To calculate the inverse of a signal or a filter, it is sufficient to take the inverse of its Fourier Transform. The signal used here, is the impulse response of a second order AR filter. The AR filter is selective, that is to say that its gain is only important in a limited bandwidth.
Outside this resonance frequency the modulus of the Fourier Transform tends to zero. Its inverse thus becomes very large. The relationship between extreme values at the resonance frequency and those at the Shannon frequency is important, as we can establish from the figure, which shows the log of the complex gain. The relationship is of the order of 10000.
We show the impulse response and the inverse filter in time which principally shows the higher frequencies.
The seismic trace is the result of the convolution of the impulse response of the reflectivity function by the signal. The deconvolved trace is obtained by convolution of the trace by the inverse filter of the wavelet. When the trace has no additional noise, the result is perfect.

If we add noise to the trace by choosing the signal-to-noise ratio, the deconvolution is no longer efficient. To obtain an acceptable result, it is sufficient to add a constant to the modulus of the Fourier Transform of the signal. This constant, defined by the regularization parameter, avoids division by very small numbers.