Reflection moveout

Moveout is the Doodlebugger's term (still in use today) for the progressive delay of reflection arrivals with increasing source-receiver offset. Historically, a series of approximations have addressed this problem:

Hyperbolic Moveout

For a homogeneous isotropic overburden above a horizontal reflector, the P-wave moveout times are given by

$ t^{2}(x)=t^{2}(0)+{\frac {x^{2}}{v^{2}}}. $

(1)

where $ v $ is the overburden velocity. This hyperbolic equation is just a statement of the Pythagorean Theorem.

A simple application of Taylor's theorem results in a generalization of this:

$ t^{2}(x)\approx t_{0}^{2}\left[1+{\frac {x^{2}}{t_{0}^{2}v_{NMO}^{2}}}\right]. $

(2)

This is just a first-order expansion of $ t^{2}(x) $ in normalized small offsets $ {\frac {x^{2}}{t_{0}^{2}v_{NMO}^{2}}} $. This hyperbolic moveout equation is just mathematics, with almost no physics at all. However, it is useful because:

$ \;\;\;\;\;\; $a. It reduces to (1) in the case of homogeneous overburden.

$ \;\;\;\;\;\; $b. It obeys the Scalar Reciprocity Theorem. (For example, an expansion of $ t(x) $ in terms of $ {\frac {x}{t_{0}v_{NMO}}} $ is not the same when source and receiver positions are interchanged (so that $ x\rightarrow -x $), so such an expansion would not be valid.)

Equation (2) is valid for inhomogeneous, anisotropic media, for sufficiently short offsets. It does not assume straight rays, only short offsets. The parameter $ v_{NMO} $ is empirically determined, by "flattening the gather".

The first analytic treatment of moveout which addressed inhomogeneity of rock properties was due to Dix. For 1D (layered) isotropic media, with sufficiently short offsets:

$ v_{NMO}^{2}=v_{RMS}^{2}\equiv {\frac {1}{t_{0}}}\sum _{i}^{n}v_{i}^{2}t_{i} $

(3)

with the sum over all layers above the reflector. $ v_{i} $ is the (constant) velocity of layer $ i $, and $ t_{i} $ is its two-way vertical travel time. It does not assume straight rays, but observes Snell's law at every layer boundary. (There is an implicit assumption here that the layers are coarse layers, so that the wave equation applies, rather than the equation of motion.) This equation underlies the so-called Dix formula for the interval velocities $ v_{i} $.

If the reflector dips, further considerations occur.

Nonhyperbolic Moveout

The hyperbolic equations (2) and (3) were used by the Doodlebuggers to find most of the oil reserves we know today. But, when the utility of AVO became apparent, the industry began to use longer offsets (for AVO leverage), and so these approximations were not sufficient to flatten these longer gathers. So, the obvious thing to do was to make a higher-order Taylor expansion:

$ t^{2}(x)\approx t_{0}^{2}\left[1+{\frac {x^{2}}{t_{0}^{2}v_{NMO}^{2}}}+c_{2}\left({\frac {x^{2}}{t_{0}^{2}v_{NMO}^{2}}}\right)^{2}...\right] $

(4)

where the fourth-order coefficient $ c_{2} $ is a second moveout parameter. Like (2), equation (4) is mostly just mathematics, with the only physics in it being the restriction to even powers of $ x $. It is valid for inhomogeneous, anisotropic media, for sufficiently short offsets (generally somewhat longer than those adequately described by (2)).

If the subsurface is assumed to be a sequence of coarse isotropic horizontal layers, then:

$ c_{2}=-{\frac {t_{0}\sum v_{i}^{4}t_{i}-(v_{NMO}^{2}t_{0})^{2}}{(2v_{NMO}^{2}t_{0})^{2}}} $

(5)

i.e., it does not have to be empirically determined, but can be calculated from the interval velocities found from equation (3). This shows that non-hyperbolic moveout occurs with isotropic layering. However, it has been found that equation (5) usually does not adequately predict the value of $ c_{2} $ found empirically from flattening the gathers, using equation (4).

Further, equation (4) is unstable, overly sensitive to the furthest offsets (because of the quartic powers of x), so a physically-based modification was proposed:

$ t^{2}(x)=t_{0}^{2}\left[1+{\frac {x^{2}}{t_{0}^{2}v_{NMO}^{2}}}+{\frac {c_{2}\left({\frac {x^{2}}{t_{0}^{2}v_{NMO}^{2}}}\right)^{2}}{1+A\left({\frac {x^{2}}{t_{0}^{2}v_{NMO}^{2}}}\right)}}\right] $

(6)

where $ A $ is to be chosen so that, at the furthest offsets, the second term in the denominator dominates the "1", and the factor $ x^{2} $ cancels two of the factors $ x^{2} $ in the numerator, and the times increase with the horizontal velocity. However, equation (6) requires that the processor must determine the third moveout processor from the data; this is normally not feasible. Hence, equation (6) has not been very useful.

Anisotropic Moveout

This impasse was resolved by recognizing the ubiquity of anisotropy. For a homogeneous polar anisotropic overburden above a horizontal reflector, the P-wave short-spread moveout times are given by equation (2), with

$ v_{NMO}^{2}\approx v_{0}^{2}(1+2\delta ) $

(7)

where $ v_{0} $ is the overburden vertical velocity, and $ \delta $ is the near-offset anisotropy parameter. For this same case of homogeneous polar anisotropic overburden, with longer offsets, the moveout is given approximately by:

$ t^{2}(x)\approx t_{0}^{2}\left[1+{\frac {x^{2}}{t_{0}^{2}v_{NMO}^{2}}}-{\frac {2\eta \left({\frac {x^{2}}{t_{0}^{2}v_{NMO}^{2}}}\right)^{2}}{1+(1+2\eta )\left({\frac {x^{2}}{t_{0}^{2}v_{NMO}^{2}}}\right)}}\right] $

(8)

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \eta\equiv \frac {\epsilon - \delta} {1+2\delta} is a combination of previously defined anisotropic parameters. This resembles equation (6), but with only one non-hyperbolic parameter required, hence it is feasible for processors to use.

This idea is commonly used to describe non-hyperbolic moveout in inhomogeneous 1D media, in the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t^2(x)\approx t_0^2\left [1+\frac{x^2}{t_0^2v_{NMO}^2}-\frac{2\eta_{eff}\left (\frac{x^2}{t_0^2v_{NMO}^2}\right )^2}{1+(1+2\eta_{eff})\left (\frac{x^2}{t_0^2v_{NMO}^2}\right )}\right ]

(9)

where the effective non-hyperbolic moveout parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \eta_{eff} is given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \eta_{eff}=\frac{t_0\sum v_{0i}^4(1+2\delta_i)^2t_i-(v_{NMO}^2t_0)^2}{8(v_{NMO}^2t_0)^2}+ \frac{t_0\sum \eta_i v_{0i}^4(1+2\delta_i)^2t_i} {(v_{NMO}^2t_0)^2}

(10)

(compare with (5)), which reduces to (8) for a single layer. Here,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): v^2_{NMO}\equiv \frac{1}{t_0}\sum_{1}^{n}v^2_{0i}(1+2\delta_i)t_i

(11)

with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): v_{0i} the vertical velocity in layer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): i , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_i its short-spread anisotropy parameter.