Graphical method for interval velocity measurement

A wave of velocity v from a point source at location $(x,z)\,=\,(0,z_s )$ passes any point (x,z) at time t where  

$$v^2 \ t^2 \eq x^2\ +\ ( z \ -\ z_s )^2$$ (39)

In equation (39) x should be replaced by either half-offset h or midpoint y. Then t is two-way travel time; the velocity v is half the rock velocity; and $(z \,-\, z_s )$ is the distance to an image source.

Differentiating (39) with respect to t (at constant z) gives

$$\begin{eqnarray} v^2 \ 2 \ t\ \ \ &=&\ \ \ 2 \ x\ {dx \over dt } \\ v^2 \ \ \ &=&\ \ \ {x \over t }\ {dx \over dt }\end{eqnarray}$$ (40) (41)

Figure 14 shows that the three parameters required by (41) to compute the material velocity are readily measured on a common-midpoint gather.

 

tangent Figure 14 A straight line, drawn tangent to hyperbolic observations. The slope p of the line is arbitrary and may be chosen so that the tangency occurs at a place where signal-to-noise ratio is good. (Gonzalez)

Equation (41) can be used to estimate a velocity whether or not the earth really has a constant velocity. When the earth velocity is stratified, say, v(z), it is easy to establish that the estimate (41) is exactly the root-mean-square (RMS) velocity. First recall that the bit of energy arriving at the point of tangency propagates throughout its entire trip with a constant Snell parameter $p\,=\,dt/dx$.

The best way to specify velocity in a stratified earth is to give it as some function v' (z). Another way is to pick a Snell parameter p and start descending into the earth on a ray with this p. As the ray goes into the earth from the surface $z \,=\, 0$ at $t \,=\, 0$,the ray will be moving with a speed of, say, v (p,t). It is an elementary exercise to compute v (p,t) from v' (z) and vice versa. The horizontal distance x which a ray will travel in time t is given by the time integral of the horizontal component of velocity, namely,  

$$x \eq \int_0^t \ v (p,t) \ \sin \, \theta \ dt$$ (42)

Replacing $\sin \, \theta$ by pv and taking the constant p out of the integral yields  

$$x \eq p \ \int_0^t \ v(p,t)^2 \ dt$$ (43)

Recalling that $p\,=\,dt/dx$,recopy (41) and insert (43):

$$\begin{eqnarray} v_{\rm measured}^2\ \ \ &=&\ \ \ {x \over t }\ {dx \over dt } \... ...m measured}^2\ \ \ &=&\ \ \ {1 \over t }\ \int_0^t \ v(p,t)^2 \ dt\end{eqnarray}$$ (44) (45)

which justifies the assertion that  

$$v_{\hbox{measured}} \eq v_{\hbox{root-mean-square}} \eq v_{\rm RMS}$$ (46)

Equation (45) is exact. It does not involve a ``small offset'' assumption or a ``straight ray'' assumption.

Next compute the interval velocity. Figure 15 shows hyperboloidal arrivals from two flat layers.

 

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Figure 15 Construction of two parallel lines on a common-midpoint gather which are tangent to reflections from two plane layers. (Gonzalez)

Two straight lines are constructed to have the same slope p. Then the tangencies are measured to have locations ( x₁ , t₁ ) and ( x₂ , t₂ ). Combining (44) with (42), and using the subscript j to denote the $j^{{\rm th}}$ tangency ( x_j , t_j ), gives  

$$x_j \ {dx \over dt } \eq \int_0^{{t}_j} \ v(p,t)^2 \ dt$$ (47)

Assume that the velocity between successive events is a constant $v_{\rm interval}$, and subtract (47) with j+1 from (47) with j to get  

$$( x_{j+1} \ -\ x_j )\ {dx \over dt } \eq ( t_{j+1} \ -\ t_j ) \ v_{\rm interval}^2$$ (48)

Solving for the interval velocity gives  

$$v_{\rm interval}^2 \eq { x_{j+1} \ -\ x_j \over t_{j+1} \ -\ t_j } \ {dx \over dt }$$ (49)

So the velocity of the material between the $j^{{\rm th}}$ and the $j+1^{{\rm st}}$ reflectors can be measured directly using the square root of the product of the two slopes in (49), which are the dashed and solid straight lines in Figure 15. The advantage of manually placing straight lines on the data, over automated analysis, is that you can graphically visualize the noise sensitivity of the measurement, and you can select on the data the best offsets at which to make the measurement.

If you do this routinely you quickly discover that the major part of the effort is in accurately constructing two lines that are tangent to the events. When you run into difficulty, you will find it convenient to replot the data with linear moveout $t' \ =\ t\,-\,px$.After replotting, the lines are no longer sloped but horizontal, so that any of the many timing lines can be used. Locating tangencies is now a question of finding the tops of convex events. This is shown in Figure 16.

 

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Figure 16 Measurement of interval velocity by linear moveout. (Gonzalez)

In terms of the time t', equation (49) is  

$$v_{\rm interval}^2 \eq {1 \over {\Delta t \over \Delta x} }... ...q {1 \over {\Delta t' \over \Delta x \ } \ +\ p}\ {1 \over p }$$ (50)

Earth velocity is measured on the right side of Figure 16 by measuring the slope of the dashed line, namely $\Delta t' / \Delta x$, and inserting it into equation (50). (The value of p is already known by the amount of linear moveout that was used to make the plot).