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Convolutional equivalent layer for magnetic data processing

We present a fast equivalent layer method for processing large-scale magnetic data. We demonstrate that the sensitivity matrix associated with an equivalent layer of dipoles can be arranged to a Block-Toeplitz Toeplitz-Block (BTTB) structure for the case where observations and dipoles are aligned on a horizontal and regularly-spaced grid. The product of a BTTB matrix and an arbitrary vector represents a discrete convolution and can be efficiently computed via 2D Fast Fourier Transform. In this case, the matrix-vector product uses only the elements forming the first column of the BTTB matrix, saving computational time and memory. Our convolutional equivalent layer method uses this approach to compute the matrix-vector products in the iterative conjugate gradient algorithm with the purpose of estimating the physical-property distribution over the equivalent layer for large data sets. Synthetic tests of total-field anomaly data show a decrease in both floating-point operations and computation runtime of our method compared to the classical approach of solving the least-squares normal equations via Cholesky decomposition. Faster results are obtained for millions of data, showing drastic decreases in computer memory usage and runtime, allowing to perform magnetic data processing of large data sets on regular desktop computers. Our results also show that, compared to the classical Fourier approach, the magnetic data processing with our method requires similar computation time, but produces significantly smaller border effects without using any padding scheme and is also more robust to deal with data on irregularly spaced points or on irregularly observation surfaces. A test with irregularly spaced field data over the CarajౠProvince, Brazil, confirms the efficiency of our method by estimating the physical-property distribution over the equivalent layer and computing the upward-continuation.