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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

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

We solve this by minimizing the objective function

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

subject to the constraint

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

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

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


$\displaystyle \bf v^T \frac{\partial{ \bf v }}{\partial \beta }=0$    , (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

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


$\displaystyle \bf v^T {\bf W}^T {\bf W} \frac{\partial{ \bf v }}{\partial \beta }=0$    . (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

$\displaystyle \bf v^T {\bf W}^T {\bf W} = \lambda \bf v^T$    , (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

$\displaystyle {\bf W}^T {\bf W} \bf v = \lambda \bf v$    , (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.

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