|Elastic wavefield directionality vectors|
My assumption is that if in any location within the modeled wavefield a single wave of a particular type (P or S) exists, then the particle displacement there will have a distinct direction vector in space. In other words, the spatial components of the displacement vector will be linearly polarized when viewed along the time axis. As a measure of this polarization, which I call the ``Polarization Coefficient'', I use the absolute value of statistical correlation between the displacement components:
The logic behind this is that if the energy going through a certain model point is indeed polarized, then there will be a linear dependence between the displacement components, and
will tend to 1. Otherwise, it will tend to 0. However, this measure is unstable where one of the displacement components is much smaller than the other (as shown in the following figures), where we would expect the polarization to actually be very specific (i.e. - along the larger component). The calculation of the polarization coefficient should be applied only within specific time windows during wavefield propagation, since conceptually, if a wavefield is complicated enough and is propagated for a certain length of time, then there will be many waves propagating in many directions, and no polarization can be expected at any one location for the entire propagation time.
Equation 25 can be used to produce a displacement correlation field. This field, in turn, will tell us to what degree the wavefield is polarized within a particular time window during propagation.
Determination of correlation time window size
The size of the time window within which the degree of polarization should be estimated is a major factor in the entire method, and can be considered as its Achilles' heel, it being a distinct point of failure (or success). On a conceptual basis, I think that the time window size should be related to the time period of the most energetic wave in the wavefield. Tying the window size to this parameter should enable us to decide on the resolution at which we wish to observe the wavefield's polarity, as a function of multiples of the most dominant frequency. However, this is useful only if there is a distinct dominant frequency. Though this may be the case for the source wavefield, the receiver wavefield may have a more variable spectrum. It is always possible to overestimate the correlation time window size, but that may result in a displacement correlation field with unuseful low values. This point remains unclear, and will likely require a lot of tuning to get right. At this stage, I define the time window size to be:
Estimation of angle between polarization direction and vertical direction
Only where there is a large polarization coefficient is it worthwhile to attempt to estimate the angle of the polarization direction to the vertical axis, within that time window. The slope of the line with the best fit to the crossploted displacements is estimated by linear regression. The angle to the vertical is the arctangent of this slope. Figure 2 explains this idea.
Figure 2. Crossplot of 2D displacement components of a nearly linearly polarized wave at some model location. The red dots represent samples of the displacements within a time window. The blue arrow indicates the polarization vector acquired by linear fitting of the displacement values.
Considering the displacement values and within a time window as two independent series, the standard linear regression equation can be applied to determine the slope of the line that best fits these series in the least-square sense:
where and are the averages of the displacement values in the and directions respectively, within the the time window of length time steps.
The angle between the linear fitting to the vertical axis is
Notice that the polarization line could ``point'' toward either direction, which is why I defined . This means that for , the polarization is in the or in the direction. In the case of a P wave, the interpretation is that within this time window, the wave at this particular location is propagating in the or in the direction. In the case of an S wave, the interpretation is that the wave is propagating in the or in the direction.
|Elastic wavefield directionality vectors|