Correction of source profiles for Doppler effects

Einar Kjartansson

Introduction

Motion of sources or receivers, relative to the medium of wave propagation, causes a shift in the frequency of the measured waveforms, Doppler shift.

**Figure 1:** Autocorrelation for a 6 second linear sweep from 5 to 100 Hz.
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**Figure 2:** Wave distortion caused by Doppler shift
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The speed of marine seismic vessels is typically around 1/600 of the speed of sound in water. The resulting frequency change is small, but can result in significant relative phase shifts in vibroseis data where the source lasts several seconds.

Figure 1 shows the autocorrelation for a 6 second linear sweep from 5 to 100 Hz.

**Figure 3:** A correction filter, computed using least-squares.
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**Figure 4:** Filter in Figure 3 applied to waveform in Figure 2.
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If the source is moving away from the receivers at a velocity of 2.5 m/s, it will move 15 m in 6 seconds. In a medium with a velocity of 1500 m/s this results in a 10 ms longer observed sweep.

Figure 2 shows the result of correlating the same sweep as used in Figure 1 with a modified sweep that has been stretched by 10 ms.

A correction filter that is computed using least-squares to predict the original sweep from the stretched version, is shown in Figure 3. Figure 4 shows the result of applying this filter to the waveform in Figure 2.

Distortion caused by source motion

Doppler effects in seismic reflection data are caused both by motion of sources and receivers. For horizontal reflectors, both contributions are of the same magnitude but opposite sign, so the two cancel each other. It is relatively straightforward to correct for the effects of receiver motion, by resampling the data to what would be recored using stationary receivers. This is possible because data is usually not spatially aliased in the receivers.

This is generally not the case for the sources. It is possible to apply filters to correct for Doppler effects due to moving sources. The correction depends on the direction of the wave or ray at the source. There is no effect for vertical rays, as the wave then travels perpendicular to the motion of the source. Through the use of dipfilters, waves leaving the source at different angles may be processed separately.

This requires decomposing the data according to the direction of wavemotion, as the size of the correction depends on the angle of the wave when it leaves the source.

**Figure 5:** Notation for geometry
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Referring to Figure 5, we can denote the distance travelled by the ray from the source at B to the receiver at C by d :

d = T V = b + c

(1)

where T is total traveltime and V is the wave velocity. The distance separating the source and receivers is a. The law of cosines can be used to express any of the angles in a triangle in terms of the sides, for example:

$\begin{displaymath} cos B = \frac{ a^{2} + c^{2} -b^{2} } { 2 a c } \end{displaymath}$

(2)

$\begin{displaymath}cos C = \frac{ a^{2} + b^{2} -c^{2} } { 2 a b } \end{displaymath}$

(3)

Solving equations 1 and 3 together, one obtains:

$\begin{displaymath}b = \frac{ d^{2} - a^{2} } { 2d - 2a cosC } \end{displaymath}$

(4)

Equations 4, 1 and 2 can be used to compute the direction of the wave at the source (cosB) from offset (a), wave velocity (V), total traveltime T, and the direction of the wave at the receivers (cosC).

Application

A procedure has been developed and tested, to correct for Doppler effects in marine Vibroseis data. It consists of three programs.

A correction for the receiver motion is applied as a time dependent lateral shift that is applied before crosscorrelation.
A set of correction filters is computed. Those filters depend on the sweep function used (the sweep is an input). A set of about 100 filters should suffice to avoid artifacts caused by using a discrete set of filters.
Correction for source motion is applied after the data has been correlated. A two dimensional Fourier transform is applied to each shot record. The data is then decomposed into several different dips in the f-k domain, and transformed back. The dipfilters are designed such that the original data is obtained by stacking all the dips. The amount of correction, as a function of dip, time and offset is computed using the expressions in the previous section and used to select a correction operator that is then applied. The final result is obtained by stacking all the corrected dip-panels.

About this document ...

Correction of source profiles for Doppler effects

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The translation was initiated by Einar Kjartansson on 1998-06-21

Einar Kjartansson
1998-06-21