What is Convolution in geophysical signals Processing?

Convolution in geophysical signals Processing.

Convolution is one among operations that is utilized in geophysical signal processing. However the term seems to not be clearly understood among us due to its formal similarity with other functions such as Cross - correlation. Let's try to discuss in a simplified way through this post, how this term is really about in digital signal processing.

First of all, It would be a good approach that in order to understand this term, we have to focus our thoughts on a system that can feed (input) a certain variable and then produce the result as output variable.

In Digital signal processing, a linear system can change the signal feed into it and then give out another signal as output. The input signal can be decomposed into a set of impulses, each of which can be viewed as a scaled and shifted delta function, while the output signal resulting from each impulse is a scaled and shifted version of the impulse response. Then the overall output signal can be found by adding these scaled and shifted impulse responses.

Before proceeding, furthermore in our discussion as described above, it would be interesting if you clearly understand some terminologies related to the above brief description as explained here.

Input Signal: Input signal is the signal when feeded into the linear system and convolved with impulse response produces the output signal. In most Digital Signal Processing applications, the input signal is hundreds, thousands, or even millions of samples in length.

Output Signal: Output Signal is outcome signal produced by convolution of input signal and the impulse response.

Impulse Response: The term impulse response always refers to the output signal produced by a Linear system when the input signal is delta Function, see figure 1 below. However the term impulse response takes many different titles which reserve the same meaning depending on applications and the system itself being considered.

Let us take an example, If the system being considered is a Filtering system (Filter), the impulse response is either termed as Filter kernel or the convolution kernel (kernel). The same applies If it is in image processing, the impulse response is called the Point spread function.

Figure 1: Simple layout for how impulse Response is obtained.

In geophysical measurement of various physical parameters such as travel times, acceleration due to gravity, magnetic susceptibility just to mention a few. The physical instruments are to take such measurements, since the instrument itself is considered as a measurement system, then the impulse response can be termed as instrument response (resolution function).

In other words, if we know a system's impulse response, then we can calculate what the output will be for any possible input signal.

What is Convolution?

Convolution can be defined as a formal mathematical operation that takes two signals and produces a third signal. It is like multiplication, however there is a slight difference in mode of operation and notation between these two terms such as convolution in time domain corresponds to multiplication in frequency domain.

Convolution is used in the mathematics of many fields, such as probability and statistics. Also In linear systems, It is used to describe the relationship between three signals of interest which are the input signal, the impulse response, and the output signal.

Let's say if we have a Linear system, and an input signal, x(n) enters such a linear system with an impulse response, h(n), then the resulting output signal, y(n), can be expressed by the equation below as the input signal convolved with the impulse response. See figure 2 below

x(n) * h(n) = y(n)...............(1)

Figure 2: Simple convolution layout

Let us continue to get a better understanding and clear the doubt regarding the term convolution, if now we suppose to have time - an invariant system, and let say y(n) is a system’s impulse response. The other input x(n) is then passed through this system, one sample at a time. At each instant t , the corresponding input sample x(t) , will generate an impulse response x(t)y(n).

Then the response will also be delayed by the same amount, x(t)y(n−t). When this is done for all samples x(t) , then we will have a sequence, obtained by adding all of the impulse responses.

x(n)∗y(n) = ∑t=−∞ to ∞ x(t)y(n−t)................(2)

Convolution is represented by the star, *. However a star is used frequently in a computer program to represent multiplication, but a star in an expression means convolution and not multiplication.

Towards the end of this discussion, now we have the sense that, if we consider the expression 2 above, in earthquake seismics, the signal (seismogram) recorded by a seismograph can then be described as the outcome from the convolution of ground motion x(n) and instrument (seismograph) response y(n), See figure 3 below

Figure 3: (a) Simple convolution in waveform (b) Simple layout of (a).

Another good example on convolution model in seismic reflection is detailed by Reynold (1997), that seismic trace as the recorded output wavelet (W) by seismograph is obtained as convolution between Earth's reflectivity series (R) and Seismic source wavelet (S) with noise components, see figure 4, below.

Then Output wavelet (W) = R * S

Figure 4: How convolution is performed (Reynolds, 1997)

The simplified version of figure 4, is to just take Long multiplication between the Reflectivity series and source wavelet as described here below. You have to use the Long Vertical Multiplication method.

1 -½ ½ × 1 ½ ½ = 1 0 ¾ ¼

It is the point that we can view the mathematical model of convolution as applied in digital signal processing to analyze how each sample in the input signal contributes to many points in the output signal.

However the length of the signals are not restricted by convolution, but treated by expression that the Length of output signal is equal to the length of the input signal, plus the length of the impulse response, minus one.

Such as if the length of input signal is 81 samples, while that of impulse response is 31 samples, then, The length of each output signal is: 81 + 31 - 1 = 111 samples long.

Since most Digital Signal Processing applications, the input signal is hundreds, thousands, or even millions of samples in length. This means that from our example above, the input signal runs from sample 0 to 80, the impulse response from sample 0 to 30, while the output signal runs from sample 0 to 110.

I hope now, you have grasped the basic idea regarding the term convolution. Next post we will try to discuss the "convolution theorem" as an extension of this post. Just stay tuned!

At the end of the day, it is clear that one of the most difficult tasks you will encounter in geophysical Signal Processing, will require you to have the conceptual fit on mathematics in order to communicate these ideas.

Good Luck!

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