Case 2: spatially smooth mass density variation

Assume that the mass density $\rho({\bf \underline{x}})$ is smooth in some sense on the scale of the dominant pressure wavelengths of concern. This leads us to examine the relative magnitude of the two terms on the left hand side of (13):

 

$$\vert \rho\d_i\ddot{u}_i \vert \,\,\,\mbox{ and }\,\,\, \vert \ddot{u}_i\d_i\rho\vert \,.$$ (18)

It is convenient to normalize the two terms by $\vert\rho\ddot{u}_i\vert$ and scale by the Fresnel length L_f, since spatial changes in properties are only meaningful with respect to the spatial wavelengths and frequency components in the propagating wavefield:

 

$$R_1 = \frac{\vert \d_i\ddot{u}_i\vert}{\vert \ddot{u}_i\vert... ...}\,\,\, R_2 = \frac{\vert\d_i\rho\vert}{\vert\rho\vert} L_f \,.$$ (19)

Examining these two terms, it is apparent that (1) is valid when

 

$$\frac{ \vert\d_i\rho\vert }{\vert\rho\vert} \ll \frac{\vert \d_i\ddot{u}_i\vert}{\vert \ddot{u}_i\vert}$$ (20)

and

 

$$v_2^2({\bf \underline{x}}) = \lambda({\bf \underline{x}}) / \rho({\bf \underline{x}}) \,.$$ (21)

As a matter of practical interest, consider the following example. Assume a propagating wavefield in a region with a phase velocity v₂ of 3 km/s, an average density $\tilde{\rho}$ of 2.0 g/cc, and a dominant frequency content of 30 Hz. The dominant spatial wavelength will be on the order of 100 m, and we will assume that the wave energy will be coherent over a quarter-wavelength Fresnel zone of L_f = 25 m. A reasonable ``background'' density gradient characteristic of the Gulf of Mexico would be about 0.2 g/cc/km. Using these values, and approximating

$$R_2 \approx \frac{\vert \Delta\rho\vert}{L_f} \cdot \frac{L_... ... = \frac{\vert \Delta\rho\vert}{\vert \tilde{\rho} \vert} \,,$$ (22)

where $\tilde{\rho}$ is an average over the Fresnel zone and $\Delta\rho$ is the total density contrast over the Fresnel zone, I evaluate the density gradient term R₂ to be $\approx 0.25$%. Similarly, for the displacement gradient term,

$$R_1 \approx \frac{\vert \Delta\ddot{u}_i \vert}{\vert \tilde... ...vert \ddot{u}_{max} \vert}{ 1/2 \vert \ddot{u}_{max} \vert} \,,$$ (23)

or, $R_1 \approx 200$%. For this realistic exploration example, $R_2 \ll R_1$ by about three orders of magnitude! In fact, scaling this result implies that a density gradient of up to 16 g/cc/km over a 25 m Fresnel zone, or a total density contrast of 0.4 g/cc/L_f, will result in an error of less than 10% if the variable velocity acoustic equation (1) is used:

$$\nabla\rho\leq 16 \mbox{ g/cc/km } \,\,\,\mbox{ implies } \,\,\, R_2/R_1 \leq 10 \mbox{\%} \,.$$ (24)

On the other hand, this result also implies that very large contrasts in density and very high frequency data may be sufficient to invalidate the use of (1). The validity condition (20) is likely to be violated at strong geologic discontinuities such as shale/carbonate boundaries or at basement volcanic unconformities given sufficiently high frequency data.