DATA ROTATION

Figure 1 shows a portion of a three-component crosshole record. The first event on each component, with its apex near 0.4 seconds, is a direct P-wave arrival. The event having a minimum traveltime of approximately 0.8 seconds is a converted SV arrival.

 

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Figure 1 A portion of a three-component crosshole record.

We would like to rotate these data to a coordinate system where the P and SV particle motion directions lie along two of the axes. Such a rotation separates the two events and could also reveal SH wave energy, having a particle motion perpendicular to both P and SV.

The transformation from X, Y, and Z components to P, SV, and SH wavetypes is described by the equation:

$$\pmatrix{P\cr SH\cr SV\cr} = \pmatrix{\sin\theta \cos\phi ... ...ta \sin\phi & \sin\theta \cr } . \pmatrix{X \cr Y \cr Z \cr},$$ (1)

where $\theta$ and $\phi$ are two angles, the azimuth and the dip angle, that parameterize the raypath of a wave arriving at a receiver. For more details on the transformation, see Karrenbach and Cole (1989).

We need to know $\theta$ and $\phi$ for each receiver in order to perform the rotation. To compute them, we rotate a small window of data (centered around one of the events) using all possible combinations of the two angles. Then we select the combination of angles that maximizes the power of the appropriate wavetype after rotation.

Figure 2 shows the arrival direction picks for the P event. Since this is a crosshole experiment, we expect the source-receiver azimuth to be constant. It is generally consistent; the small variations could be due to raypath effects. We expect the dip angle to vary depending on the relative depths of the source and receiver. Without knowing more about the experiment geometry, we can't confirm that the observed dip angles are reasonable.

 

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Figure 2 For each trace, the azimuth and dip angles that maximize the power of the P event after rotation have been picked.

Figure 3 shows the data from Figure 1 after rotation using the picked arrival directions. We have done a pretty good job of separating the P and SV events. Unfortunately the SH section doesn't seem to show any SH events, only residual energy left after rotation.

 

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Figure 3 The data from Figure 1 after rotation. The P and SV events have been separated fairly well. The SH section contains little of interest.