Apparent velocity: v x ,v y ,v z Consider a wavefront arriving at an array of detectors. For simplicity, use only two dimensions, (x, z),

2.9 Apparent velocity: v x ,v y ,v z Consider a wavefront arriving at an array of detectors. For simplicity, use only two dimensions, (x, z),

and let the wavefront make an angle θ with respect to the horizontal (θ is called the emergence angle of the wave). The detectors are spaced at intervals of ∆x at z = 0 (Figure 2.11). If the medium

42 CHAPTER 2. VELOCITY immediately beneath the receivers has the acoustic velocity, v 0 , then the wavefront arrives at x + ∆x

later than it does at x by the delay

Thus it appears to move along the array at the speed

The quantity v x is called an apparent velocity because the wavefront appears to move at that speed even though its true speed is v 0 . Apparent velocities are one of the four fundamental observables in seismic data, the others being position, time and amplitude. The apparent velocity is never less than the real velocity and can range up to infinity. That is v 0 ≤v x ≤ ∞. Since infinities are cumbersome to deal with it is common to work with the inverse of v x , called the time-dip, ∆t/∆x or horizontal slowness, s x . Another common convention for horizontal slowness is to call it p which signifies the rayparameter.

Now, enlarge the thought experiment to include an array of receivers in a vertical borehole and spaced at ∆z. Reasoning similar to that used before shows that the arrival at z − ∆z is delayed from that at z by

Therefore, the vertical apparent velocity, v z , is

Using simple trigonometry, it results that

If this argument is extended to 3D, the result is that there is an apparent velocity in the y direction as well and that the sum of the inverse squares of the apparent velocities is the inverse square of the real velocity:

We observe apparent velocity, or equivalently time-dip, by simply choosing an event of interest on

a seismic record and measuring the slope, ∆t/∆x. It will be seen shortly that this is a measurement of the ray parameter of the ray associated with the chosen event. Knowledge of the ray parameter essentially determines the raypath, provided that the velocities in the subsurface are known. This allows the ray to be projected down into the earth and to possibly determine it’s reflection point. This technique, called raytrace migration, will be discussed in section 4.2.2.

Another way to measure apparent velocities is with a multi-dimensional Fourier transform. In 2-D for example, the f-k transform represents a seismic record on a grid of horizontal wavenumber, k x , and temporal frequency, f . Each point in the (k x ,f ) plane has a complex number associated with it that gives the amplitude and phase of a fundamental Fourier component: e 2πi(k x x−ft) . In 3D these fundamental components are monochromatic (i.e. a single f ) plane waves so in 3D or 2D they are called Fourier plane waves. These waves are infinite in extent so they have no specific arrival time. However, it does make sense to ask when a particular wave crest arrives at a particular

43 location. Mathematically, a wave crest is identified as a point of constant phase where phase refers

2.9. APPARENT VELOCITY: V X ,V Y ,V Z

to the entire argument of the complex exponential. If the Fourier transform has the value Ae iφ at some (k x ,f ), then the Fourier plane wave at that point is Ae 2πi(k x x−ft)+iφ . The motion of a point of constant phase is tracked by equating the phase at (x, t) with that at (x + ∆x, t + ∆t) as in

(2.39) from which it follows that

2πi(k x x − ft) + iφ = 2πi(k x (x + ∆x) − f(t + ∆t)) + iφ

Thus the ratio of f to k x determines the horizontal apparent velocity of the Fourier plane wave at (k x ,f ). Radial lines (from the origin) in the (k x ,f ) plane connect points of a common apparent velocity. The advantage of the Fourier domain lies in this simultaneous measurement of apparent velocity for the entire seismic section. The disadvantage is that the notion of spatial position of a particular apparent velocity measurement is lost.

If equation (2.38) is evaluated for the Fourier plane wave e 2πi(k x x+k y y+k z z−ft) the result is

Now, for any monochromatic wave, we have λf = v and, using k = λ −1 , then equation (2.41) leads to

(2.42) This very important result shows that the coordinate wavenumbers, (k x ,k y ,k z ), are the components

k 2 =k 2 x +k 2 y +k 2 z .

of a wavenumber vector, #k. Using a position vector, #r = (x, y, z), the Fourier plane wave can be written compactly as e 2πi( k· r−ft) . Equation (2.42) can be expressed in terms of apparent wavelengths

(e.g. k x =λ −1 x , etc.) as

which shows that, like apparent velocities, the apparent wavelengths are always greater than the true wavelength. An important property of the wavenumber vector is that it points in the direction of wave propagation (for an isotropic, homogeneous medium). An easy way to see this is to take the gradient of the phase of a Fourier plane wave. Since the wavefront is defined as a surface of constant phase, then wavefronts can be visualized as contours of the phase function ˜ φ = πi(#k · #r − ft). The gradient of this phase function points in the direction normal to the wavefronts, that is the direction of a raypath. This gradient is

where ˆ x, ˆ y, and ˆ z are unit vectors in the coordinate directions. Calculating the indicated partial derivatives leads to

∇˜ # φ=k x x+k ˆ y y+k ˆ z z = #k. ˆ

(2.45) This equation can be achieved more simply by writing # ∇ = ˆr ∂ ∂r , where r = (x 2 +y 2 +z 2 ) and ˆ r

is a unit vector pointing to a particular location, so that

∂#k · #r φ=ˆ r

=ˆ r #k

#k.

∂r

44 CHAPTER 2. VELOCITY

∂ zˆ ∂ φ ˜ ∂ y+ yˆ ∂ φ ˜ z + = x ∇ ˜ φ # ∂ φ ∂ x ˜ ˆ

medium 1

medium 2

Figure 2.13: Snell’s law results from the physical requirement that the apparent velocity along an interface be conserved.

So, the inverse apparent velocities are proportional to the components of the wavenumber vector that points in the direction of wave propagation.