Journal of the Earth and Space Physics, Vol. 46, No. 4, Winter 2021, P. 65-77 (Research) DOI: 10.22059/jesphys.2020.282486.1007124

A Semi-Automatic 2-D Linear Inversion Algorithm Including Depth Weighting Function for DC Resistivity Data: A Case Study on Archeological Data

Sets of Pompeii

Varfinezhad, R.1 and Oskooi, B.2*

1. Ph.D. Student, Department of Earth Physics, Institute of Geophysics, University of Tehran, Tehran, Iran 2. Associate Professor, Department of Earth Physics, Institute of Geophysics, University of Tehran, Tehran, Iran

(Received: 8 June 2019, Accepted: 21 Jan 2020)

Abstract In this paper, a new simple, efficient and semi-automatic algorithm including depth weighting constraint is introduced for 2-D DC resistivity data inversion. Inversion procedure is linear; however, DC resistivity data inversion is generally nonlinear due to the nonlinearity of Maxwell’s equation relative to resistivity (conductivity). We took the advantage of the 2-D forward operator formula obtained based on integral equations (IE) by Perez- Flores et al. (2001), for the inversion algorithm. Inversion algorithm is iterative and regularization parameter and depth weighting exponent are the critical parameters that have default values of 0.1 and 1, respectively. The presented technique was used only for dipole-dipole array by Perez-Flores et al. (2001), but here in addition to improving results for dipole-dipole array, its productivity is demonstrated for other geo-electrical arrays such as Wenner alfa, Wenner Schlumberger. Three synthetic data sets computed by Res2dmod software are utilized to investigate the performance of the algorithm through comparing the results with Res2dinv software output sections. Finally, the algorithm is applied on an archeological data set of Pompeii, which was collected by dipole-dipole array. IE inversion algorithm lead to satisfactory inversion models for both synthetic and real cases which reconstruct the subsurface better than or as well as that of the software. Keywords: DC resistivity, Depth weighting, Integral equation, Inversion, Res2dinv.

1. Introduction The direct current (DC) resistivity method is one of the most common methods for the exploration of the earth’s subsurface due to being a cost-effective and well-established method (Wei et al., 2013). The purpose of electrical resistivity tomography (ERT) surveys is to determine the subsurface resistivity distribution both horizontally and vertically by making measurements on the ground surface. Introducing an efficient algorithm for inversion of resistivity data is necessary because quantitative interpretation cannot be made from the non-inversed data and interpretation based on pseudo-section is qualitative. The DC resistivity quantitative interpretation has been developed largely over the years and various techniques have been proposed for DC resistivity inversion (e.g., Loke and Dahlin, 1997; Jackson et al., 2001; Pérez-Flores et al., 2001; Loke et al., 2003; Günther et al., 2006; Boonchaisuk et al., 2008; Li et al., 2012; Wei et al., 2013; Timothy et al., 2015). The numerical calculation of the electric field started in the late 1960s using

the techniques of integral equations (Dieter et al., 1969), which is the case here, then other numerical techniques such as finite element (FE) (Coggon, 1971) and finite difference (FD) (Mufti, 1976) were introduced to DC resistivity forward modeling. The Integral Equation (IE) method is a powerful tool in electromagnetic (EM) modeling for geophysical applications especially for those models that have a background conductivity with simple structure (Zhadanov, 2009). The main advantage of the IE method in comparison with the FD and FE methods is the fast and accurate simulation of the response in models with compact 2-D or 3-D bodies in a layered background (Varfinezhad and Oskooi, 2019). Resistivity data inversion is generally nonlinear, but here, a linear method proposed by Perez-Flores et al. (2001) is used. During the past few decades, there have been different algorithms for resistivity data inversion, most of which solved the problem in nonlinear form. General formula of their objective function is as follows:

*Corresponding author: boskooi@ut.ac.ir

66 Journal of the Earth and Space Physics, Vol. 46, No. 4, Winter 2021

min →‖ ( ( ) − )‖ + ‖ ( − )‖ where d, F(m), m0 and m are observed data vector, calculated forward response, initial model and updated model obtained from adding updating term dm to m0; and are data and model weighting matrices and is regularization parameter. Smith and Vozoff (1984) used just the first term without data weighting matrix and instead of the second term they used Truncated Singular Value Decomposition (TSVD) as another way of regularization. Narayan et. al (1994) used both terms but with some differences. Model weighting matrix is not used and regularization parameter adoption is different. Sasaki (1994) applied model weighting matrix, which was the second smoothness operator; Gunther et al. (2006) as well as Perez-Flores et al. (2001) utilized both data and model weighting matrices, which were data covariance matrix and smoothness operator, respectively. However, Perez-Flores et al. solved a linear problem that was based on integral equation. In addition to smoothness constraint. It can be said that these weights act as depth weighting function, but their mathematic formulas are not known. A 2-D inversion scheme with lateral constraint and sharp boundaries was introduced by Auken and Christiansen (2004). They believed that quasi-layered model can show actual geology more accurately in sedimentary environments. In general, the algorithm used by Loke and Barker it for RES2DINV software which is the most versatile one for different cases. Depth weighting function introduced by Li and Oldenberg (1996 and 1998) for 3-D inversion of magnetic and gravity data that were utilized to compensate for the natural decay of the kernel matrix values with increasing depth. The exponent of this function generally depends on the depth of the anomaly; therefore, a reliable priori information about possible depth range of the anomaly is required. Li and Oldenberg (1996 and 1998) suggested exponent 2 and 3 for gravity and magnetic methods, respectively. In the absence of this, constraint cells that are near to the surface have larger weights in the inversion procedure. Depth weighting is going to be inserted into the inverse algorithm as weighting matrix with clear determination of its exponent.

In this paper, the 2-D formula of DC resistivity kernel obtained by Perez-Flores et al. (2001) is going to be manipulated, but they used it just for dipole-dipole array. Here we extend it for other geoelectric arrays. Weighted damped least-squares solution including depth weighting function as weighting matrix is adopted for inversion algorithm. Regularization parameter and the exponent of depth weighting are the critical parameters for this algorithm which are going to be addressed how to be determined. This technique is applied on different synthetic and real data sets and the results will be compared with Res2dinv software, which is the most widespread and standard software for 2-D DC resistivity data. Exact synthetic data are calculated by Res2dmod. 2. Methodology 2-1. 2-D forward operator 3-D formula of DC resistivity Kernel obtained by Perez-Flores et al. (2001) is as Equation (1): = C ( − ) −| − | − ℎ= . = . where C = ( )( )

(1)where , and are vectors defining coordinates of cell centers, current and potential electrodes, respectively, is dipole separation and n is the values multiplied by dipole separation to increase the distance between current and potential electrodes in order investigate greater depths. Vectors ,

and are generally defined as: = + + = + + (2) = + + In fact, the integral form of the interested forward problem can be considered as a Fred-Holm Integral Equation of the first kind (IFKs). Integrating the Equation (1) in y direction from -∞ to ∞ leads to the 2-D form of IFKs (Varfinezhad and Oskooi, 2019): ( ) = ( . . ) ( . ) (3)

where represents current and potential

A Semi-Automatic 2-D Linear Inversion Algorithm Including Depth … 67

electrodes, is the logarithm of apparent resistivity values, ( . ) are coordinates of points of the interested area, is kernel and

is the model. Dividing the subsurface to × cells and discretizing the previous equation gives rise to the following matrix equation (Varfinezhad and Oskooi, 2019): = (4)

where is the 2-D forward operator, and readers are referred to the forward modeling paper by Varfinezhad and Oskooi (2019) for efficient calculation of forward operator. 2-2. Inversion algorithm Solving Equation (4) in order to find the model parameters is made by inversion. For an initial model and from Equation (4), forward response is computed as: = (5)

Subtracting Equation (5) from (4), we have: ( − ) = − ∆ = ∆ ∆ = −.∆ = − (6)

By multiplying on the both sides of Equation (6), Equation (7) is obtained: ( − ) = ( − ) (7)

is the weighting matrix. Updated term ∆ = − is calculated as: ( − ) = ( ) ( )( − ) (8) Regularizing Equation (8) by taking advantage of the zeroth-order Tikhonov regularization technique leads to Equation (9): = + ( + ) ( ) ( − ) (9)

and are identity matrix and regularization parameter, respectively. representing depth weighting matrix and is defined as:

= (10)

where is the z coordinates of cell centers and is depth weighting exponent and we are trying to address how this to be chosen. The algorithm is started with an initial model that assumed here to be a homogenous model with apparent resistivity equal to the background value of observed data, but it should be said that other initial models derived from any other geophysical methods or a priori information can also be used, which is not the case in both synthetic and real cases of this paper. Iterative inversion procedure stops after four iterations, and it rarely needs to be changed to the other values, but desired solution is always captured during 10 iterations. and are mostly constant values and only for some cases replacement of other values are required. These replacements are also very easy to be made as will be shown in the following sections. Ultimately, in addition to simplicity, effectiveness, the IE code is a semi-automatic code. 3. Synthetic models Three different synthetic models are considered to investigate the efficiency of the suggested technique, and Res2dmod software is used for calculating model forward responses to produce exact synthetic data. Inversion result derived from the IE technique are compared with the results of Res2dinv software to demonstrate its productivity. Res2dinv results are shown in MATLAB to have an identical representation system for both methods and comparisons can be made better. For all cases, if the profile length is L, and a is the smallest electrode spacing, then the number of cells in x and z directions (nx, nz) and the cell lengths lx and lz are determined as nx=L/a, nz=L/(4a) and lx=lz=a. 3-1. Four conductive anomalies (Dipole- Dipole array) First, the synthetic model is composed of four conductive anomalies with the same resistivity value of 20 Ω.m surrounded by a homogenous background with resistivity 100

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