Deconvolution

Convolution was the forward operationsource(t) * refectivity(t) = signal(t)

Deconvolution is the reverse operationreflectivity(t) = signal(t) *-1 source(t)

Deconvolution

Objectives of deconvolution Make source into spike Reshape source signal Remove multiples

How do we do this

We need to decide two things How small to compress the source How long a source we want to compress

We will use the cross-correlation and auto-correlation functions to do this

Autocorrelogram

Cross-correlation just like convolution, but we use a + instead

of a

Autocorrelation Cross-correlate the signal with itself

( ) ( )a t b d +

= +

( ) ( )a t a d +

= +

Cross and Auto correlations

Cross correlation tells us how much two signals are alike (we get closer to a spike if they are similar)

Autocorrelation tells us how much a signal is like itself

Noise autocorrelates to a spike since it is, by definition, random and should be nothing like itself after shifting.

Correlation examples

Boxcar with boxcar (same as convolution because they are symmetric)

Boxcar with right triangle Noise with noise (gives single spike)

Autocorrelogram

Here is an example of an autocorrelogramof a seismic signal

Autocorrelation

Autocorrelation of basic wavelet

Primary with multiple

Primary with 2nd multiple

Signal

Primary2nd multiple

1rst multiple

Autocorrelogram

Here is how we would estimate the predicted wavelet size and how much of it to compress from the autocorrelogram

Compress source to size of first wiggle

Define source length to exclude multiples

0.1s

Decon out the multiples first

Plus shape the primary

Shape the primary first

Plus decon out the multiples

Decon in 1 stp

Starting wavelet length

Ending wavelet length

signal

Spike plus multiples

Original spike

Note sharper reflections

multiples

original

beforeAfter, reflections are sharper

Vibroseis source signal

Here is what a vibroseis source signal looks like. It is often called the sweep because it sweeps through the frequency range.

Low frequencies to start High-frequencies at end

4 second Sweep

Autocorrelation of vibroseis source

The autocorrelation of the vibroseis source gives a klauder wavelet, which is compact

Note symmetrical shape; 4 seconds of sweep is compressed into 0.1 s wavelet

0.1 s

Vibroseis recording

So to get the vibroseis source out of the vibroseis data, we simply cross-correlate it with the original vibroseis sweep signal

4-s sweep

Recorded signal

Cross-correlation of sweep and signal, each reflector now has the shape of a Klauder wavelet. This is the signal.

Vibroseis recording4-s sweep as recorded from the sweep signal in the vibroseis truck

Recorded signal recorded at each geophone

Cross-correlation of sweep and signal.This is often done in the field as the signal is recorded in the truck. Because the vibroseis correlation compressed the data, some data space savings is achieved.

DeconvolutionDeconvolutionHow do we do thisAutocorrelogramCross and Auto correlationsCorrelation examplesAutocorrelogramAutocorrelogramVibroseis source signalAutocorrelation of vibroseis sourceVibroseis recordingVibroseis recording