DYNAMICS AS PERTURBATION OF THE KINEMATICS

Before going ahead, it is important to answer the following question: is the reflection response of an interface separating two layers equivalent to the sum of the responses of the two interfaces that separate each layer from the background medium, as the separation between the layers goes to zero? This question is illustrated in Figure . The answer is yes, but only if we take into account all the multiple conversions and internal reflections inside the background sandwiched layer.

I found however, after several numerical simulations (see Figure ), that under some conditions (on the deviation of the perturbations from the background) a good approximation can be achieved by the consideration of only the primary P energy. A theoretical consideration of this approximation is painful for the elastic case, but its possible to have a good feeling of the limitations involved by the analysis of the simple case of normal incidence. The reflection coefficient when the two layers are in direct contact is

$$R = { I_2 - I_1 \over I_2 + I_1 },$$

where I_x is the impedance of medium x (P velocity $\times$ density).

When the two layers are separated by the background layer whose thickness approaches zero, the reflection coefficient will be (if we consider only primaries)

$$R^{\prime} = { I - I_1 \over I + I_1 } + { I_2 - I \over I_2 + I },$$

where I is the impedance of background medium. This last expression can be put into the following form:

$$R^{\prime} = F({\scriptstyle {I_1 \over I},{I_2 \over I}}) \: R,$$

where

$$F({\scriptstyle {I_1 \over I},{I_2 \over I}}) = {2 \: [ {I_... ...I_2 \over I}] \over [1 + {I_1 \over I}] [1 + {I_2 \over I}] }.$$

The elastic response from an interface separating two layers (I₁ and I₂) is modeled as the superposition of the elastic responses from the two interfaces that separate each layer from an hypothetical layer (I).

The ratio between modeled and actual normal incidence reflection coefficient, as a function of the background impedance, for fixed impedance between the two layers.

Figure  shows how the ratio (F) between the two reflection coefficients behaves for fixed I₁ and I₂, and variable I. If the impedance of the background medium lies between the impedance of the two media, the ratio is very close to unity for any practical case (for instance, F_max= 1.03 for I₂/I₁=2).

The differences between the reflectivities for non-normal incidence were computed for some specific cases corresponding to a background P velocity of 2500 m/s and a maximum variation of 500 m/s around it for the perturbation layers. The results are shown in Figure . All graphs start at p^1/2=0.012 (which corresponds to an angle of 21 degrees in the background layer) because the errors are too small for smaller values. For figures (a), (b), (e) and (f) (in which the velocity of the slowest medium (1) is held constant), the error starts to grow near the critical angle of medium 2 (when its velocity is larger than the background) or near the critical angle of the background (when medium 2 is slower than the background). On the other hand, when the velocity of the fastest medium (2) is fixed (figures (c), (d), (g) and (h)), the error begins to increase near the critical angle of medium 2. We conclude then that the approximation is valid, up to the critical angle of the faster of the three media involved.

All figures correspond to v_min=2000m/s, v_max=3000m/s, v_back=2500m/s, $\rho_1=\rho_2=\rho_{back}=1.0$g/cc (v refers to P velocities, $\rho$ to density, and indexes 1, 2 and back, to media 1, 2 and background respectively). Different curves correspond to increasing P velocity of medium 1 or 2, according to the following convention: continuous=2167, dot=2333, dot-dash=2500, fine-dash=2667, large-dash=2833. (a) $v_1 = v_{min} = {\textstyle const.}$, $\sigma_1=0.1$, $\sigma_2=0.3$,$\sigma_{back}=0.22$ ($\sigma$ represents Poisson's ratios), and v₂ has a different value for each curve (according to previous convention). (b) Same as a, except that $\sigma_{back}=0.3$. (c) Same as a, except that $v_2 = v_{max} = {\textstyle const.}$ and v₁ varying. (d) Same as c, except that $\sigma_{back}=0.3$. (e), (f), (g), and (h), are equivalent to the previous four, with the only difference of $\sigma_1=0.3$, $\sigma_2=0.1$.