Wave Gradiometry / Wave Gradient Analysis

(1) Theory.      (2) Application to the USArray.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

(1) The theory:

This web page gives a brief review about the basic idea of the wave gradiometry method. Refer to the references listed at the bottom of this page for more details.

(1.1) Wave function.

The wave function (of displacement, velocity or acceleration)  at location  may be written as:

                               (1)

where  carries amplitude variations as a function of locations, while   represents the phase variation as a function of time  and location  with  and  the slowness in the  direction and  direction, respectively.

 

(1.2) Spatial gradients.

Take derivatives of the equation (1) with respect to  and  respectively, we have:

                                             (2a)

                                             (2b)     

where  and  are amplitude variations in the  direction and  direction, respectively

 

(1.3) Find ,,  and .

For a better way to find these parameters, read the paper in the reference “Wave Gradiometry in the Time Domain”). A simple way to find  and  as well as spatial gradients of “G” is discussed as below. Fourier transforms equation (2a) and (2b) and divides both sides by U:

                                                       (3a)

                                                       (3b)

Now the four parameters can be found by identifying the real and imaginary parts of the left-hand-side of equation (3a) and (3b), respectively.

                                                            (4a)

                                                           (4b)

                                                                 (4c)

                                                                 (4d)

 

(1.4) Major Products.

Now ray parameter , wave directionality ,  geometrical spreading   and radiation pattern  can be found by:

                                                                                      (5a)

                                                                          (5b)

.                                                                (5c)

.                                                                  (5d)

Where “r” is the epicenter distance. The geometrical spreading  and radiation pattern  should be understood as amplitude variations in the radial and azimuthal directions, respectively.

(2) Application to the USArray (Click).

 

References:

(1)   Langston, C. A.(2007), Spatial Gradient Analysis for Linear Seismic Arrays, BSSA, Vol. 97, No. 1B, 265-280.

(2)   Langston, C. A.(2007), Wave Gradiometry in Two Dimensions, BSSA, Vol. 97, No. 2, 401-416.

(3)   Langston, C. A.(2007), Wave Gradiometry in the Time Domain, BSSA, Vol. 97, No. 3, 926-933.

(4)   Langston, C. A. C. Liang (2007), Wave Gradiometry for polarized waves, in review.

(5)   Liang, C., C. A. Langston (2007), Wave Gradiometry for USArray, in review.

(2) Application to the USArray: