Convolution Product. Change of wave shape as a result of passing through a linear filter

Transcription

1 Convolution Product Change of wave shape as a result of passing through a linear filter e(t): entry signal (source signal) r(t): impulse response (reflectivity of medium) (a) The spikes are sufficiently separated that the convolution just results in a duplication of the input wavelet at the spike times and with the spike amplitudes. The convolution process just involves multiplying every sample of the spike series by the input wavelet and adding all the results. (b) Here the spikes are closer together and interference occurs in the resulting trace.

2 Convolution Product Change of wave shape as a result of passing through a linear filter Continuous Discrete e(t): entry signal (source signal) r(t): impulse response of the filter (reflectivity of the substrata)

3 Convolution Product Convolution in the time domain is represented in the frequency domain by a multiplying the amplitude spectra and adding the phase spectra. The way to do 1- One spike 2- Multi spikes

4 Deconvolution Product Deconvolution is the reversal of the convolution process DECONVOLUTION If the wavelet were known, the input spike series could be discovered by the deconvolution process. The convolutional model of the seismic trace states that the trace we record is the result of the earth's reflectivity (what we want) convolved with the source wavelet (and it's ghosts), multiples, the recording system and some noise.

5 Deconvolution Product Deconvolution is the reversal of the convolution process Spiking or Whitening deconvolution S = R * E [seismic trace = reflectivity * source] Deconvolution operator D D * E = δ (spike) D * S = D * R * E = D * E * R = δ * R = R Time-variant deconvolution D changes with time to account for the different frequency content of energy that has traveled greater distances Predictive deconvolution The arrival times of primary reflections are used to predict the arrival times of multiples which are then removed

6 Spiking Deconvolution 3/4 3/4-1/2

7 Spiking Deconvolution Recorded Waveform Deconvolution Operator Output 1/ /4-1/2 Recorded Waveform Deconvolution Operator Output 1-1 3/4-1/2 1/

8 Spiking Deconvolution Recorded Waveform Deconvolution Operator Output 1-1 3/4-1/2 1/ Recorded Waveform Deconvolution Operator Output 1-1 3/4-1/2 1/

9 Spiking Deconvolution Source wavelet becomes spike-like Original section Deconvolution: Ringing removed

10 Predictive Deconvolution If we know the source pulse = Then cross-correlating it with the recorded waveform gets us back (closer) to the reflectivity function

11 Predictive Deconvolution If we don t know the source pulse 1) Then autocorrelation of the waveform gives us something similar to the entry signal plus multiples 2) Cross-correlating the autocorrelation with the waveform then provides a better approximation to the reflectivity function.

12 Predictive Deconvolution Predictive deconvolution requires that the autocorrelation of the source wavelet is known (rarely true in practice). The autocorrelation of the seismic trace is used as an approximation instead. The autocorrelation function is critical in picking the deconvolution parameters of gap (also called minimum autocorrelation lag) and operator length (sometimes called maximum autocorrelation lag).

13 Predictive Deconvolution Predictive deconvolution requires that the autocorrelation of the source wavelet is known (rarely true in practice). The autocorrelation of the seismic trace is used as an approximation instead.

14 Deconvolution SURFACE-CONSISTENT: deconvolution is commonly applied to land seismic data and in AVO processing. The technique ensures that traces from the same surface source and receiver location (or CMP, offset in addition) have the same, consistent, operator applied.

15 Wiener Filtering Predictive Deconvolution The Wiener filter is that which best (in a least squares sense) shapes a given wavelet to a desired wavelet. Without going into the mathematics it turns out that the filter is found by dividing the cross-correlation of the input with the desired output by the auto-correlation of the input. This solution sets up a series of simultaneous equations which are solved rapidly in the computer by matrix inversion using the Levinson algorithm. A certain percentage of noise (called white noise or white light) is added to stabilise the inversion program.

16 Spectral Whitening Spectral Whitening (sometimes called balencing or broadening) is a process usually applied for improving the resolution and appearance of seismic data without overly boosting noise. Inverse Filtering would tend to produce the green line. The commonest form of spectral whitening is to split the dataset into several narrow frequency bands by bandfiltering, equalising the sections by AGC (or some other scaling function) and add the resulting sections together.