Joint two-dimensional electrical resistivity and seismic S-wave travel times inversion to characterize near-surface heterogeneities

Establishing the precise relationship between electrical resistivity and seismic shear (S) wave velocity in heterogeneous near-surface materials is a fundamental problem in geophysics and can complement petrophysical measurements for improved subsurface characterization. The relevant data from two-dimensional (2-D) electrical resistivity and seismic refraction investigations of the near-surface have jointly been inverted leading to accurate models. Nevertheless, Joint 2-D resistivity-velocity inversion is a difficult task since there is no established analytical relationship between resistivity and velocity. There are different approaches to 2-D joint inversion of disparate data with varying degrees of success. These can be classified into (1) petrophysical approach and (2) structural (or geometrical) approach. The petrophysical approach are based on the fact that for some specific geological environments, multiple geophysical parameters can be correlated via physical or empirical relationships. In the structural approach, both methods of geophysical are sensing the same underlying geology which in turn structurally controls the distribution of petrophysical properties. In this paper, we select the structural approach and posit that petrophysical information may be derived from the resultant models.
When there is no special analytical relationship between the physical properties that have been extracted by different geophysical methods, we can estimate the models that there are good structural agreement between the physical properties, by means of joint inversion techniques. Gallardo and Meju (2003) by introducing cross-gradientsfunction that is defined in the form the cross product of the gradients, estimated the structural resemblances between the resulting images from joint inversion. The cross-gradients function is incorporated as a constraint in a nonlinear least squares problem formulation, which is solved using the Lagrange multiplier method. When the value of this function is zero, images of models will completely be similar in structure. Being zero of this function requires that the simultaneous spatial changes of different geophysical models, independent of the amplitude, should be collinear. In term of geology this means that if the changes of the physical properties for both methods are collinear then can characterize boundaries of layers and structures using the images obtained from structure‐coupled inversion.
For the DC resistivity, forward problem is used by the procedure given by Perez-Flores et al. (2001). In this approach, the resistivity forward calculation is stated as a linear problem, which is based on the nonlinear integral equations for electromagnetic inverse problems of Gomez-Trevino (1987). The resistivity response of the model using the forward codes of Perez-Flores is approximated because of its computational speed of this approach we have employed it in the two dimensional joint inversion procedure. For estimating the first-arrival times from source to receiver, we adopt the approach of pseudo-bending technique (Um and Thurber, 1987). Compared to the earlier bending methods (Julian and Gubbins, 1977), the pseudo-bending technique is much rigorous and faster. For a cell with constant velocity, the Jacobian matrix is simply the length of the linear segment of the ray in the cell in transit, which is calculated impressively by ray tracing through the field of travel times that are generated during the forward modeling process.
Incorporating auxiliary factors such as, Levenberg-Marquardt (LM) stabilization factor and smoothness constraints in the inverse problem assures convergence and stability of solutions much more and can resolve the nonuniqueness that has been taken place because of data error. Joint two-dimensional (2-D) inversion scheme is used to a test data sample (to validate inversion technique) and also to a field data set that has been recorded along a profile. The obtained results from this method is compared with the conventional separate inversion results and we conclude from this comparison that the joint inversion scheme is more powerful than the traditional separate inversion in illustrating the structural similarities between seismic velocity and resistivity models. As a result, this gained structural conformity of the cross-gradients inversion models can help us in better characterization of heterogeneous near-surface materials.