f Inverse Q filtering with pareto-levy scaling

Hargreaves (1992) solves the time-varying inverse Q filter problem by stretching the data logarithmetically and performing the time-stationary inverse in the stretch domain. Although numerically more efficient than previous inverse Q algorithms (Bickel and Natarajan, 1985), Hargreaves observed that his algorithm appears to overcorrect early events and leave later events undercorrected. I have found that the residual errors are the result of time-varying bandwidths in tbc stretch domein and imperfections in Carpenter (1966) scaling which, at best, approximates the time-frequency response of Futterman's (1962) approximation to the constant Q attenuation law. The figure below illustrates the improvement obtained for late events when the bandwidth is increased and Carpenter scaling is replaced witti Pareto-Levy scaling (Mandelbrot, 1960) - the exact time-frequency scaling exact for the Kjartanssian (1979) constant Q model.