SÍMBOLOS DE MÉXICO

for our heterogeneity model comes from th e tran sm itted mode, the sum of the in­ cident and unconverted scattered modes, which is characterized by both a phase shift and change in am plitude relative to the incident mode. As dem onstrated by Snieder (1988d) for m edia sm oothly varying with respect to dom inant wavelength, th e phase is th e more sensitive of the two param eters to heterogeneity and, in our model, causes equivalent tim e delays/advances of up to one half period for wavefronts passing through the regions characterized by negative/positive velocity p ertu rb atio n s. This effect is most pronounced for the higher-order tran sm itted modes (ZÜ3 and up) since these contain a higher proportion of their energy over the range of depths where the m agnitude of heterogeneity is greatest. The effects on th e am plitude of the tran sm itted mode are a little less dram atic (variations of a few percent) b u t nonetheless interesting. In all cases, the am plitude of the tra n s­ m itted m ode is greater along the positive x-axis (lowest velocities) th an along the negative x-axis (highest velocities). For the low order tran sm itted modes, R l , R 2, th e m axim um am plitude occurs along the positive x-axis and is greater than th a t of th e incident mode (i.e. the am plitude which would be observed in the absence of any heterogeneity). For higher-order tran sm itted modes however the am pli­ tu d e is greatest in directions perpendicular to the x axis, and less than th a t in th e incident m ode over the entire range of azim uths. This contrast in behaviour between low and high order tran sm itted modes stem s, again, from the fact th a t higher modes carry a greater proportion of their energy over the interval with the g reatest velocity perturbations. As a result th e higher modes are more sensitive to th e heterogeneity and actually undergo significant reflection from the large radial gradients in heterogeneity along the positive and negative x-axes. Thus energy propagating in this direction is trap p ed within the internal boundaries of the h e t­ erogeneity, which results in a reduction in the am plitude of the tran sm itted mode observed in th e far-field (see for example the radiation p a tte rn for R6 —► Ä6, figure 5.9). In contrast, th a t portion of the incident wavefield propagating along th e y-

axis encounters no large im pedance contrast and so remains unaffected. The lower order modes do not experience significant reflection and hence only the effects of

focussing and defocussing, as predicted from geom etrical optics, are observed.

5 .7 4 Model 2

T he next model exhibits the same azim uthal variation in m aterial properties, th a t is a cos0 dependence but differs in th a t the heterogeneity shifts to greater depths with radial distance from the source (see figure 5.7). The ‘p e ak ’ in figure 5.8a is located at 10 km depth (as in model 1) at th e inner boundary, r = 0.5 km, but falls to 20 km depth at the outer boundary r = 9.5 km. Viewed in 3- D this may appear as a som ewhat unlikely representation of the real earth b u t it serves to illustrate, by comparison with model 1, the m anner in which th e scattered wavefield is affected by heterogeneity which extends system atically through a range of depths.

T he scattered energy spectra for modes 771, 773 and 776 are shown in figure 5.10 and do indeed deviate som ewhat from the corresponding plots for model 1 where th e heterogeneity largely confined to a specific depth level. Energy scattered from incident 771 into 771 and 772 is now much decreased since much of the heterogeneity now occurs at depths below which the fundam ental mode carries significant energy. T he scattered energy spectra for incident 773 differs prim arily in the relative d istri­ bution of energy in adjacent modes; and it is apparent th a t this particular choice of heterogeneity model has served to reduce significantly th e energy scattered into 774 in favour of 775. A similar effect is noted for incident 776, where th e relative proportions of energy in the modes 774 and 778 is up considerably a t th e expense of adjacent modes 775 and 777. R adiation p attern s (not shown) are sim ilar in general to those for model 1 and indicate as expected th a t the m ajor contributing modes to the scattered field contain most of their energy in azim uthal order m = 1. T he principal differences arise, again in modes where the dom inant contributions come from first order scattered converted modes.

5. 75 Model 3

O ur final model is more complicated in its horizontal description and comprises equal contributions from cos0, cos20 and cos3# terms. The net effect of this config-

— Rl —RI — Rl —LI Rl —R6 R 3 -R 4 — R 3 -R 3 — R 3 -L 4 — R 6 -R 6 — R 6 -L 5 _{— R 6 -R 7}

F ig u r e 5.9. Far-fielcl radiation p attern s for a selection of scattered modes for incident modes

R l , R3

R6

Incident Mode: R1

Incident Mode: R3

Incident Mode: R6

Modal Order

Figure 5.10.

Transmitted scattered energy spectra for incident modes 721, 773