Compensating for attenuation by inverse Q filtering
Carlos A. Montaña, Gary F. Margrave
Three different approaches for inverse Q filtering are reviewed and assessed in terms of effectiveness in correcting amplitude and phase, computational efficiency and numerical instability. The starting point for the three methods is the linear, frequency-independent Q theory, in which an attenuated trace can be forward modeled as the product of a wavelet matrix, an attenuation matrix and the earth reflectivity series. In the first method the reflectivity series is solved by inverting a matrix similar to the attenuation matrix with minimized nearly-singular characteristics. The second approach is based on the downward continuation migration method and is a highly efficient and numerically stable method. The last assessed method uses the generalized nonstationary Fourier integrals to apply the inverse Q filter. The performance of the three kinds of filters is assessed by assuming that the exact value of Q is available. One synthetic trace is created with a pulse source to avoid introducing error by ignoring the conmutator term which is required when the Q filter is applied before a spiking deconvolution to remove the source signature. Several attenuated traces are forward modeled for different Q values, and used to test the filters. Two attributes are used to quantify the similarity between the expected and the real output: the L2 norm of the difference between the expected and the real output and the maximum crosscorrelation and its corresponding lag, computed for windowed fragments of the traces.