The WKBJ approximation requires that the minimum wavelength involved in the problem is bigger than the lateral gradient between the constituent structures. If the lateral heterogeneous model is made up of two or more juxtaposed vertically heterogeneous structures, this means that two adjacent structures have to be very close in the parameter’s space to allow the application of the WKBJ approximation.
Focusing the attention on a model made up by only two structures (1D, laterally homogenous), this problem is solved introducing between them a set of substructures that have the goal to “smooth” the gradient of the lateral variation, so that the new laterally varying model presents weak lateral heterogeneities, where weak is meant in the sense of the wavelength.
Let us suppose that the two structures present both a different stratification (different layer's thickness) and different elastic and anelastic parameters. We need to determine the distance at which the two structures have to be placed (length of the smoothing zone) and the number of substructures that have to be interposed between them.
6.1.1.1 Computation of the length of the smoothing zone
The WKBJ approximation implies that:
c p
p
(6.1)
where p is a structural parameter and the gradient is computed along the direction of the
lateral heterogeneity.
With respect to the development by La Mura (2009), where condition (6.1) is applied when p is the density, the P-wave velocity and the S-wave velocity, respectively, here we consider p=c, i.e. the phase velocity. This choice is justified by the fact that modal summation technique works in the (,c ) space.
Let us assume a reference system with a downward z-axis and the x–axis (direction of the lateral heterogeneity) positive from left to right. Using p = c, (6.1) can be written as:
quantity we seek for, is the distance at which the two structures must be placed so that the lateral heterogeneity can be considered weak or, equivalently, the length of the zone needed to smooth the lateral heterogeneity.
where the phase velocity difference in (6.3) is computed between the homologous modes and frequencies. Taking the maximum over m and f, we have:
maximum difference between phase velocities, i.e. the greater the difference, the larger the length of the zone boundary needed to smooth the heterogeneity.
Numerical test show that a satisfactory lower boundary value that can be used to satisfy the double inequality in (6.4), without loosing too much in the accuracy of the computed seismograms, is 5.
6.1.1.2 Computation of the number of substructures
L* must be filled with a set of contiguous laterally homogeneous structures, obtained by means of linear interpolation of the elastic parameters characterizing the two structures initially adjacent. The number n of these substructures is determined applying a modified
sequence of juxtaposed, vertically heterogeneous subregions that are laterally homogeneous, in order to determine how dense a sequence of subregions is needed to obtain “satisfactory”
accuracy level in the computed time series. In fact, quantitatively accurate statements concerning the necessary density of laterally-homogeneous subregions along a propagation path really require the comparison of theoretical seismograms computed with fixed source and for successively higher densities of subregions.
To express the subregion density required to model accurately any given lateral heterogeneity, Schwab (1994) introduced the parametermin /( h )max, where min is the minimum wavelength involved and ( h )max is the maximum step size in the staircase modelling with subregions. On a qualitative level he asked how large the parameter min/( h )maxmust be before the wave motion is effectively unable to sense the difference between the true structure and the approximation. Comparing time series computed for successively higher densities of subregions, Schwab found that a subregion division with:
20
is necessary to be sure to keep the noise level caused by the subregions approximation well below the “satisfactory” accuracy level. The value 20 in (6.5) has been chosen as a conservative value, but in some cases a lower value can be sufficient. In general, the Schwab criterion can be expressed as:
10
We modified criterion (6.6), formulated for the thickness of the layers, adapting it to the phase velocities. Since c / f , and substituting h with c , (6.6) becomes:
Finally, the number, n, of substructures to be interposed between the two initial structures is given by:
min
) ( ) ( max max
c
f c f
n m f cm m
10 1 2 (6.9)
and the length of each substructure is given by L / n.
6.1.2 3D grid
Once the smoothing procedure (see paragraph 6.1.1) is applied to all the structures in contact that model the study area, the 3D model is simply constructed by distributing all the set of vertically heterogeneous structures and substructures on a regular grid, associated with a Cartesian reference system (x-axis is longitude and y-axis is the latitude). Each knot of the grid is therefore occupied by a vertically heterogeneous anelastic structure (1D structure).
The grid can be rectangular or squared and the grid steps (x and y) are determined as the minimum substructure length along x and y, respectively. This choice assures the validity of the WKBJ approximation.
An example of the grid is shown in Figure 6.1.
Figure 6.1 - Example of a grid; the star stands for the source, the triangle stands for the receiver; the coloured bullets stand for the different vertically heterogeneous sections; the arrows show the Cartesian axes x and y.
Figure 6.2, if the point (x, y) is within the cell whose corner points are (xi, yi), (xi+1, yi), (xi,
Differentiating (6.10) we obtain the first derivatives:
)
An analogous method for interpolation of the phase attenuation C2(x,y) is used.
Figure 6.2 - Schematic representation of a cell of the grid with the coordinates of the corner points.