Parametric velocity fitting

It is desirable to start the Monte Carlo process at an initial estimate of interval velocity that is in the right ``ball park'' relative to the semblance information. Such an initial estimate can save time in minimizing the number of Monte Carlo random walks needed to obtain satisfactory convergence, and provides a basis for v_i and $\bar{v}_r$ velocity constraints. To derive an initial velocity estimate, I use the following parametric form of rms velocity:

 

$$\bar{v}_s\sim v_o + \alpha t^{\beta} \;.$$ (3)

This equation is a general power-law fit to many typical classes of rms velocity functions, ranging from constant velocity, to constant gradient, to polynomial fits.

The parameter v_o is the $\bar{v}_r$ value at time zero (although the t value could be shifted if required), and for marine data may have a range of 1.4 - 1.6 km/s. The parameter $\alpha$ is related to the rms velocity gradient with traveltime. In fact, for the special case that $\beta$ = 1, $\alpha$ is precisely the gradient. It has a typical range of 0.1 - 1.0 km/s/s. The parameter $\beta$ is related to the rate of increase or decrease in velocity with time, and is ``loosely'' related to the ``curvature'' of the velocity function with time. For a constant velocity medium, $\alpha$ = 0, and for a constant gradient medium, $\beta$ = 1. Typical marine sediment velocity profiles have an exponent of $\beta \sim 0.5$.

Given the parametric form (3), an analytic expression for the interval velocity can be easily obtained after a little algebra, by assuming (3) is an rms average:

 

$$v_i\sim \sqrt{ v_o^2 + 2v_o\alpha(1+\b)t^{\b} + (1+2\b)\alpha^2 t^{2\b} } \;.$$ (4)

In practice, I make a coarse discretization of the parameter space $\{v_o,\alpha,\b\}$, perform an exhaustive search over that domain for the rms curve (3) that, when integrated against the velocity semblance scan, maximizes total semblance. This process gives an optimal three parameter set $\{v_o^*,\alpha^*,\b^*\}$, which, when substituted into (4) yields the initial estimate of interval velocity. Figure a demonstrates the initial parametric fit on a typical semblance scan. The curve that lies closest to the semblance peaks is the $\bar{v}_s$ curve given by (3) evaluated at the $\{v_o^*,\alpha^*,\b^*\}$which maximize semblance. The smooth curve of higher average velocity is the parametric interval velocity curve v_i given by (4) evaluated at $\{v_o^*,\alpha^*,\b^*\}$.

The implementation of the parametric velocity search is diagrammed below:

INITIAL PARAMETRIC VELOCITY ESTIMATE
====================================
	* Choose parametric form:  Vrms = Vo + a * t**b
		- Vo = Vrms(t=0)
		- a   vertical velocity gradient
		- b   velocity "curvature"

1.4 <  Vo  < 1.6  
			0.0 <   a  < 1.0  
			0.0 <   b  < 2.0
	* Exhaustive search over finite parameter domain {Vo,a,b}
	* Retain Vrms*(Vo*,a*,b*) that maximizes semblance
	* Vrms* -> Vint*  is analytic
	* RESULT:
		- a Vrms* parametric fit that is globally "optimal"
		- a Vint* model that is physically reasonable
		- a starting point for MC random walks and constraints

The parametric fit in Figure a seems remarkably reasonable. The amount of velocity misfit is subtle compared to the Monte Carlo fit shown in Figure b. This can be seen by examining a section stacked with parametric $\bar{v}_s$ velocities. Figure shows a section stacked with a 1-D velocity function, which is an optimal Monte Carlo fit at midpoint 7 km. Figure has been stacked with the smoothed 2-D parametric velocity field. Figure is the optimal stack using the smoothed 2-D Monte Carlo velocity picks, to be discussed in the next section.

The 1-D velocity stack in Figure is coherent at the west where the single Monte Carlo velocity analysis was done, but fades to the east due to significant lateral velocity variation that is not accounted for by the 1-D stacking velocity. The 2-D parametric velocity stack is very coherent across the entire line in Figure . Finally, the 2-D Monte Carlo velocity stack is the best, as shown in Figure . The Monte Carlo stack is superior in its resolution of shallow fault structure and fault diffractions, as well as within the fault block at 12.5 km and 2 seconds depth. However, for a ``quick and dirty velocity field,'' the parametric search can be very fast and effective.

 

cmpfit
Figure 2 (a) Initial parametric velocity fit, (b) Monte Carlo nonlinear velocity fit.

 

stack1
Figure 3 Stacked section using a 1-D velocity function, obtained by an optimal Monte Carlo fit at midpoint 7 km.

 

gstack
Figure 4 Stacked section using 2-D smoothed parametric velocities, obtained by the controlled parameter search method.

 

mcstack
Figure 5 Optimal stacked section using 2-D smoothed Monte Carlo velocities, obtained by the nonlinear random walk search method.