Geophysical Ray Tracing - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 316

Fluids, by contrast, are characterized by viscous losses, and is
approximately proportional to f 2 (Clay, 1990). Toksoz and Johnston (1981), for
example, give an approximate law for seismic damping at low frequencies,
A = A 0 e - x cos 2 f(t - x/c) (6.54)
= f 2 /cf 0 (6.55)

where is an absorption coefficient, f represents frequency (in Hertz), c is the
sound speed, and the positive transition frequency f 0 is empirically determined
constant of the formation. We stress that more detailed models taking into
account fluid saturation, pore pressure, rock properties, and so on, are available
in the literature, e.g., Yew and Jogi (1976), White (1984), De la Cruz and
Spanos (1985), McCann and McCann (1985), and Hassanzadeh (1991).
The use of Equations 6.54 and 6.55, for illustrative purposes only, does not
imply any agreement or disagreement with present research views on dissipation
modeling. Our objective is a tutorial one, emphasizing the development of high
order corrections to the eikonal equation, which will be different for different
dissipation models. Because our kinematic wave formalism for damped
oscillations was developed entirely in terms of the imaginary frequency function
i (k), we need to determine the i (k) consistently with Equations 6.54 and 6.55.
Now , s i nc e f = /2 and = ck, we can write rewrite Equation 6.55 in the
wavenumber-explicit form = 2 /(4cf 0 ) = c 2 k 2 /(4cf 0 ) = ck 2 /(4f 0 ). Then,
because x = ct, the exponential term in Equation 6.54 can be manipulated to give
an expression for the imaginary frequency

i (k), that is,

exp (- x) = exp {-{ck 2 /(4f 0 )} x}

= e x p { - { c 2 k 2 /(4f 0 )}t}

= e x p {

i (k)t} (6.56)

where we have followed the sign convention of Chapter 1 (refer to the
discussion immediately following Equation 2.30). Thus,
i (k) = - c 2 k 2 /(4f 0 ) < 0 (6.57)
as required. For this model,
i k (k) = - c 2 k/(2f 0

) < 0 (6.58)

i kk (k) = - c 2 /(2f 0

) < 0 (6.59)

i kkk (k) = i kkkk (k) = i kkkkk (k) = ... = 0 (6.60)
so that simplifications to the general high-order kinematic wave modulation
equations of Chapter 2 are definitely possible. In the next section, we will show
how formulas such as these will lead to improved modulation laws that apply
even when Fermat' s Principle does not.