convergência do método.

Avaliamos a performance do método híbrido por meio de experimentos numéricos com perfis de eletroresistividades reais e sintéticos, formados por conjuntos de SEVs obtidas com o arranjo Schlumberger. Os dados de campo foram coletados nas proximidades de Conceição do Coité, estado da Bahia, Brasil.

Palavras-chave: fatoração incompleta de Cholesky, modelagem bidimensional de resistividade, inversão geofísica, algoritmos genéticos, inversão linearizada, otimização híbrida.

INTRODUCTION

Inversion of resistivity sounding is a non-linear problem that estimates the spatial distribution of resistivities of the subsoil materials from apparent resistivity data measurements. Local and global optimization algorithms have been reported in geophysical data inversion by many authors (TARANTOLA; VALETTE, 1982; ROTHMAN, 1985; SEN; BHATTACHARYA; STOFFA, 1993; CHUNDURU et al., 1997). In case we begin the inversion using a starting model located near to a local or a global minimum, gradient methods can be very useful to find an optimal solution. Otherwise, global optimization algorithms such as simulated annealing or genetic algorithms can be used. The major drawbacks associated with local and global algorithms are the requirement for a priori information and the computational cost, respectively. Several different hybrid optimization approaches can be proposed to overcome these drawbacks (CHUNDURU et al., 1997; PORSANI et al., 2000).

To develop an efficient hybrid optimization scheme, it is important to choose efficient global and local algorithms. For geophysical inversion, successful attempts were made by several authors (CARY; CHAPMAN, 1988; PORSANI et al., 1993; LIU; HARTZELL; STEPHENSON, 1995). A very good explanation about the advantages and drawbacks of local, global and hybrid algorithms was presented by Chunduru and others (1997). Also to develop an efficient hybrid inversion algorithm for 2D resistivity inversion, a fast forward modeling algorithm is required. For the 2D inversion of field resistivity sounding data we have implemented a 2D finite-difference algorithm for computation of the forward modeling that uses an incomplete Cholesky factorization scheme (MEIJERINK; VAN DER VORST, 1977) coupled with the preconditioned conjugate gradient method (GREENBAUM, 1997).

Electrical resistivity inversion methods aim to determine the distribution of subsurface resistivity by measuring the distribution of electrical potential from a set of current electrodes at the earth surface. For a Schlumberger configuration of electrodes, the apparent resistivity satisfies the equation

where Df is the electrical potential difference between two electrodes located at M and N, and I is the current generated by two electrodes located at A and B. The axis x is set along the electrodes.

The one-dimensional method of Vertical Electrical Sounding (VES) for horizontally layered media is well known¹. The free parameters of this model are the resistivity r_i (1 < i < n) and the thickness h_i (1 < i < n) of each layer, and are represented by the vector m. The center of electrode configuration is fixed, and the spacing s = AB/2 is the only independent variable. One can evaluate the apparent resistivity r_a(m, s) in closed form (KOEFOED, 1979).

The two-dimensional model accounts for both lateral and vertical variations of resistivity. In this case, the apparent resistivity r_a also depends on the position x where the VES is performed. We partition the domain into N rectangular blocks. The components of the free parameter vector m are the resistivity of each block. Unlike the 1D model, the apparent resistivities r_a (m, x, s_i) are approximated by a numerical method. We employ a finite-difference method to evaluate the scalar electrical potential f, as described in the following section.

FINITE-DIFFERENCE MODELING

Assuming that the electric conductivity s of the medium varies only along the axis x and the depth z, the electrical potential generated by a pointwise source at (x_f, 0, 0) is a solution of the Poisson equation

where d(•) is the Dirac delta and Ñ is the gradient vector operator. A Fourier transform in the y direction yields

Equation (3) is discretized using an NxM non-uniform rectangular grid. We evaluate the finite-difference solution _i,j » _k(x_i, z_j) in its interior domain of validity according to Dey and Morrison (1979):

The boundary condition at the top layer is

We stretch the grid in geometric progression near the lateral and lower boundaries, imposing the following condition (DEY; MORRISON, 1979):

where K_0,1 are the modified Bessel functions (ABRAMOWITZ; STEGUN, 1970). We employed growth factors of 2.529 and 2.215 in the horizontal and vertical directions, respectively (MEDEIROS, 1987).

Let x = (₁₁, ..., _1N, ₂₁, ..., _MN)^T and b = (b₁₁, ...b_MN)^T. Equations (6)-(10) yield a linear system of the form Cx = b. The capacitance matrix C is symmetric, positive definite, and satisfies C_i,j = 0 if |i - j| ¹ 0,1, M.

The Cholesky factorization C = LL^T leads to a lower triangular matrix L such that L_i,j ¹ 0 if |i-j|< M in general. However, the observed values of |L_i,j| are relatively small if |i-j|¹ 0,1, M. For instance, Figure 1 displays the absolute values of the diagonals of L resulting from the model with M = 10, N = 20, and a medium composed of two homogeneous layers with the same thickness. The resistivities of the upper and lower layers are r₁ = 10Wm and r₂ = 500Wm, respectively.

We consider an incomplete Cholesky factorization C » HH^T where H is a lower triangular matrix satisfying H_i,j = 0 if i-j ¹ 0,1, M. Since H preserves the sparsity pattern of C, the matrix HH^T is a suitable preconditioner for iterative methods for solving Cx = b (MEIJERINK; VAN DER VORST, 1977).

Once

_k(x_i, z_j) is approximated, the electrical potential f(x_i, 0, z_j) is estimated by numerically integrating Equation (5) according to Dey and Morrison (1979). We employ the following frequency values: k = 0.001, 0.002, 0.004, 0.008, 0.015, 0.03, 0.06, 0.09, 0.12, 0.15m^-1. To evaluate the apparent resistivity, we place the potential electrodes at the surface nodes of the grid, so that the difference of electrical potential needed in (1) can be computed from f(x_i, 0,0)(1 < i < M) and a prescribed spacing s.

A Preconditioned Conjugated Gradient Algorithm

The incomplete Cholesky factorization approximates the solution x of Cx = b by the solution x₀ of H^T x₀, where Hy = b. To further improve this estimate solution we employ the preconditioned conjugated gradient (PCG) method (GREENBAUM, 1997). In the following algorithm, n_iter is the maximum number of iterations, || r_l || = , and tol is the error tolerance.

Steps of the preconditioned conjugated gradient algorithm

• calculate r₀ = b - Cx₀;

• solve H^Tz₀=y for z₀, where Hy = r₀, and set p₀= z_0;

• for l = 0, 1, ..., n_iter (|| r_l || > tol)

- calculate x_l+1 = x_l + a_lp_l, where a_l = ;

- calculate r_l+1 = r_l - a_lC_Pl;

- solve H^Tz_l+1 = y for z_l+1, where Hy = r _l+1;

- calculate p_l+1 = z_l+1 + b_lpl, where b_l = ;

A similar algorithm has been used in 3D electroresistivity modeling (ZHANG; MACKIE; MADDEN, 1995). Figure 2 compares the CPU processing time of the PCG method above and a direct method based on the Cholesky factorization (CF). The computations were performed in a RISC 6000 IBM and the model problem is the same as in Figure 1, with N = 100 and M = 5, 10,..., 40. We set a tolerance, tol = 10^-10, for the PCG method. In this example the PCG algorithm becomes a better alternative when N×M is greater than 4000, which is a suitable resolution for two-dimensional inversion of real data.

LINEARIZED INVERSION

Let 1 < p < 2. The L_p norm of an M-dimensional vector v = (v₁,...v_M) ^T is given by ||v||_p = .

Let us introduce an iterative scheme to minimize the objective function proposed by Scales and Gersztenkorn (1986):

where r_a(x_i, s_i) and r_a(m, x_i, s_i) are the observed and theoretical apparent resistivities, respectively. Note that E(m) is the L_p norm of the error of the theoretical apparent resistivities to the power p. We linearize r_a(m, x_i, s_i) by Taylor's series about an estimate free parameter vector m_k:

Let d_k,i = r_a(x_i, s_i) - r_a(m_k, x_i, s_i),r_k,i = |r_a(x_i, s_i) - _a(m, x_i, s_i)|^p-2(1 < i < M), and .

Substituting (12) into (11), we find a quadratic function of m, whose minimum satisfies

Where Dm = (m_k+1 – m_k). By using a regularization factor l (MENKE, 1989) we compute the new solution m_k+1 as

In particular, the method with p = 2 and l = 0 corresponds to the plain least squares method. The row i of the sensitivity matrix G_k is weighted by the i-th diagonal component of the matrix R_k, which is a function of the deviation between the observed resistivity values, and the ones computed from current model m_k (PORSANI; NIWAS; FERREIRA, 2001).

To increase the robustness of the algorithm, we apply a logarithmic scaling to the free parameters and to the field data (RIJO et al., 1977). Moreover, given a tolerance parameter e, we set r_k,i = e^p-2 if |r_a(x_i, s_i) - _a(m, x_i, s_i)|< e.

We employ a harmonic measure of fitness (PORSANI et al., 2000)

The ratio F varies within [-1,1], and approaches 1 as r_a(m_k, x_i, s_i) approaches r_a(x_i, s_i)(1 < i < M). The components of the sensitivity matrix are approximated by forward differences (MCGILLIVRAY; OLDENBURG, 1990). We employ a conjugated gradient method to evaluate m_k+1 from (14). Let A = R_kG_k + lI. We have that:

which motivates modifying the conjugated gradient algorithm to avoid the computation of

R_kG_k:

Steps of the conjugated gradient algorithm for L_p inversion

• s₀ = Dd_k - G_kx₀;

• r₀ = R_ks₀ -»x₀;

• p₀ = r₀ and q₀G_kp₀;

• for l = 0, 1, ..., n_iter (|| r_l || > tol).

- x_l+1 = x_l + a_lp_l where a_l = ;

- s_l+1 = s_l - a_lq_l;

- r_l+1

R_ks_l+1 - lx_l+1;

- p_l+1 = r_l+1 + b_lp_l, where b_l;

- q_k = G_kp_k;

When l = 0, the algorithm designed by Gersztenkorn, Bednard e Lines (1986) for 1D inversion of the acoustic wave equation is recovered.

NUMERICAL EXAMPLES

Inversion of synthetic data

We consider the model of a buried dike outlined in Figure 3. The vertical electrical soundings are performed throughout 21 stations with a set of 19 s-values. Noise is introduced when AB/2 = 17.5m, 47.5m, 87.5m and 107.5m.

The horizontal grid employs 252 nodes. Five nodes are distributed in geometric progression on both ends, while the increment between interior nodes is 5m. The vertical grid employs 26 nodes with non-uniform spacing.

In the experiment it is assumed that the location and size of the blocks are known. The initial solution is m₀=r(1,1,1,1)^T , r =500Wm. Figure 4 compares the performance of inversions in the norms L₁ and L₂ without regularization (l = 0).

Figure 5 shows three VES corresponding to stations 1, 7 and 11, inverted using L₁ and L₂ norms. We compare results of apparent resistivities resulting from two extreme scenarios: when the dike width is zero (r₀) and when the width is infinite (r_¥). These scenarios yield horizontally layered media, and can be considered as lower and upper bounds of the influence of the dike; that is, r₀ does not take the dike into account, while r_¥ is driven by the resistivity of the dike and the upper layer (FERREIRA, 1999).

Both inversions delivered exact block resistivities when outlier noise is removed (note that F(m₁₀) = 1). Otherwise, the resistivities were accurately computed in the L₁ norm (Table 1).

Inversion of field data

Our next experiment concerns field data acquired near Conceição do Coité, Brazil (PINHEIRO NETO, 2000). This area has an aquifer whose average yield is 1.78 m³/h with up to 7278mg/l of total dissolved solids.

Twenty VES were acquired, and they are shown in Figure 6(a). In order to fit data to the finite-difference grid, we interpolated the VES curves to evaluate the apparent resistivity with an initial spacing AB/2 = 7.5m and uniform increments of 5m.

The media parametrization is based on a partition into 27×5 blocks. We estimated the thickness of each layer by the average thickness calculated at each station by using 1D VES inversion. The initial model had r = 40Wm in the first four layers and r = 300Wm in the bottom layer. The horizontal grid was similar to the horizontal grid used in the synthetic model. We employed 302×9 nodes.

We performed 10 iterations of linearized inversion in the norms L₁ and L₂ with the same regularization factors l = 0.001 and l = 0.1 in the L₁ and L₂ norms, respectively. The percent relative errors with respect to the interpolated data were similar and under 45% as illustrated in Figure 7.

The region of low resistivity near station S5 of the computed models (Figure 6) is consistent with the presence of a water well near this station. The low resistivity between stations S14 and S16 is consistent with the evidence of salinization between stations S12 and S17.

GENETIC AND HYBRID ALGORITHMS

Genetic algorithms (GA) employ the concepts of survival of the fittest, crossover, and mutation to generate a set of free parameter vectors that progressively approach field data. These methods fit into the class of global, probabilistic optimization methods. Genetic algorithms are based on the principle of natural selection and genetics. Detailed descriptions of GA are given by Holland (1975) and Goldberg (1989), and theory and examples of geophysical applications can be found in Sen and Stoffa (1995). Basically, in the GA the model free parameters are coded in binary form. The algorithm starts with an ensemble of random models, and a new ensemble is generated similarly to the biological mechanism of reproduction that exists in nature. The models are chosen for reproduction with a probability proportional to their fitness value, and pairs of models are selected at random and exchange part of their binary chain. The crossover points are selected at random and all the bits to the right side are interchanged with a crossover probability, generating new models. To assure genetic variability in the population, a mutation process is adopted by changing at random a bit inside the binary chain based on a fixed probability. The new set of models are accepted with an update probability by comparing them with the models in the previous generation. The process of selection, crossover and mutation is applied until the fitness values converge, i.e., until the mean fitness approaches the highest fitness value in the population.

We start by randomly selecting a set (or population) of free parameter vectors m_g,j, 1 < j < P, and g = 0. We refer to each m_g,j as a model. In the second step, we evaluate the fitness F(m_g,j) of each model according to equation (15). Then, we perform the following genetic operations.

• Selection: we select a limited number of models in pairs for reproduction. They are selected by a non-uniform probability function given by

where T = T₀g^g is associated with the temperature in the simulated annealing method. The temperature is used to de-emphasize the differences in the fitness values of the models in the initial generations and to exaggerate their differences at later generations (STOFFA; SEN, 1991).

• Crossover: each pair exchanges free parameter data with a fixed probability P_x; two new models are generated. Each component (1 < i < M) of a model m_g,,j is restricted to a prescribed resolution; that is,

• Mutation: a random change with fixed probability P_m may take place in each member of all pairs. Mutation helps to preserve the population diversity and leads to new search regions.

• Update: each new model is compared with a randomly chosen current model m_g,j. If F()>F(m_g,j), then m_g,j is replaced by according to a fixed probability P_u.

These steps create a new generation m_1,j(1 < j < P). We can go back to the second step, and repeat the process until the g-th generation has a model m_g,j such that F(m_g,J) is sufficiently close to one, or until g reaches the maximum number of generations NG.

We combine the genetic with the linearized inversion methods, generating a hybrid (GA-LI) algorithm (PORSANI et al., 1993). As shown in Figure 8, the hybrid algorithm starts with an initial ensemble of randomly selected 2D resistivity models. Synthetic 2D VES corresponding to each model are computed and compared with the data to generate the fitness function for each model. The fitness functions from the current generation are compared to those from the previous generation and kept subject to an update probability. We next find the best model in each generation and apply the LI method. At the end of each GA iteration, we set

apply the iterative method (14) to m₀, and if the resulting model m_k satisfies F(m_k) > F(m_g,J), it is accepted into the population replacing m₀. The algorithm then proceeds as in AG. The genetic operators of selection, crossover and mutation are applied to the models to provide the next generation of 2D resistivity models for evaluation.

INVERSION OF FIELD DATA

This section illustrates the improvement of the hybrid approach over genetic algorithms. We consider the same settings as in the experiment with linearized algorithms.

The probabilities associated with the genetic algorithm are set similarly to earlier cases (CHUNDURU et al., 1995; SEN; STOFFA, 1995): P_c = 0.6, P_m = 0.01 and P_u = 0.95. The resolution of the free parameters is shown in Table 2. Moreover, T₀ = 5 and g = 0.98.

Both algorithms employ 200 generations with a fixed population of 250 models. The hybrid algorithm performs ten iterations of the linearized inversion algorithm, under the L₂ norm. Notice that the hybrid approach led to a considerable decay of the relative error (Figure 9). The best models of genetic and hybrid algorithms are shown in Figure 10.

DISCUSSIONS AND CONCLUSIONS

This article extends previous work in 1D resistivity inversion for 2-D inversion of sets of vertical electrical sounding. We incorporate the linearized inversion approach into a genetic algorithm. The best model, found at the end of each generation of the GA, was improved by using the LI method. By doing so, we found that a combined GA-LI approach performs better than a pure GA, and better than a pure LI run. The hybrid algorithm was tested to simultaneously invert families of synthetic and measured VES data using a 2D resistivity model. The GA-LI algorithm accelerates the convergence to the global optimum.

Our experience using the hybrid GA-LI algorithm indicates that employing linearized inversion on initial steps of the GA-LI algorithm may overemphasize a local search, specially if the best models are near local optima. On the other hand, a typical GA performance curve grows faster in the first generations, which suggests that this method is efficient on identifying the neighborhood of the global optimum. The growth is slower in the following steps and tends to saturation. Therefore LI refinement is more appropriate in later steps. A key question is when linearized inversion should take place. Another question is whether hybrid methods can be improved with more complex local search methods (for instance, multiple re-weighted least-square methods). These questions contribute to a deeper understanding of 2D inversion of geophysical problems.

Acknowledgements

The authors are thankful to Dr. Vicente Pinheiro Neto for providing the set of VES data used to test the algorithm. The third author is supported by the PRODOC fellowship granted by CAPES, Brazil.

Recebido em 23 out. 2003 / Aceito em 6 may, 2004

Received oct. 23, 2003 / Accepted may 6, 2004

NOTAS SOBRE OS AUTORES

Niraldo Roberto Ferreira é graduado em Engenharia Elétrica na Universidade Federal de Pernambuco em 1977. Mestrado em Geofísica pela Universidade Federal da Bahia, 1994. Doutorado em Geofísica pela Universidade Federal da Bahia, 1999. Atualmente professor da Escola Politécnica da UFBA.

Milton José Porsani é B. C. em Geologia pela USP, 1976. Licenciado em Geologia pela Faculdade de Educação da USP, 1977. Mestre em Geofísica pela UFPA, 1981. Doutor em Geofísica pela UFBA, 1986. Pós-doutorado em Geofísica, Institute for Geophysics at University of Texas at Austin, EUA, setembro/92 a outubro/93. De 1986 até o presente é Pesquisador do CPGG/UFBA onde coordena o Programa de Exploração de Petróleo. Em 1990 foi contratado pela UFBA mediante concurso público para professor do Departamento de Geologia e Geofísica Aplicada do IGEO. Desde 2000 é professor Titular na matéria Exploração de Petróleo. Pesquisador do CNPq, nível I-B. Tem atuado no desenvolvimento de métodos e algoritmos de filtragem e processamento de dados sísmicos e na inversão de dados sísmicos e elétricos.

Saulo Pomponet de Oliveira é graduado em Matemática Aplicada e Computacional pela Universidade Estadual de Campinas, Campinas, SP, em 1997. Mestrado em Matemática Aplicada, Universidade Estatual de Campinas, Campinas, SP, em 1998. Doutorado em Matemática Aplicada, University of Colorado, Denver, Estados Unidos, em 2003. Atualmente bolsista recém-doutor (PRODOC-CAPES) do programa de pós-graduação em geofísica da Universidade Federal da Bahia.

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1

Cf. Porsani and others (2001) and the references therein

Publication Dates

• Publication in this collection
  24 Jan 2007
• Date of issue
  Dec 2003

History

• Accepted
  06 May 2004
• Received
  23 Oct 2003