Hunting for microseismic reflections using multiplets

Singular value polarization analysis

Since S first arrivals are clearer and stronger than P first arrivals, we use S arrivals to estimate a vector in the general direction of P arrivals. The idea is to find the direction most perpendicular to the strongest (direct) S arrivals. We setup a $3 \times n$ matrix of shear first arrivals, $\bf W$ , by windowing around the first shear arrival. As we seek a vector $v$ that is perpendicular to the shear arrival direction (which is a P-arrival) we want

$${\bf W} \bf v \approx 0$$ (1)

We solve this by minimizing the objective function

$$J=\parallel \bf v^T {\bf W}^T {\bf W} \bf v \parallel_2$$ (2)

subject to the constraint

$$\bf v^T \bf v =1$$ (3)

Let $\alpha$ and $\beta$ be two spherical surface coordinate parameters over which we will minimize. Taking partial derivatives of the constraints yields:

$$\bf v^T \frac{\partial{ \bf v }}{\partial \alpha }=0 \phantom{\mbox{\ \ \ ,}}$$ (4)

and

$$\bf v^T \frac{\partial{ \bf v }}{\partial \beta }=0$$ (5)

which says that $\bf v^T$ is perpendicular to the two partial derivatives. Next, taking partial derivatives of the sum of squares expression gives

$$\bf v^T {\bf W}^T {\bf W} \frac{\partial{ \bf v }}{\partial \alpha }=0 \phantom{\mbox{\ \ \ .}}$$ (6)

and

$$\bf v^T {\bf W}^T {\bf W} \frac{\partial{ \bf v }}{\partial \beta }=0$$ (7)

Therefore $\bf v^T {\bf W}^T {\bf W}$ is also perpendicular to both partial derivatives and consequently must be parallel to $\bf v^T$ . This means that

$$\bf v^T {\bf W}^T {\bf W} = \lambda \bf v^T$$ (8)

where $\lambda$ is the eigenvalue of the matrix ${\bf W}^T {\bf W}$ that will make the least squares expression a minimum. Transposing we get

$${\bf W}^T {\bf W} \bf v = \lambda \bf v$$ (9)

which is a classic eigenvector problem for the matrix ${\bf W}^T {\bf W}$ . Since the right singular vectors of $\bf W$ are the same as the eigenvectors of ${\bf W}^T {\bf W}$ , we used the LAPACK routine SGESVD to find our desired P-wave direction vector.

Hunting for microseismic reflections using multiplets