# 0.5 Comparing seismic imaging algorithms

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Here we compare two different methods for interpreting data from a seismic survey.

The data from a seismic survey appears in the form of time series vectors, representing samples of the received acoustic signals over a fixed period of time. In our simulation example, the samples come every 4 ms (Fs = 250 Hz) and the vectors are 512 samples (2.048 s) in length. There is one time vector for each source/receiver combination, so for our example, we have 32 sources and 128 receivers for a total of 4096 time vectors. Therefore, our raw data will be represented as a 512x4096 matrix which we will load into MATLAB.

At this point, it is necessary to process this matrix and map this data onto a grid so that the shape of the formations we are studying may be determined. The basic technique involving ellipses has been described above, so all the program needs to do is to draw the ellipses, weighted by the magnitude of the time samples, and add the results for each time vector.

We consider now, two possible algorithms for drawing the necessary ellipses. The first method will traverse every pixel on the grid and at each point plot the value of the correct sample from the time vector. The second method will traverse the time vector and for each sample will plot an ellipse of appropriate magnitude.

What do we need to consider when comparing these two methods? First of all, there are 4096 vectors so this problem can become computationally very costly. For each method, we seek to find a standard run-time and evaluate different methods of optimizing this run time. Additionally, we hope to resolve as clear a picture as possible, so we should compare the graphs of the final answers to see which resolution is clearer.

## Method 1 – traversing the grid

For every point, we can calculate the total distance required to travel from the source to that point and then to the receiver using the distance formula on the (X,Y) coordinates of the point, the source, and the receiver.

$D()=\sqrt{(Y()-{Y}_{S}())^{2()}+(X()-{X}_{S}())^{2()}}()+\sqrt{(Y()-{Y}_{R}())^{2()}+(X()-{X}_{R}())^{2()}}()$

We may now divide by velocity to get the time in seconds, and then we may again divide by the sample period to get an index for the time series vector that corresponds to this point.

${t}_{i}()=\frac{D()}{V(){T}_{s}()}$

Note that this value for time will not be an integer, so we must interpolate using the time series indices above and below it.

${t}^{+}()=\mathrm{ceil}({t}_{i})$

${t}^{-}()=\mathrm{floor}({t}_{i})$

A simple linear interpolation method will give us an appropriate value that is weighted based on how close t is to t-minus and t-plus.

$\mathrm{Mag}()=\mathrm{TimeSeries}({t}^{-})({t}^{+}()-{t}_{i}())+\mathrm{TimeSeries}({t}^{+})({t}_{i}()-{t}^{-}())$

We finish by applying these equations to every point on the grid. This will trace out the ellipse patterns we desire for a given source-receiver pair.

## Method 2 – traverse the time vector

Taking one source-receiver pair's time vector, we first find the distance between the source and receiver. This distance is the minimum distance a signal must traverse. Since each sample in a time vector is 4ms, and we know that a wave travels at 1500m/s, the n th sample in a time vector takes n *4ms to travel n *0.004s*1500m/s. If the n th sample distance less than the distance between the source and receiver, it is ignored since it is bad data. Usually these samples have a received signal value of 0 anyway.

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