Adaptive Cross Approximation for Compressing the Jacobian Matrix in the Gauss-newton Inversion

Among offshore hydrocarbon exploration technologies, the controlled source electromagnetic (CSEM) method have gained a lot of interest in both academia and industry because of its ability to detect hydrocarbon reservoirs [1]. In order to maximally extract information from the data, a full nonlinear inversion approach is employed [2]. In such an approach, the investigation domain is subdivided into pixels and by using an optimization process, the conductivity distribution in the domain can be generated and hence the location, the shape and the conductivity of the reservoir can be inferred. Most of these nonlinear methods use iterative schemes where the conductivity distribution is updated in each iteration based on a search direction computed from a gradient of the cost function. Therefore, in these derivative-based approaches the Jacobian matrix plays a key role. The elements of this matrix are the derivative of the simulated data with respect to the pixel conductivities. Its size is equal to the number of measurement data times the number of unknown pixels. In the CSEM data inversion, the size of the data set and the inversion region can be very large. Hence, the storage of the Jacobian matrix requires a huge amount of memory. This is one of the bottlenecks of using gradient-type nonlinear inversion approaches. Moreover, because the Jacobian matrix is a dense matrix, the arithmetic operation of a matrix-vector multiplication can be very expensive as the size of the Jacobian matrix increases. To reduce the size of the Jacobian matrix, we usually invert a subset of the data at the risk of missing important data points. An alternative way is to compress the Jacobian matrix based on the fact that the electromagnetic field has a limited spatial bandwidth. We can either use physics-based techniques that rely on the field kernel or use pure numerical methods that tend to be very expensive to compute. In this work, we use the adaptive cross approximation (ACA) technique introduced by Bebendorf [3] to compress the Jacobian matrix. The ACA technique converts the Jacobian matrix into two smaller rectangular matrices. This approach reduces both memory usage and CPU time of the Gauss-Newton inversion approach as well as stabilizes the inversion process. To demonstrate the Gauss-Newton inversion using the compressed Jacobian matrix, we employ inversion examples in two-and-half dimensional (2.5D) geometries.