castle - a theory of normal moveout

GEOPHYSICS, VOL. 59, NO. 6 (JUNE 1994); P. 983-999, 11 FIGS., 1 TABLE.

A theory of normal moveout

Richard J. Castle*

ABSTRACT

Three geophysical principles are shown to be suffi-cient to determine the most general, practical normalmoveout (NMO) equation. The principles are reci-procity in a common midpoint (CMP) gather, finiteslowness, and exact constant velocity limit. The re-sulting equation is the shifted hyperbola NM0 equa-tion that has three parameters. Comparisons at bothnear and far offsets between the shifted hyperbolaNM0 equation and the results for layered mediaassign geophysical meaning to the parameters. Two ofthe parameters, zero offset time and NM0 velocity,are constants and control the very near offset behav-ior. The third parameter is dimensionless and controlsthe far offset behavior of the NM0 curve, but it maybe a function of offset so as to exactly fit any travel-time curve. The parameters may be found by a linearleast-squares fit to data. The theory applies to alloffsets for nonturning wave reflections in an isotropicearth for both P-waves and converted (P - SV)waves.

INTRODUCTION

The normal moveout (NMO) equation in use throughoutthe industry today,

(1)

originated with Dix (1955) as a small offset approximation fora horizontally layered-earth model, as shown in Figure 1. Inthis equation, t is the traveltime from the source to thereflector and back to the receiver, is the two-way verticaltraveltime from the surface to the reflector,x is the distancefrom the shot to the receiver, and

where is the interval velocity of the kth layer and isthe vertical traveltime in the kth layer.

Geometrically, this NM0 equation describes a hyperbolathat is symmetric about the t-axis and has asymptotes thatintersect at the origin of the coordinate system (x = 0,t = 0). Figure 2 illustrates this geometry. The solid curve,labeled “true,” is a ray-traced traveltime curve from alayered-earth model. The dashed curve, labeled is theDix NM0 curve for the model, where the rms velocity of themodel has been used in evaluating equation (1). The asymp-totes of the Dix NM0 curve are shown as dashed lines. Thatthe Dix NM0 equation is a small offset approximation isevident from this figure.

Bolshix (1956), again working with a layered-earth model,obtained the NM0 equation

where the time-weighted moments of the velocity distribu-tion are given by

refer to the standard NM0 formula as the Dix NM0 equationand the Dix formula as the relation between interval velocities andrms velocities for a horizontally layered medium.

Presented at the 58th Annual International Meeting, Society of Exploration Geophysicists. Manuscript received by the Editor May 13, 1993;revised manuscript received October 22, 1993.*Chevron Petroleum Technology Company, P.O. Box 446, La Habra, CA 90633-0446.© 1994 Society of Exploration Geophysicists. All rights reserved.

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and = Equation (3) is presented as the first fourterms of a Taylor’s series expansion of traveltime as afunction of offset. Bolshix does not give a formula forcalculating all the terms of this series, although he doesindicate how they might be derived one at a time.

Taner and Koehler (1969) give the following equation for.

FIG. 2. Geometry of Dix NM0 equation.

and the coefficients are given by equation (4). AlthoughTaner and Koehler give explicit expressions for only the firstfive they give a recursion relation that can be used to findthe rest.

Equation (5) is an exact series expansion of t However,this radius of convergence of this series is unknown.

If Bolshix’s equation is squared and then compared withthe equation of Taner and Koehler, it is seen that the twodisagree in the sixth order term. It will be shown later in thispaper that the sixth order term in Bolshix’s equation is inerror.

Malovichko (1978, 1979), apparently unaware that therewas anything wrong with Bolshix’s equation, recognizedthat Bolshix’s equation constituted the first four terms ofGauss’s hypergeometric series, which has a known analyticsum. Assuming that if Bolshix’s equation were extended tomore terms, it would still closely mimic Gauss’s hypergeo-metric series, Malovichko derived the shifted hyperbolaNM0 equation

Geometrically, this NM0 equation describes a hyperbolathat is symmetric about the t-axis and has asymptotes thatintersect at = t = This curve is a shifted hyperbolain the sense that it is a Dix NM0 curve shifted in time by Figure 3 illustrates this geometry. The solid curve, labeled“true,” is the same ray-traced traveltime curve shown inFigure 2. The dashed curve, labeled “shifted,” is the shiftedhyperbola NM0 curve for the model. The asymptotes areshown as dashed lines.

Comparing Figures 2 and 3, it is evident that the shiftedhyperbola NM0 equation is a better fit to the ray-tracedcurve than is the Dix NM0 equation. It is also clear thatthe shifted hyperbola NM0 equation, like the Dix NM0equation, is a small offset approximation. However,Malovichko’s derivation of this equation, unlike Dix’s deri-vation of the Dix NM0 equation, does not give an under-standing of the nature of this approximation.

That a shifted hyperbola is a more accurate NM0 equationthan the Dix NM0 equation was shown by de Bazelaire(1988) using arguments from geometrical optics. These argu-ments, however, do not give formulas that relate the subsur-face geology to the constants in the NM0 equation.

A Theory of Normal Moveout 985

Castle (1988) derived Malovichko’s result from first prin-ciples and showed that when represented as a series in t equation (10) is exact through fourth order in offset, whilethe Dix NM0 equation is exact only through second order inoffset.

Ohanian (1993) extended de Bazelaire’s analysis and ob-tained Malovichko’s result.

In this paper, I argue that the mathematical form of themost general practical NM0 equation can be determinedfrom a few simple geophysical principles or Theidea of practicality is central to the argument; I do not seeka solution to the traveltime problem for any arbitrary earthmodel. NM0 is a concept peculiar to the problem of stackingdata and to making some comment as to the subsurfacevelocity structure.If this cannot be done with a simpleformula, then NM0 and stacking become impractical andprestack migration is used. Thus, my approach is to look forthe most general NM0 formula that meets a few simplegeophysical requirements that I feel are necessary for NM0to be useful.

My conclusion is that the most general, practical NM0equation is the shifted hyperbola equation. This conclusionreflects common experience. Imagine poking a stick to thebottom of a still stream. One sees the stick bending at theair-water interface, and the bottom of the stick appearscloser than it really is. This closer appearance is just the shiftterm in the shifted hyperbola NM0 equation, and the factthat the stick still appears straight, in the water layer, meansthat the raypaths still appear straight, hence the hyperbolicterm. Careful calculation and knowledge of velocity func-tions are necessary to get the shifted hyperbola parameters,but the form of the equation is foretold by the stick in thewater.

In the next section, What Must an NM0 Equation LookLike?, the argument for the general form of an NM0equation is presented. This section also shows how toproduce more elaborate shifted hyperbola NM0 equationsby allowing the parameter S to be a function of offset. Thesection, Stacking and Velocity Analysis, examines stackingand velocity analvsis for shifted hvperbolas. The final sec-

FIG. 3. Geometry of shifted hyperbola NM0 equation.

tion, Converted Waves, shows how to apply shifted hyper-bola theory to converted waves.

Mathematical derivations are in the appendixes. The long-est, Appendix A, NM0 for a Layered Earth, contains thederivation for a layered-earth model of an exact series ex-pansion (Result 4) of traveltime as a function of offset; this isa new result. The shifted hyperbola NM0 equation is thenderived as a small offset approximation to this series, and theDix and shifted hyperbola NM0 equations are comparedwith the exact series (Result 5). Appendix B, Radius ofConvergence for NM0 Power Series, is a discussion ofthe radii of convergence of the NM0 series derived inAppendix A. Appendix C, Residual Normal Moveout, pre-sents an addition theorem for NM0 (Result 6), and Appen-dix D, The Effect of a Datum Shift, gives a formula forrelating velocities measured at different datums (Result 7).Both of these results are new.

WHAT MUST AN NM0 EQUATION LOOK LIKE?

NM0 is a practical pursuit with three objectives: to alignevents on a CMP gather to improve the S/N of the stackeddata, to form a stack that is a good approximation to a zerooffset section, and to provide a measurement of the subsur-face velocity. An NM0 formula is an approximate equationthat relates offset and traveltime. To be useful, the approx-imation must be good enough to adequately stack data. Thefollowing three geophysical assertions about the nature ofNM0 equations provide a basis for determining the mathe-matical form of NM0 equations.

Assertion 1, Reciprocity

On a common midpoint (CMP) gather, the traveltimecurve of a reflection event is symmetric about the time axis.

Mathematically this means that time is an even function ofoffset.

Assertion 2, Finite slowness

For all offsets and all times, the slowness dt/dx is finite.Another way of saying this is that the apparent velocity is

never zero.

Assertion 3, Constant velocity limit

In the limit as the earth velocity approaches constantvelocity, an NM0 equation must approach the exact result,

lim tv

where is the zero-offset time and v is the velocity of themedium.

This assertion reflects the fact that although we stack dataover areas where the velocity varies by factors of two orthree, the rate of change of velocity is slow. For this reason,true NM0 curves look a lot like constant velocity NM0curves. If this were not true, NM0 and stacking as is nowpracticed by the industry would not work.

Let us assume that an NM0 equation can be written as

= (15)

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This form satisfies the reciprocity assertion. Furthermore,assume that both and are polynomials:

The reason for considering an NM0 equation of the form ofequation (15) where f and g are both polynomials is practicalfor it allows the coefficients to be found from a linear least-squares fit to data.

The finite slowness assertion places a condition on therelationship between f and g. From equations (15), and(17) the slowness is

implies t it follows from equation (15) that = and, hence,

For the slowness to remain finite in this limit, we require

(20)

In other words, the highest power of cannot exceed thehighest power of t.

The first approximation of the kind we have been consid-ering is

(21)

which can be solved for t to get the shifted hyperbola NM0formula,

where the standard shifted hyperbola NM0 equation param-eters are given in terms of a, b, and c. This equation satisfiesall three assertions. A direct derivation of the shifted hyper-

bola NM0 equation for a layered-earth is given in AppendixA, where it is shown how to calculate the parameters fromthe velocity function.

The next order of approximation is

which can be solved for the NM0 equation

where h is some function of x2. This equation does notsatisfy Assertion 3 because as the velocity model approachesconstant velocity, the cube root does not approach a squareroot. The problem is that the coefficient of t3 in equation (26)is assumed to be nonzero when solving for t to getequation (27), but this coefficient must go to zero forconstant velocity. Mathematically, this can be handled bytaking limits, but for a NM0 formula to be practical,arithmetic must suffice for its evaluation. Geophysically,Assertion 3 is stating the empirical fact, based on decades ofstacking data, that reflection traveltime curves, mostly, looka lot like hyperbolas. Thus, a practical NM0 formula willreduce to a hyperbola for constant velocity.

This objection occurs for all higher order approximationsthan the shifted hyperbola. Thus we conclude:

Result 1

The general form of a practical NM0 equation is that of ashifted hyperbola.

In Appendix A it is shown that for a layered earth, theshifted hyperbola NM0 equation is exact through fourthorder in offset for t2 as a series expansion in offset. Thus, theshifted hyperbola formulation of NM0 is a better small offsetapproximation, t = + + + error, than is theDix NM0 formulation, = + + error.

That a shifted hyperbola is a good representation of faroffset behavior can be seen from Figure 4. In this figure,assume that is near critical, so that sin Changing a few degrees either way does not change thetraveltime t = cos much, so that can be takenas some constant when is near critical. However, smallchanges in induce large changes in the traveltime in thesecond layer, But,

where = Thus the total traveltime is a shiftedhyperbola:

FIG. 4. Two-layered earth model for long offset discussion.


A little thought shows that the same argument holds for layers, the only change is that the raypath for is no longerstraight.

We have seen that for a layered-earth the shifted hyper-bola representation of works for both both small andlarge offsets. However, and are functions of offset.For example, in Figure 4 the velocity is for small offsetsand for offsets such that is near critical. Result 5,Appendix A, shows that for “reasonable” offsets, thisbehavior can be described by the following form of theshifted hyperbola formula:

where = is a constant. Using the binomial theoremto expand the square root and keeping terms only throughsecond order in offset gives the small offset behavior of theshifted hyperbola equation:

(29)

Thus, the small offset behavior of equation (28) does notdepend upon S. The role of S is to prescribe the offsetbehavior of the terms in the shifted hyperbola equa-tion; hence, although constant S in the shifted hyperbolaformula fits large offsets better than the Dix NMO formula,we can fit a traveltime curve exactly by allowing S to varywith offset. From equation I define

(30)

where t(x) is the exact traveltime curve.The most general form of the shifted hyperbola NMO

equation is

(31)

where the offset dependence is explicit. From Result 1 thisequation applies to all earth models. We now assume that theconclusion of the immediately preceding argument aboutlayered-earth models is true, in general, to obtain the follow-ing:

Result 2

The offset dependence of and in equation (31) is

(32)

where the function S(x) is given by

and the constant is defined by the small offset limit,

0 l

In this result, in the layered-earth calculations has beenreplaced by because there is no reason to assume thatthis constant is in general rms velocity.

Inserting the exact series from Result 4 ofAppendix A into equation (30) gives S as a function of offsetfor a layered-earth:

S(x) =

- - 2k

00

k = l 2k

To second order in offset, equation (37) is

s + =

where

and the are given by equation (4).Figure 5 shows the results of a modeling experiment that

FIG. 5. Error curves for four NMO equations.

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illustrates the concepts discussed above. The model is asubsurface with velocity function

v(z) = 1800 + 0.62,

where is depth and mks units are used. The depth of thereflector is 1500 m. In the figure, the abscissa is the ratio ofoffset to depth of the reflector and the ordinate is thedifference between the true traveltime and the NM0 equa-tion. The true traveltime was calculated using equation (2) inSlotnick (1959, 252). Four NM0 equations are shown: theDix NM0 equation, labeled Dix; the shifted hyperbola NM0equation with constant S, labeled S; the shifted hyperbolaNM0 equation with S given by equation (38), labeled and the exact series expansion, Result 4 of Appendix A,truncated after the x6 term, labeled TS.

First consider the truncated exact series. It is almost exactto an offset about equal to the depth of the reflector, afterwhich it precipitously diverges from the true traveltime. Thisrapid divergence occurs because the slowness of thetruncated series grows as x5. It is this effect that inspiredAssertion 2 above (C. W. Frasier, private communication).

The Dix NM0 curve is almost exact to an offset of abouthalf the depth of the reflector, while the shifted hyperbolaNM0 curve with constant S is about exact to an offset equalto the depth of the reflector. In fact, the shifted hyperbolacurve is a better approximation than the Dix curve at alloffsets. However, neither of these curves diverges from thetrue travel as fast as the truncated series because theirslownesses approach a constant for large offset.

This shifted hyperbola NM0 curve with S given byequation (38) is about exact to an offset of about 1.5 times thedepth of the reflector, after which it diverges. The pole atx = does not cause the NM0 curve to become un-bounded since t = Thus defining S by equation(38) produces an NM0 equation that is close to exact foroffsets about half the depth of the reflector larger than forconstant S, but then diverges more quickly. The reason forthe quick divergence is that the series in equation (37) aretruncated to form Also, see Appendix B for adiscussion of the radii of convergence of these series.

STACKING AND VELOCITY ANALYSIS

The preceding section illustrated the use of the shiftedhyperbola NM0 equation to predict the traveltime curve ofa given layered-earth model. We now turn to the practicalstacking problem: given a traveltime curve, how well canthe curve be described by the shifted hyperbola NM0equation, and how good a measurement of rms velocitycan be made? In the following argument, no assumptionabout the subsurface is made when discussing the fitting of ashifted hyperbola NM0 curve to data, but a layered, isotro-pic earth is assumed when discussing the measurement ofV

The starting point in equation (28), which can be written as

(42)

where V is rms velocity. Equation (42) is the fundamentalequation for performing a least-squares fit of a shiftedhyperbola to data. If we set S = 1, then equation (42)

becomes = which fits a Dix hyperbola tothe data, with the stacking velocity and zero offset timegiven by V = and = respectively. For DixNMO, the stacking velocity and rms velocity are taken to bethe same.

If S is an unknown constant, then

which fits the data to a shifted hyperbola withThe shifted hyperbolaparameters are given by

constant S.

For very long offsets it may be necessary to allow S to bea function of offset. Such an S must be a function of an evenpower of offset to satisfy Assertion 1. It might seem that S =

+ would be a good first choice for S as a functionof offset. However, if this choice of S is inserted intoequation (28), then for large offsets, the functional form ofthe NM0 equation is

which is ungeophysical. Equation (38) suggests

S

when equation (42) becomes

(51)

Solving for t:

It follows directly that

The limit of equation (52) for small offset is


which, when compared with equation (29), gives

for the rms velocity.This technique for deriving NM0 equations may be ex-

tended to S as a rational polynomial of higher orders of offsetif necessary.

Result 3

The shifted hyperbola NM0 equation in the form given byequation (42) may be fit to data in the following manner:

1) If constant S is adequate, use equations (43)-(49) to fita shifted hyperbola to data.

2) If S must be a rational polynomial in x2, as shown inequation (50), use equations (51)-(55) to fit a shiftedhyperbola to data.

3) If S as a rational polynomial of higher orders of offset isrequired, then the following technique, which was usedtoa)

c)

d)

e)

derive equations may be used:Write S as a rational polynomial of even powers ofoffset. The highest power of offset in both thenumerator and the denominator must be the samefor the shifted hyperbola NM0 equation to begeophysical in the far offset limit.Insert S-into equation (42). Multiplying the resultingequation by the denominator of S and collectingterms will result in an equation that can be fit to thedata by least squares.Write the equation in the previous item as + bt +c = 0 and use the quadratic formula to get the NM0formula.Setting offset to zero in this NM0 formula gives thezero offset time.Expand the NM0 formula as a series expansion inoffset, keeping only the constant and the x2 terms.Comparing this result with equation (29) gives arelation for .

6 and 7 illustrate these results with a simpletwo-layer earth model. The earth model is shown in Figure 6.This model was ray traced to determine the true NM0curve. Offset was evenly sampled from zero to 20 km inincrements of 100 m. Since the depth to the reflector is 2 km,the maximum offset corresponds to an offset to depth ratio of10: 1. A Dix NM0 hyperbola, a shifted hyperbola withconstant S, and a shifted hyperbola with S given byequation (50) were fit to this NM0 curve using the tech-

niques of Result 3. The results are shown in Figure 7. Plottedin this figure is the difference between the true traveltime andthe traveltime predicted by the least-squares fit of the data tothe various NM0 hyperbolas. Figures 7a-7c show the indi-vidual results, and Figure 7d shows the composite of allthree. Note the different vertical scales. From Figure 7, it isclear that the progression Dix, to constant S shifted hyper-bola, to shifted hyperbola with S given by equation (50), is asignificant improvement in matching the data at each step.The Dix hyperbola does not produce an acceptable stack;the shifted hyperbola with constant S would produce amarginally acceptable stack; and the S(X) shifted hyperbolaproduces a good stack.

In addition to producing a stack, the fitting of an NM0curve to data is used as a measurement of rms velocity. Theresults of the velocity measurements for the data from themodel in Figure 6 are shown in Figure 8. The ordinate is themeasured rms velocity from the fit of a Dix NM0 hyperbola,labeled Dix, from a shifted hyperbola with constant S,labeled S, and from a shifted hyperbola with S given byequation (50), labeled S(X). The abscissa is the maximumoffset used for the fit. The progression Dix, to constant Sshifted hyperbola, to shifted hyperbola with S given byequation (50), is a significant improvement in measuring rmsvelocity.

CONVERTED WAVES

To apply shifted hyperbola theory to converted waves, theequivalence between the traveltime of the reflected rayshown in Figure 9a and the transmitted ray shown inFigure 9b is used. Because of this equivalence, the travel-time of the reflected ray in Figure 9c is exactly twice that ofthe reflected ray in Figure 9a. Hence Result 2 holds forconverted waves with

2

(56)

where and are the P-wave zero offset time and rmsvelocity, and are the S-wave zero offset time and rmsvelocity, and = + is the converted wavezero-offset time. The relation for the velocity moments ofAppendix A is

FIG. 6. Two-layered earth model for examplesquaresfitting of NM0 curves.

of least-

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FIG. 7. Error plots for fitting NM0 curves to two-layeredmodel data: (a) Dix NM0 equation; (b) shifted hyperbolaNM0 equation with constant S; (c) shifted hyperbola NM0e uation with S a rational polynomial in x 2 ; (d) composite ofa - c . FIG. 9. Converted wave raypaths.


CONCLUSIONS

It is possible to determine the most general practical NM0equation from the three geophysical requirements of reci-procity, finite slowness, and correspondence with the limitof a constant velocity earth. The result, equation (31), is theshifted hyperbola NM0 equation. The three offset depen-dent coefficients, and in this equationmay be expressed, equations (32)-(34), as functions of twoconstants, zero-offset time and NM0 velocity, and the singlefunction S(X) given by equation (35). Equation (35) may beviewed as a prescription for finding the shifted hyperbolaNM0 equation that will exactly match any given traveltimecurve.

The shifted hyperbola NM0 equation may be fit to datausing linear least-squares techniques by choosing S(x) to bea rational polynomial of powers of x2, where the highestpower of x is the same in both the numerator and thedenominator. Result 3 is a prescription for formulating theleast-squares problem and for finding the zero-offset timeand NM0 velocity. For a layered medium, the NM0 veloc-ity is an estimate of rms velocity.

NM0 for converted waves may also be done using shiftedhyperbola theory.

REFERENCES

Abramowitz, M., and Stegun, I. A., 1964, Handbook of mathemat-ical functions with formulas, graphs, and mathematical tables:National Bureau of Standards Applied Mathematics Series, 55,U.S. Government Printing Office.

de Bazelaire, E., 1988, Normal moveout revisited-Inhomogeneousmedia and curved interfaces: Geophysics, 53, 142-58.

Bolshix, C. F., 1956, Approximate model for the reflected wavetraveltime curve in multilayered media: Applied Geophysics, 15,3-14, (in Russian).

Castle, R. J., 1988, Shifted hyperbolas and normal moveout: Pre-sented at the 58th Ann. Internat. Mtg., Soc. Expl. Geophys.,Expanded Abstracts, 894-896.

Dix, C. H., 1955, Seismic velocities from surface measurements:Geophysics, 20, 68-86.

Malovichko, A. A., 1978, A new representation of the traveltimecurve of reflected waves in horizontally layered media: AppliedGeophysics, 91, 47-53, (in Russian).

- 1979, Determination of the zero offset effective velocity andthe degree of velocity nonhomogeneity from a single reflectedwave traveltime curve in the case of a horizontally layered media:Applied Geophysics, 95, 35-44, (in Russian).

Ohaman, V., 1993, Virtual images and normal moveout: Presentedat the 63rd Ann. Internat. Mtg., Soc. Expl. Geophys., ExpandedAbstracts, 1126-l 129.

Slotnick, M. M., 1959, Lessons in seismic computing: Soc. Expl.Geophys.

Taner, M. T., and Koehler, F., 1969, Velocity spectra-digitalcomputer derivation and applications of velocity functions: Geo-physics, 34, 859-881.

Todorov, P. G., 1979, New explicit formulas for the coefficients ofp-symmetric function: Proc. Am. Math. Soc., 79.

Todorov, P. G., 1983, Inverting power series by means of the diBruno precise formula: Plovdivski Universitet . Nauchni Trudove,21, 103-108 (in Bulgarian).

APPENDIX ANM0 FOR A LAYERED EARTH

This section is divided into three parts. The first presentsthe derivation of the shifted hyperbola NM0 equation for thehorizontally layered earth model. The second part comparesthe Dix NM0 equation and the shifted hyperbola NM0equation. The last shows that there is a natural, dimension-less parameter describe NMO.

Derivation for horizontally layered earth model

Figure 1 shows the raypath for a reflection in a horizon-tally layered earth. The shot is at S, the reflection point is R,and the geophone is at G. The shot to receiver offset isx andthe shot to receiver midpoint M lies directly above thereflection point. Hence, the line MR is perpendicular to theline SMG. Since the angle of reflection equals the angle ofincidence, the ascending raypath is the reflection of thedescending raypath about the line MR. What I propose to dois calculate the traveltime from S to R to G as a function ofthe offset x. Starting with the traveltime equations in para-metric form, an exact NM0 equation is derived wheretraveltime is given as a series expansion in offset. Thisequation is a new result. Next, the shifted hyperbola NM0equation is derived as an approximation to the exact result.

An exact series expansion for NMO.-The starting point forthis derivation is the parameteric equations for traveltimeand offset as functions of ray parameter, Slotnick (1959):

where t is traveltime, x is offset,p is ray parameter, is thethickness of layer k, is the velocity in layer k, and isthe number of layers. These equations are valid for all valuesof p such that 1 for all Using the binomialtheorem to expand the denominators of these equationsgives

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and the symbol (2j ! is defined as

In these equations, is the two-way vertical traveltime tolayer N,

1

and is the 2jth time-weighted moment of the velocitydistribution,

where the last equality follows from equation (A-10). Toevaluate the coefficients in equation equation (A-4)must be inverted forp as a function of For compactness,define

so that equation (A-4) becomes

(A-13)

(A-14)

whence, according to equation 12 of Todorov (1983),

So far all the series we have considered have converged aslong as critical refraction did not occur. The radius ofconvergence of the series in equation (A-15) is less general.The discussion of this radius of convergence and the con-vergence of the remaining series in this section will be put offuntil Appendix B so as not to distract the reader from thelogic of the derivation of the shifted NM0 equation.

The coefficients are given by

=

Equations (A-3) and (A-4)are parametric equations fornormal moveout. Expanding t in a Taylor’s series about zerooffset, x = 0, yields

and the symbol is defined by


recursively and, thus, eliminate the need to figure out whatthe

Differentiating equation (A-15) gives

where is the smallest non-negative integergreater than or equal to and + is definedby equation (A-17). Evaluating equation (A-22) at = 0gives

Inserting this result into equation (A-12):

(k k even =

0 k odd.(A-24)

Thus, from equation (A-9), we get the following theorem:

Result 4

An exact normal moveout equation for the horizontallylayered earth model is

(A-25)

(A-26)

Equation (A-25) is a complete solution to the normalmoveout problem for a horizontally layered-earth. As acheck, this result is now compared with the results publishedby Taner and Koehler (1969). Using equations (A-16) and(A-13):

1 1

(A-27)

0

(A-28)

(A-29)

so that the first four terms of equation (A-25) are _ _

1 l

(A-30)

The first three terms of Bolshix’s (1956) NM0 equation,equation (3), agree with this result. The last term ofBolshix’s equation is incorrect, being only part of the lastterm of equation (A-30).

If equation (A-30) is now squared and only terms throughx6 kept, then the following equation is produced:

This agrees with the first four terms of the equation derivedby Taner and Koehler (1969); the conversion from theirnotation to mine is

The shifted hyperbola NM0 equation.- The shifted hyper-bola NM0 equation is now derived as an approximation tothe exact result given in equation (A-25). This approximationis exact through fourth order and a very good approximationto the sixth’ order term as well. The meaning of “very goodapproximation” will be given in the next section. Thestarting point is to look at those terms in equation (A-25)which depend only upon and

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where the last equality defines the coefficients . The seriesin equation (A-37) is reminiscent of Gauss’s hypergeometricseries,

The ratio of the coefficients for the series in equations (A-37)and (A-41) are:

The first five are given in Table A-l. From this table it isseen that the series in equations (A-37) and (A-41) are thesame through the x2 term. Therefore, if the series in equa-tion (A-37) is replaced with 1; 2; then the resultingexpression for t will be the same as in equation (A-41)through the term. 1; 2; can be summed to asimple analytic form, Abramowitz and Stegun [1964, 556,equation 15.1.13]:

Thus

Equation (A-45) is the shifted hyperbola NM0 equation.

Comparison with Dix NM0 equation

Let us now try to estimate the nature of the differencebetween the shifted hyperbola NM0 equation and the DixNM0 equation. The shifted hyperbola NM0 equation isequation (A-37) with the series replaced by 1; 2; which has the simple analytical form shown in equation(A-43). If instead of using this simple analytical form for

1; 2; the series expansion equation (A-41) is used,then the shifted hyperbola NM0 equation is

The first four terms of equation (A-47) are

The first four terms of the exact NM0 expansion are given inequation (A-30). Let be the difference

Thus the shifted hyperbola NM0 equation is exact throughfourth order inx. Also, vanishes identically fora constant velocity medium and is small for media with smallaccelerations, which is true of most geologies. Hence, the


shifted NM0 equation is also a “very good approximation”to the sixth order term.

To compare the shifted and Dix NM0 equations, squareequation (A-48):

+ + + (A-50)

where the first four terms are correct, as was shown inequation (A-31), and the last term is the error in the sixthorder term of the shifted hyperbola NM0 equation. The DixNM0 equation is

. (A-51)

These two equations show the differences between theshifted and Dix NM0 equations. Both are exact throughsecond order in x; however, the Dix NM0 equation endswith second order in x, while the shifted hyperbola NM0equation is exact through fourth order in x and, for mostgeologies, presents a“very good approximation” to thesixth order term as well.

The results of this and the previous section are summa-rized by the following theorem:

Result 5

For a horizontally layered-earth model, the shifted hyper-bola NM0 equation,

and

2

where

is exact through fourth order in offset and, for most geolo-gies, a very good approximation through sixth order in offset.

The natural parameterization of NM0

Result 4 presented an exact NM0 equation. In this result,traveltime is expressed as a series expansion in even powersof offset. Thus, the dimensions of the coefficients are differ-ent for every term. What I propose to show in this section isthat there is a natural, dimensionless variable that can beused for the expansion parameter, resulting in a seriesexpansion for NM0 in which all the coefficients have thedimensions of time.

The offset dependence of every term in the series ofequation (A-25) is of the form

The quantity has dimension of time. To get adimensionless quantity we need to divide by time. The onlynatural time variable is to. Thus, define the dimensionlessparameter by

to (A-52)

whence equation (A-25) becomes

(A-53)

where

(A-54)

This is the natural parameterization of NMO. It can be givengeophysical meaning by the following relationship:

offset

depth to reflector

A P P E N D I X BRADIUS OF CONVERGENCE FOR NM0 POWER SERIES

What I wish to calculate here is an upper bound for theradius of convergence of the exact normal moveout equationfor the horizontally layered-earth, equation (A-25) of AppendixA. This equation results from the parametric equations (A-3)and (A-4) which are valid for all values ofp up to the criticalangle. The critical step in going from the parametric equationsto the NM0 equation is the inversion of the power series foroffset as a function of ray parameter to the power series for ray

parameter as a function of offset. Explicitly, given that equa-tion (A-14) is valid for allp such that the rays do not go critical,what is the radius of convergence of equation (A-15).

The starting point is equation (A-2). Equation (A-14) is thisequation with the right-hand side expanded using the bino-mial theorem. The expansion is valid as long as the ray doesnot go critical; that is, c 1 for all k. To discuss theradius of convergence of equation (A-14), we must let x and

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p be complex. Since a power series and the function itrepresents are the same inside the radius of convergence ofthe series, we see from equation (A-2) that the radius ofconvergence of equation (A-14) is

1

max

where is the largest of the The radius of convergence, of equation (A-15) the

smaller of the two following conditions: (1) smallest value of for which the has a pole, or (2) the radius of

the largest disk, centered at the origin, such that the mappingof the disk by equation (A-14) lies entirely withinthe disk Letp = Since a is constrainedto be finite by equation (B-l), we are interested in values ofx given by

The result of this limit is a bound on the radius of conver-gence of equation (A-15)

APPENDIX CRESIDUAL NORMAL MOVEOUT

Suppose that NM0 has been removed from an event usingthe NM0 function

but that NM0 should have been removed using the function

In this section, I propose to calculate the residual normalmoveout (RNMO) which, when applied to data that has hadNM0 removed according to will produce the sameresult that would have been obtained if the original data hadhad NM0 removed using

The RNMO will be viewed as being composed of twoparts: a constant term, and an offset dependent term,

so that

(C-1)

The constant term is given by

= = +

where = (0) and = (0).The offset dependent term is

or

The quantity inside the square root may be closely approx-imated by a perfect square:

To show that the second term on the right-hand side of thisequation is negligible when compared to the first, the twoterms are written in terms of the natural NM0 parameter.

The first term on the right-hand side of equation (C-5) is


Combining equations (C-6) and (C-7) gives the right-handside of equation (C-5) in terms of the natural NM0 param-eter:

then the right-hand side of equation (C-5) is approximately aperfect square:

Inserting equations (C-2) and (C-13) into equation (C-l)produces the following theorem about the form of RNMOcurves:

Result 6

The residual normal moveout left in the data after remov-ing NM0 using the NM0 curve

where is the natural NM0 parameter.The geometrical relationships expressed by this theorem

are illustrated in Figure C-l. Of particular interest in thisfigure is the relation that the magnitude AT of the RNMO onthe far offset trace is equal to This relation is aconsequence of the following observations:

1) The NM0 curve and the data intersect at the pointlabeled A on the far offset.

2) Removing NM0 using moves the data at point A onthe far offset to the point labeled B at zero offset time

l3) To flatten the data curve, the data at point B must be

moved to point C at a zero-offset time of =

This relation between and the RNMO on the far offsettrace, the value of the RNMO on the far offset trace, and the

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value of the RNMO at some other offset is sufficient infor-mation to calculate the constants for the RNMO curve.

In practice the concept of RNMO is most useful when thevelocity function used to initially remove NMO, the functionT1 (X) in the above result, is close to being correct. Thisimplies, and is implied by, the condition that inFigure 10 is small, typically not more than 200 ms. If this istrue, then 0 because only shifts the NM0curve and does not change its shape. Thus for most practicalapplications of RNMO, the approximation = 0 can bemade. Making this approximation leads to several simpleresults.

If srnmo = 0, then from equation (C-2),

(C-14)

From Figure C-l, is equal to minus the RNMO on thefar offset trace:

-AT, (C-15)

where AT is positive if Thus the RNMO curve,from equation (C-l) and Result 6, is

FIG. C-l. Geometry of residual normal moveout.

The geometry of this RNMO curve is shown in Figure C-2.The parameter may be calculated from the value of theRNMO curve at the far offset,

Solving this equation for

Using these results the proper NM0 curve, to haveused to originally stack the data can be calculated:

FIG. C-2. Residual normal moveout for = 0approximation.

A Theory of Normal Moveout 9 9 9

APPENDIX D

THE EFFECT OF A DATUM SHIFT

One result of the shifted hyperbola NM0 equation is asimple expression for the change in rms velocity implied by achange of datum. The situation envisioned here is a datum shiftby an amount so that the new time coordinate is related tothe old by

From equation (A-46)

where the primed coordinate system is the datum-shiftedsystem. The NM0 curve in the datum-shifted system is which leads to the following result.

Result 7

which impliesThe effect of a datum shift, on the rms velocity implied

by an NM0 curve is

(D-1)

castle - a theory of normal moveout

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