DUALITIES INVOLVING REFLECTORS AND

REFLECTION-TIME SURFACES

M. Tygel∗, J. Schleicher∗∗, and P. Hubral∗∗

∗ Dept. Appl. Math., IMECC Univ. Est. de Campinas

(UNICAMP), C.P. 6065, 13081-970 Campinas, S.P., Brasil

∗∗ Geophys. Inst., Univ. Karlsruhe, Hertzstr. 16, Geb. 42,

76187 Karlsruhe, Germany

Karlsruhe, January 30, 2015

——————————————————————————–

Submitted to: Journal of Seismic Exploration

1

Abstract

Fundamental geometrical relationships between the 3-D reflection-time surface on the

one hand and the 3-D subsurface reflector on the other hand can be expressed in form of

certain duality theorems. For instance, for each point on the reflector, there is a diffraction-

time surface that is tangent to the reflection-time surface, and for each point on the reflection-

time surface there is an isochrone that is tangent to the reflector. Besides providing a simple

proof of these facts, the first duality theorem also states that the changes of the isochrone

with varying time and of the diffraction-time surface with varying depth are reciprocal to

each other. The second duality theorem expresses a relationship between the curvatures of

the reflection and diffraction time surfaces and of the reflector and isochrone. This allows

to represent the Fresnel-zone matrix and the Fresnel geometrical-spreading factor (a concept

only recently introduced by the authors) as a function of the difference in spatial second

derivatives between (a) the diffraction and reflection time surfaces at the tangency point or

(b) the isochrone and reflector surfaces at the reflection point. The duality theorems are of

fundamental importance in migration and demigration as well as in seismic modeling, reflection

imaging, and traveltime inversion in general.

Introduction

The 3-D seismic reflection method uses data collected from an organized areal or linear

distribution of source-receiver pairs (the seismic experiment). Its general aim is to invert

reflection data for properties pertaining to the subsurface region, which is illuminated by the

seismic experiment. One of the main objectives is to map or image (depth migrate) subsurface

interfaces (i.e., discontinuities of the medium properties such as density and/or velocity) that

reflect the seismic energy back to the measurement surface. This paper provides new interesting

results on the relationship between reflections and subsurface reflectors.

2

The positioning in depth of the seismic reflectors with no regards to the wave amplitudes

involved is called a kinematic depth migration or map migration. When amplitudes (e.g.,

geometrical-spreading factors, reflection/transmission coefficients, attenuation) are taken into

consideration, that is, when changes in amplitudes are quantitatively controlled during the

migration, we talk about amplitude-preserving depth migration. In particular, when the depth-

migrated seismic signals are freed from geometrical-spreading effects (and other amplitude

factors are not affected), we call the process a true-amplitude depth migration. It is well known

that an amplitude-preserving or even a true-amplitude migration can only be performed as

a prestack migration because the common-midpoint (CMP) stack, although improving the

signal-to-noise ratio, is not an amplitude-preserving process.

The two main geometric concepts for migration are the surface of maximum convexity,

now commonly known as diffraction-traveltime or Huygens surface and the surface of equal

reflection times, now commonly known as aplanatic surface or isochrone (?). Both surfaces

are the basis to the methods of diffraction-stack migration (?; ?) or Kirchhoff migration (?).

The same kind of surfaces play also a fundamental role in the method of discontinuities (?; ?).

Given a dense distribution of source-receiver pairs on a measurement surface and a subsurface

target reflector below an inhomogeneous velocity overburden, it is well known that the Huy-

gens surface pertaining to a reflection point on the target reflector and its primary-reflection

traveltime surface are tangent surfaces in the seismic record section (or record space). This

is true irrespective of the measurement configuration that is used. It was already mentioned

by Hagedoorn (1954) that both surfaces are closely related. In this work, we will further in-

vestigate this relationship, which we call duality. This duality between the reflection time and

Huygens surfaces on the one hand and isochrone and subsurface reflector on the other hand

can be expressed in form of two herein called duality theorems. We will observe that both the

first and second derivatives of these surfaces are related to each other. Their curvatures are

also closely related to the Fresnel zone at the reflection point.

3

Moreover, the (high-frequency) diffraction-stack (or Kirchhoff) migration result placed

into the reflection point is proportional to (and to a great extent controlled by) the Fresnel

geometrical-spreading (FGS) factor (?). In this work, we will show that this (generally com-

plex) factor accounts for the contribution to the overall reflection-ray geometrical-spreading

factor which is due to the target reflector only. Except for a multiplicative quantity (which is

closely related to the Beylkin determinant (?; ?)) that depends on the model and measurement

configuration, the FGS factor can be expressed as the difference in second derivatives between

the diffraction and reflection traveltime surfaces at the point of tangency (or stationary point).

These derivatives are taken with respect to the 2-D configuration parameter that specifies the

location of source-receiver pairs on the measurement surface, evaluated at the tangency point.

In this way, the FGS factor can be found from a traveltime analysis without knowing anything

about the reflector at the reflection point. This property is inherently made use of in true-

amplitude migration, where the geometrical-spreading factor of a reflected ray needs to be

eliminated from the migration amplitudes without knowing the reflector. On the other hand,

the FGS factor can also be expressed in terms of quantities pertaining to the subsurface region

(or model space), a result that is required, e.g., for a true-amplitude demigration of imaging

in general. Both results are consequences of the mentioned duality between the reflection-time

and Huygens surfaces on the one hand and the subsurface reflector and isochrone on the other

hand. This duality also allows to represent the FGS factor as the difference in spatial second

derivatives between the isochrone and reflector surfaces at the reflection point. This is partic-

ularly revealing for the understanding not only of migration and demigration processes, but

also in seismic reflection imaging, modeling and traveltime inversion in general.

4

Basic assumptions

To make the migration problem mathematically tractable, a number of simplifying as-

sumptions has to be made. These refer to the subsurface model (and involve, e.g., restrictions

of the spatial variation of medium properties and interface shapes) as well as to the seismic

wave propagation that takes place. In the following, we assume the subsurface to consist of

3-D isotropic elastic layers in which the density and seismic velocities vary smoothly. The

layers are separated by smoothly curved interfaces, across which the density and or velocities

may have zero- or first-order discontinuities or jumps. Finally, the wave-propagation events

that are of interest in this work are, for simplicity, taken to be the elementary P-P reflec-

tions, simply called primary reflections. The above conditions justify the assumption that the

wave propagation can be described by zero-order ray theory (see, e.g., Cerveny, 1985, 1987,

or Kravtsov and Orlov, 1990), with the exception of singular regions of the ray field.

Considering the general 3-D map migration problem, we suppose an areal distribution

of (reproducible) source-receiver pairs densely distributed on the measurement surface, ΣM

(Figure 1b). The location of the source-receiver pairs is defined by the measurement configu-

ration and is described by a 2-D vector (i.e., the configuration parameter), ξ∼, varying on a

planar set, A, called the aperture of the seismic experiment. Because of the assumed dense

distribution of source-receiver pairs on ΣM , we take A to be a (full) planar domain. More

precisely, we assume that all source-receiver pairs (S,G) are uniquely described by functions

S = S(ξ∼) and G = G(ξ

∼) defined in A (for further details, see Schleicher et al., 1993).

One of the interfaces of the 3-D layered subsurface model is to be taken as the target

reflector and denoted by ΣR. We suppose ΣR to be parametrized as z = ζR(r∼), with r

∼

varying on a planar domain E. The coordinates (r∼, z) refer to a global 3-D Cartesian system

(Figure 1b), in which r∼

is a 2-D horizontal vector and the z-axis points in downward direction.

Points on ΣR are denoted by MR, i.e., a point MR has the coordinates (r∼, ζR(r∼

)).

5

The primary-reflection traveltime surface of ΣR, resulting from the chosen configuration,

will be denoted by ΓR (Figure 1a) and parametrized as t = τR(ξ∼), with ξ

∼in A. Also, (ξ

∼, t)

denote the global 3-D Cartesian coordinates of the record section, in which ξ∼

is a horizontal

vector and the t-axis points in upward direction. Points on ΓR are denoted by NR, i.e., a point

NR has the coordinates (ξ∼, τR(ξ∼

)).

Duality

One last, but important, assumption concerns the uniqueness of the one-to-one relation-

ship between points MR on the target reflector ΣR and points NR on its traveltime surface

ΓR. It is stated as follows: Each point NR on ΓR specifies one source-receiver pair (S,G) in

the chosen measurement configuration, which in turn determines one and only one reflection

point MR on ΣR. In other words, there exists a unique primary-reflection ray SMRG only.

Reciprocally, each point MR on ΣR determines one and only one source-receiver pair (S,G)

(and thus one point NR on ΓR) in the chosen measurement configuration, for which SMRG is

a primary-reflection ray. In other words, each point MR on ΣR is the reflection point for one

and only one source-receiver pair in the chosen measurement configuration.

It is useful for our purposes to recast the above one-to-one relationship in terms of the

parametrizations of ΣR and ΓR. We have

(a) For each ξ∼

in A, there exists one and only one r∼R = r

∼R(ξ∼) in E, for which

S(ξ∼)MR(r∼R)G(ξ

∼) is a primary-reflection ray. Here, MR(r∼R) denotes the point on ΣR spec-

ified by its horizontal coordinate vector, r∼R.

(b) For each r∼

in E, there exists one and only one ξ∼R = ξ

∼R(r∼) in A, for which

S(ξ∼R)MR(r∼

)G(ξ∼R) is a primary-reflection ray. Here, MR(r∼

) signifies each point on ΣR

specified by its horizontal coordinate vector, r∼.

The above condition defines a one-to-one correspondence (function) between points

6

MR on the target reflector and points NR on its primary-reflection traveltime surface ΓR.

We call this correspondence duality. Any two corresponding points MR on ΣR and NR on

ΓR are called dual points of each other. Note that, given a fixed point MR(r∼R), its dual

point NR is the point on ΓR that is defined by the configuration parameter ξ∼R(r∼R) and the

traveltime τR(ξ∼R) of the primary-reflection ray that has MR(r∼R) as its reflection point on

ΣR. Reciprocally, given a fixed point NR(ξ∼R), its dual point MR is the specular reflection

point on ΣR with the coordinates (r∼R(ξ∼R), ζR(r∼R)) that is defined by the primary-reflection

ray joining the source-receiver pair specified by NR on ΓR. As this dual property holds for

each pair of points MR(r∼) and NR(ξ∼

), we say that the reflector ΣR in (r∼, z)-space and ΓR

in (ξ∼, t)-space are dual surfaces of each other.

Diffraction and Isochrone Surfaces

For each subsurface point M = M(r∼, z), we introduce the diffraction-traveltime or

Huygens surface ΓM : t = τD(ξ∼;M) with ξ

∼in A. The diffraction traveltime τD(ξ∼

;M) is

defined by

t = τD(ξ∼;M) = τ(S(ξ

∼),M) + τ(M,G(ξ

∼)), (1)

in which τ(S(ξ∼),M) and τ(M,G(ξ

∼)) denote the traveltimes along the rays that join the

source point S(ξ∼) to the subsurface point M and point M to the receiver point G(ξ

∼),

respectively.

In the same way, for any point, N = N(ξ∼, t), in the record section, we introduce the

isochrone surface ΣN : z = ζI(r∼;N), implicitly defined by the set of points, MI = MI(r∼

, z =

ζI(r∼;N)) in the (r

∼, z)-space which satisfies the condition

τD(ξ∼;MI) = τ(S(ξ

∼),MI) + τ(MI , G(ξ

∼)) = t. (2)

The domain of definition of the Huygens surface ΓM (i.e., the set of configuration parameters

ξ∼

for which the function t = τD(ξ∼;M) is defined) depends on the point M and on the

7

macro-velocity model. In the same way, the domain of definition of the isochrone ΣN (i.e., the

set of horizontal spatial vectors, r∼, for which the function z = ζ(r

∼) is defined) depends on

the point N and the macro-velocity model. These sets can, in principle, be even void sets.

Note that both the Huygens and isochrone surfaces (1) and (2) are defined by the very

same traveltime function τD. To obtain the Huygens surface (1), one has to keep the subsurface

point M (i.e., the coordinates r∼

and z) fixed. On the other hand, to obtain the isochrone (2),

one has to keep the section point N (i.e., the coordinates ξ∼

and t) fixed. It will thus not be

very surprising that there exists a fundamental duality not only between surfaces ΣR and ΓR

but also between ΣN and ΓM .

Let us therefore now focus our attention to all Huygens surfaces ΓM defined by points

MR on ΣR and to all isochrone surfaces ΣN defined by points NR on ΓR. It is assumed that

all these surfaces are defined in (nonvoid) subdomains of A and E, respectively. For these

surfaces, the duality can now be readily extended as follows: The Huygens surface ΓM for

point MR and the isochrone ΣN for point NR are dual surfaces of each other when the points

MR and NR are, themselves, dual points.

It is important to recognize that the dual surfaces, ΣR (i.e., the target reflector) and ΓR

(its configuration-dependent primary-reflection traveltime surface) have, of course, a definite

physical meaning as the model and as the observation data. On the other hand, the isochrone

and Huygens surfaces are only to be seen as auxiliary surfaces in the (r∼, z) and the (ξ

∼, t)

domains, respectively. As well known, for each point MR on ΣR, the corresponding Huygens

surface ΓM is tangent to the reflection traveltime surface ΓR (see first duality theorem below)

at the dual point NR of MR. This important geometrical property is the basis for a (Kirchhoff-

type) diffraction-stack migration. Reciprocally, for any point NR on ΓR, the corresponding

isochrone ΣN is tangent to the reflector ΣR at the dual point MR of NR. This provides the

basis for an isochrone-stack demigration (?).

The above geometric properties of the dual points and surfaces are fundamental prop-

8

erties of reflection waves. They involve first-order derivatives (i.e., slopes) of these surfaces

in their respective domains of definition. However, as shown below, also the amplitudes of

primary reflections are related to the Huygens and isochrone surfaces. This involves second-

order derivatives (i.e., curvatures) of these surfaces in their domains of definition. In fact, the

geometrical-spreading factor of a primary reflected elementary wave can be directly inferred

from (a) the second-order derivative matrix, with respect to the configuration parameter ξ∼,

of the difference between the diffraction and reflection traveltime surfaces at the tangency

point NR, or from (b) the second-order derivative matrix, with respect to the horizontal spa-

tial vector parameter r∼, of the difference between the isochrone and reflector surfaces at the

tangency point MR. The precise mathematical formulation of the above statements will be

given in the form of two duality theorems. However, before we are ready to present these, we

find it convenient to introduce a set of useful definitions.

Some useful definitions

The fundamental function to start with is the Huygens or diffraction-time surface

τD(ξ∼;M) defined for each ξ

∼in A and for each M(r

∼, z) in equation (1). Referring to points

MR = MR(r∼, ζR(r∼

)) on ΣR, it is useful to introduce the traveltime function τDR(ξ∼, r∼),

defined for all ξ∼

in A and all r∼

in E, by

τDR(ξ∼, r∼) = τD(ξ∼

;MR) = τD(ξ∼, r∼, ζR(r∼

)) . (3)

From the assumptions above, we know that each source-receiver pair (S,G) determines exactly

one reflection point MR. This means that the horizontal coordinate r∼R that specifies the

specular reflection pointMR(r∼R, ζR(r∼R)) on ΣR for a given source-receiver pair (S(ξ∼), G(ξ

∼))

is a function of the vector parameter ξ∼

that specifies that pair, i.e., r∼R = r

∼R(ξ∼). It follows

that the reflection-time surface τR(ξ∼) is given by

τR(ξ∼) = τDR(ξ∼

, r∼R(ξ∼

)) = τD(ξ∼, r∼R(ξ∼

), ζR(r∼R(ξ∼))) , (4)

9

which describes the traveltime surface ΓR for all ξ∼

in A.

The second fundamental equation (2) implicitly defines the isochrone. Correspondingly

to equation (3), it is also useful to introduce the function τDI(ξ∼, t, r

∼), defined by

τDI(ξ∼, t, r

∼) = τD(ξ∼

;MI) = τD(ξ∼, r∼, ζI(r∼

;N)) ≡ t . (5)

We observe that τDI(ξ∼, t, r

∼) is actually a function independent of r

∼. Furthermore, we con-

clude from equation (5) that, for a given point NR, the first derivative of τDI(ξ∼, t, r

∼) with

respect to t equals unity. All higher derivatives with respect to t as well as all derivatives with

respect to r∼

vanish.

We next consider the Hessian matrices H≈D and H

≈R of the above defined traveltime

surfaces t = τD(ξ∼;MR) and t = τR(ξ∼

), respectively, with respect to ξ∼, viz.,

H≈D =

∂2τD(ξ∼;MR)

∂ξi∂ξj

(i, j = 1, 2) (6a)

and

H≈R =

∂2τR(ξ∼)

∂ξi∂ξj

(i, j = 1, 2) , (6b)

evaluated at ξ∼

= ξ∼R. Correspondingly, we also consider the Hessian matrices Z

≈I and Z≈R of

the isochrone z = ζI(r∼;NR) and the reflector z = ζR(r∼

), respectively, with respect to r∼,

viz.,

Z≈I =

∂2ζI(r∼;NR))

∂ri∂rj

(i, j = 1, 2) (7a)

and

Z≈R =

∂2ζR(r∼)

∂ri∂rj

(i, j = 1, 2) , (7b)

evaluated at r∼

= r∼R. Moreover, let H

≈DR and Λ≈DR denote the second-order Hessian and

mixed-derivative matrices of τDR(ξ∼, r∼), respectively, viz.,

H≈DR =

∂2τDR(ξ∼, r∼)

∂ri∂rj

(i, j = 1, 2) (8)

10

and

Λ≈DR =

∂2τDR(ξ∼, r∼)

∂ri∂ξj

(i, j = 1, 2) , (9)

both evaluated at ξ∼

= ξ∼R and r

∼= r

∼R. Finally, let mD and nI be the vertical stretch factors

(?)

mD =∂τD(ξ∼

, r∼, z)

∂zz = ζR(r∼

)

, (10a)

nI =∂ζI(r∼

, ξ∼, t)

∂tt = τR(ξ∼

)

, (10b) also

evaluated at ξ∼

= ξ∼R and r

∼= r

∼R. It is to be remarked that the latter quantities, Λ≈DR, mD,

and nI do not depend on the curvatures of either the reflector or the reflection-time surface at

the dual points MR and NR, but only on the macro-velocity model. It is shown in Schleicher et

al. (1993), how Λ≈DR can be expressed in terms second-derivative matrices of the traveltimes

along rays SMR and MRG. In Hubral et al. (1992), these in turn are related to ray-segment

propagator matrices that can be computed from dynamic ray tracing. Moreover, it was shown

in Tygel et al. (1994b) that mD can simply be expressed as

mD =2 cosαR cosβR

vR, (11)

where αR is the reflection angle, βR is the local dip angle, and vR is the local velocity just

above the reflector, at MR (Figure 1).

Duality theorems

Having introduced all the necessary mathematical terminology, we are now ready to state

the fundamental geometrical results concerning the reflection-time and Huygens surfaces on

the one hand, and the reflector and isochrone on the other hand, at the dual tangency points

MR and NR. These will be given in the form of two duality theorems that concern the two

vertical stretch factors mD and nI as well as the curvatures of all the involved surfaces. Proofs

11

of the duality theorems are provided in Appendix A.

First duality theorem

Given a fixed pair of dual points MR and NR, the first duality theorem consists of the

following three statements:

Ia) The Huygens surface t = τD(ξ∼;MR) for the point MR is tangent to the reflection

traveltime surface t = τR(ξ∼) at NR.

Ib) The isochrone z = ζI(r∼;NR) determined by NR and the reflector z = ζR(r∼

) are

tangent at MR.

Ic) The vertical stretch factor mD =∂τD(ξ∼

;M)

∂zof the Huygens surface at M = MR

and the vertical stretch factor nI =∂ζI(r∼

;N)

∂tof the isochrone at N = NR are reciprocal

quantities, i.e., the product of both is equal to one.

Second duality theorem

There exists also a duality relationship between the second-derivative matrices of the

reflector ΣR, isochrone , reflection-time surface ΓR, and Huygens surface at the dual points

NR on ΓR and MR on ΣR. This relationship is quantified by the second duality theorem which

states that

mD

(

H≈D − H

≈R

)

= − Λ≈DR

(

Z≈I − Z

≈R

)

−1

Λ≈

TDR , (12)

provided all matrices involved are well defined and nonsingular at the dual points. As the

Hessian matrix of a given surface is closely related to its curvature matrix, theorem (12)

essentially shows how the difference in matrix curvature between the reflection-traveltime

surface and its tangent Huygens surface at the point NR is related to the difference in matrix

curvature between the reflector and its tangent isochrone at the dual point MR. For instance,

12

one of the conclusions that may be immediately drawn from the above second duality theorem

(12) and that will be elaborated below is the following. As the Huygens and isochrone surfaces

can be computed for any dual pair (NR,MR) if the macro-velocity model is specified, one can

directly construct the reflector curvature matrix at MR once the curvature matrix of the

reflection-time surface at NR has been determined from the reflection event.

The second duality theorem (12) can be alternatively formulated in the form of two

independent claims, involving the matrix H≈DR. These are stated as

IIa) The difference between the Hessian matrices H≈D and H

≈R of the Huygens and

reflection-time surfaces is given by

H≈D − H

≈R = Λ≈DRH≈

−1

DRΛ≈TDR . (13a)

IIb) The difference between the Hessian matrices Z≈I and Z

≈R of the isochrone and

reflector is given by

Z≈I − Z

≈R = −1

mDH≈DR . (13b)

Once the results (13a) and (13b) are established, the second duality theorem (12) follows

immediately from a simple substitution of H≈DR given by equation (13b) into formula (13a).

Fresnel geometrical-spreading factor

The Fresnel geometrical-spreading (FGS) factor at the reflection point MR on ΣR is

defined as the factor LF (MR) such that (?)

LSG =L1 L2

LF (MR). (14)

In the above equation, LSG denotes the point-source geometrical-spreading factor of the

primary-reflection ray SMRG, where (S,G) is the source-receiver pair determined by the point

NR on ΓR, which is dual to the point MR on ΣR. Moreover, L1 and L2 are the point-source

13

geometrical-spreading factors for the (first and second) ray segments SMR and MRG with a

point source in S and MR, respectively. In other words, the FGS factor describes the influence

on the geometrical-spreading factor LSG of the total ray SG that is exclusively attributed

to the existence and properties of the (real or fictitious) interface at MR. By way of asymp-

totically evaluating the Kirchhoff integral (?), the FGS factor was in fact introduced as the

factor that describes the influence of the Fresnel zone at the reflection point onto the overall

geometrical spreading of a reflection ray. A thorough investigation of the role of Fresnel zones

in Kirchhoff-type integrals is given in Klimes (1994). Note that an equation similar to formula

(14), however, not involving the FGS factor but the area of the Fresnel zone at the reflection

point was derived by Sun (1995). The FGS factor assumes a key role in a number of forward

and inverse seismic modeling problems (?) and particularly in true-amplitude migration (?)

as well as in seismic imaging (?).

In this section, we will see how the FGS factor is related to the reflection-time and Huy-

gens surfaces as well as to the isochrone and reflector. We will see that besides the curvature

matrices of either pair of these surfaces, there also enters the configuration-dependent Beylkin

determinant (?; ?) into one possible representation of the FGS factor.

As shown in Tygel et al. (1994a), the FGS factor can, in our notation, be expressed as

LF (MR) =cosαR

vR| det(H

≈F )|−1/2 exp

{

iπ

2κF

}

, (15)

where H≈F is the so-called Fresnel matrix (?; ?) that defines the size of the Fresnel zone at

MR for any given frequency, and

κF = [2− Sgn(H≈F )]/2 (16)

is the additional number of caustics that the wave encounters along the total ray due to the

existence of the interface. The symbol Sgn(H≈F ) denotes the signature of the matrix H

≈F , this

being the difference between the number of its positive eigenvalues and the number of its

negative ones. Note that we are implicitly assuming that the real symmetric 2×2-matrix H≈F

14

is non-singular, i.e., it has two real and nonvanishing eigenvalues.

The Fresnel matrix H≈F is the second-derivative matrix of τDR with respect to the

coordinates of an (arbitrarily oriented) 2-D local Cartesian coordinate system within the

plane tangent to the reflector ΣR at MR. It can thus be represented in the form

H≈F = P

≈TH≈DRP≈

, (17)

where P≈

is the projection matrix that describes a projection from the 2-D horizontal coordi-

nates r∼

of the global Cartesian system to the 2-D local Cartesian coordinate system at MR.

The determinant of P≈

is simply given by cosβR (see Appendix B). Inserting the two equalities

(13a) and (13b) of the second duality theorem into formula (17), we can also write H≈F as

H≈F = P

≈TΛ≈

TDR(H≈D −H

≈R)−1Λ

≈DRP≈(18a)

and

H≈F = − mD P

≈T (Z

≈I −Z≈R)P≈

, (18b)

respectively. It is shown in Appendix B, how Hessian matrices Z≈

in arbitrary Cartesian co-

ordinates relate to curvature matrices K≈. Due to equations (11) and (B-9), expression (18b)

can be rewritten as

H≈F = −

cosαR

vR2(K

≈ I −K≈R) , (18c)

where we have introduced the curvature matrices K≈ I and K

≈R of the isochrone and reflector,

respectively, at MR.

Equations (18) lead to the following alternative and interesting representations for the

FGS factor.

(A) Representation in terms of traveltime derivatives:

LF (MR) =cosαR

vR cos2 βR| det(H

≈DS)|1/2 | det(Λ

≈DR)|−1 exp

{

iπ

2κDS

}

, (19a)

15

(B) Representation in terms of depth-surface derivatives:

LF (MR) = cos−2 βR | det(2Z≈IS)|

−1/2 exp

{

iπ

2κIS

}

, (19b)

(C) Representation in terms of depth-surface curvatures:

LF (MR) = | det(2(K≈ I −K

≈R))|−1/2 exp

{

iπ

2κIR

}

. (19c)

In the above, we have introduced for convenience the notations,

H≈DS = H

≈D −H≈R, κDS = [2− Sgn(H

≈DS)]/2. (20a)

Z≈IS = Z

≈I −Z≈R, κIS = [2 + Sgn(Z

≈IS)]/2. (20b)

κIR = [2 + Sgn(K≈ I −K

≈R))]/2. (20c) The

expressions for H≈F and LF (MR) in terms of the difference in curvature matrices K

≈ I −K≈R

of the isochrone and the reflector provides, in our opinion, a much better understanding

of these quantities than was hitherto the case. However, although representation (19c) may

seem the most intriguing one, also the other two are very useful. The matrix H≈DS is the

second-derivative matrix of the traveltime difference τD(ξ∼;M)− τR(ξ∼

) that appears in true-

amplitude diffraction-stack (DS) migration (?). In full correspondence, the matrix Z≈IS is the

second-derivative matrix of the difference ζI(r∼;N)− ζR(r∼

) that occurs in a true-amplitude

isochrone-stack (IS) demigration (?). Thus, the above representations (19) turn out to be the

most fundamental formulas upon which the computation of true-amplitude weights for both

true-amplitude diffraction-stack migration and an isochrone-stack demigration is based.

Curvature duality

Introducing the isochrone and reflector curvatures K≈ I and K

≈R as above into the second

duality theorem (12), it can be rewritten in the form of a ‘curvature duality theorem’, i.e.,

H≈D −H

≈R = −vR

2 cosαRP≈Λ≈DR

(

K≈ I −K

≈R

)

−1

Λ≈

TDRP≈

T . (21)

16

In this form, it directly shows the relationship between the curvatures of the reflection-time

and Huygens surfaces on the one side and of the isochrone and reflector on the other side.

Of course, the traveltime-derivative matrices H≈D and H

≈R are closely related to wavefront

curvatures of certain hypothetical waves (?). Equation (21) thus also relates the curvature

matrices K≈ I and K

≈R to those wavefront curvatures.

Beylkin Determinant

The FGS factor given by representation (19a) contains an up to now not sufficiently

addressed factor, the determinant of the second mixed-derivative matrix Λ≈DR. As will be

shown in this section, this factor is closely related to the familiar Beylkin determinant hB (?;

?), thus revealing the close relationship of the representation (19a) to true-amplitude migration

or migration/inversion. In this way, it also becomes clear how detΛ≈DR can be computed by

dynamic ray tracing, because the problem or how to determine hB is already solved (?).

For a dual pair of points NR on ΓR and MR on ΣR, the Beylkin determinant hB is

written as

hB = det

∇∼τD

∂∂ξ1

∇∼τD

∂∂ξ2

∇∼τD

N,M

= det

∂τD∂r1

∂τD∂r1

∂τD∂r1

∂2τD∂ξ1∂r1

∂2τD∂ξ1∂r2

∂2τD∂ξ1∂z

∂2τD∂ξ2∂r1

∂2τD∂ξ2∂r2

∂2τD∂ξ2∂z

NR,MR

. (22)

As shown in Appendix C, the above 3×3-determinant hB is simply related to the determinant

of the 2× 2-matrix Λ≈DR by

det(Λ≈DR) =

hBmD

. (23)

17

Substituting equation (23) back into the FGS factor representation (19a) yields

LF (MR) =mD cosαR

hBvR| det(H

≈DS)|1/2 exp

{

iπ

2κDS

}

, (24)

As readily seen from its definition (22), the Beylkin determinant hB only depends on the

macro-velocity model in the reflector overburden as well as on the measurement configuration

and not on the reflector itself. We conclude, once more, that the FGS factor is the quantity that

carries all the information concerning the reflector curvature that influences the geometrical-

spreading factor LSG of the overall reflection ray SMRG.

Taking determinants on both sides of equation (21) and also making use of equation

(23), we may recast the second duality theorem in terms of the Beylkin determinant as

det(H≈D −H

≈R) =

(

vR2 cosαR

)4

h2B det

(

K≈ I −K

≈R

)

−1

. (25a)

Sgn(H≈D −H

≈R) = − Sgn

(

K≈ I −K

≈R

)

. (25b) It is

now instructive to divide both sides of equation (25a) by h2B. We then recognize that the right-

hand side of the obtained equality is independent of the chosen measurement configuration.

Consideration of different configurations containing the same central ray (e.g., common shot

and common offset), give rise to useful relations between the corresponding reflection-time

and Huygens surfaces. In this way, the formulas of Shah (1973) can be generalized. Note that

this will obviously not work when hB = 0.

2-D Media

The 2-D counterparts of all the above expressions can easily be obtained from the

corresponding 3-D expressions above upon replacement of the vectors ξ∼

= (ξ1, ξ2) and r∼

=

(r1, r2) by the scalars ξ and r, respectively.

Duality

18

Introducing the notation τD for the 2-D diffraction traveltime (that leads to a Huy-

gens curve instead of a Huygens surface), we find for the second duality theorem the two

corresponding equations to formulas (13),

HD −HR = Λ2

DR

1

HDR(26a)

and

ZI − ZR = −1

mDHDR . (26b)

The overlined quantities are defined in analogy to their 3-D counterparts. For instance,

mD =∂τD∂z

. (27a)

and

ΛDR =∂2τDR

∂ξ∂r, (27b)

both evaluated, of course, at ξR and rR.

The proofs of equations (26) need not be repeated for 2-D as they are obvious from the

proofs given in 3-D in Appendix A. From equations (26), we obtain the 2-D version of the

second duality theorem

HD −HR = −Λ2

DR

mD(ZI − ZR)

−1 . (28)

Introducing the two-dimensional Beylkin determinant hB defined by

hB = det

∇∼τD

∂∂ξ ∇

∼τD

NR,MR

= det

∂τD∂r

∂τD∂z

∂2τD∂ξ∂r

∂2τD∂ξ∂z

NR,MR

, (29)

it is shown in Appendix C that the relationship between ΛDR and hB is given by

ΛDR = −hBmD

. (30)

It follows that in dependence on the Beylkin determinant, the second duality theorem reads

in 2-D

HD −HR = −h2

B

m3

D

(ZI − ZR)−1 . (31)

19

This equation relates the second derivatives appearing in the asymptotic evaluation of the

2-D diffraction-stack and isochrone-stack integral operators (?). It is thus of fundamental

importance when the corresponding integral operators are chained in order to find stacking

operators for all kinds of imaging problems. Note that in 3-D, a relationship in terms of hB

corresponding to equation (31) exists only for the determinants of the matrices in equation

(12), not for the matrices themselves. It is given in equations (25).

Fresnel geometrical-spreading factor

In 2-D media, we also have a geometrical-spreading decomposition formula similar to

expression (14), namely

LSG =LSMLMG

LF. (32)

Again, as we will see below, the 2-D FGS factor, LF (MR) is the only quantity in equation

(32) into which the reflector properties enter. It can be represented in analogy to the 3-D case

as

LF (MR) =

√

cosαR

vR|HF |

−1/2 exp

{

iπ

2κF

}

, (33a)

where again HF (that is now a scalar) is the second derivative of τDR with respect to a

Cartesian coordinate tangent to the reflector at MR. It is again HF that defines the size of

the 2-D Fresnel zone at the reflector. The Fresnel number of caustics κF is now given by

κF = (1− sgn(HF )/2 . (33b)

The 2-D equation corresponding to formula (17) reads

HF = cos2 βRHDR . (34)

Insertion of equations (26) leads to

HF =v2Rh

2

B

4 cos2 αR(HD −HR)

−1 (35a)

20

and

HF = −2cosαR cos3 βR

vR(ZI − ZR) , (35b)

respectively. The latter equation may be rewritten as

HF = −2cosαR

vR(KI −KR) , (35c)

where we have introduced the isochrone and reflector curvatures

KI = cos3 βM ZI , KR = cos3 βM ZR . (36)

The Fresnel geometrical-spreading factor LR(MR) can thus again be represented in three

different ways, namely

(A) Representation in terms of the traveltime derivatives:

LF (MR) =2

hB

(

cosαR

vR

)3/2

|HDS |1/2 exp

{

iπ

2κDS

}

, (37a)

(B) Representation in terms of the depth-curve derivatives:

LF (MR) = cos−3/2 βR(

2|ZIS |)

−1/2exp

{

iπ

2κIS

}

, (37b)

(C) Representation in terms of the depth-curve curvatures:

LF (MR) =(

2|KI −KR|)

−1/2exp

{

iπ

2κIR

}

, (37c)

where the number of ray caustics that is exclusively due to the interface is given in the differ-

ent representations by κDS = κIS = κIR with

κDS = [1− sgn(HDS)]/2. (38a)

κIS = [1 + sgn(ZIS)]/2 . (38b)

κIR = [1 + sgn(KI −KR)]/2 . (38c)

Equation (37c) expresses the dependence of the 2-D FGS factor LF (MR) on the reflector

curvature KR. Observe that it also depends on the isochrone curvature KI . Hence, if the size

21

of the Fresnel zone is unknown, knowledge of the reflector and the isochrone is needed in order

to quantify the influence which a reflector has on the geometrical spreading of the elementary

primary reflected wave following the ray SMRG.

Curvature duality

Introducing the isochrone and reflector curvatures KI and KR given in equations (36),

also the 2-D second duality theorem can be rewritten in the form of a ‘curvature duality the-

orem’, i.e.,

HD −HR = −vR

2 cosαRΛ2

DR cos2 βR(

KI −KR

)

−1

. (39a)

= −

(

vR2 cosαR

)3

h2B

(

KI −KR

)

−1

. (39b) In

this form, it directly shows the relationship between the curvatures of the reflection-time

and Huygens curves on the one side and of the isochrone and reflector on the other side.

These equations completely parallel the corresponding 3-D equations (21) and (25).

Conclusions

In this paper, we have recognized and mathematically quantified the duality between

the surfaces involved in all seismic migration, demigration and imaging. These are the subsur-

face reflector together with its configuration-dependent reflection-time surface, as well as the

Huygens surface and the isochrone. We have seen that the latter two surfaces are defined by

the same set of traveltime functions pertaining to the given measurement configuration. If we

fix the coordinates of the depth point, we obtain its Huygens surface; if we fix the time and

source-receiver coordinates, we obtain the isochrone surface. Due to this fundamental corre-

spondence, certain dualities of these surfaces can be observed. Besides the well-known facts

that (a) the Huygens surface constructed for an actual reflection point on a given reflector

22

is tangent to the corresponding reflection-time surface and (b) the isochrone constructed for

a given point on the reflection-time surface is tangent to the reflector, further fundamental

relationships exist between the first and second derivatives of these surfaces. We termed these

relationship the first and second duality theorems. The first theorem involves the variations of

the Huygens surface and isochrone in time and depth. It can be related to the stretch that is

observed in seismic depth migration. The second duality theorem involves the so-called Fres-

nel matrix that defines the size of the Fresnel zone at the reflection point. From this second

duality theorem, a new expression for the Fresnel matrix was derived that incorporates not

only the reflector but also the isochrone curvature. These (first and second) duality theorems

help to better understand the kinematics and dynamics of a variety of seismic reflection imag-

ing problems. Indeed, for a given reflector and its corresponding reflection-time surface, we

find that the contribution to the overall geometrical spreading of a primary reflection due to

the reflector curvature can be expressed as (a) a second-derivative difference of the reflection-

time and Huygens surfaces in the time section or as (b) a second-derivative difference of the

reflector and the isochrone in the depth section.

Acknowledgments

The research of this paper has been supported in part by the Commission of the Eu-

ropean Communities in the framework of the Geoscience Programme (JOULE II), by the

National Council of Technology and Development (CNPq - Brazil), by the Sao Paulo State Re-

search Foundation (FAPESP), and by a grant from CEPETRO/UNICAMP/PETROBRAS.

The responsibility for the content remains with the authors. This is Karlsruhe University,

Geophysical Institute Publication No. 664.

23

APPENDIX A

Proofs of the duality theorems

In this Appendix we prove the duality theorems stated in the text. For that purpose,

consider a fixed pair of dual points MR on ΣR and NR on ΓR (Figure 1). Let these points be

characterized by the parameters r∼R in E and ξ

∼R in A, respectively. As indicated above, both

points define the same specular reflection ray SMRG that connects the source S(ξ∼R) via MR

on ΣR to the receiver G(ξ∼R).

First duality theorem

Proof of statement Ia

Because of the above identity, referring to the vector parameter ξ∼

= ξ∼R, the diffraction

time τD(ξ∼R;MR) equals the reflection time τR(ξ∼R). To prove the tangency of both surfaces at

NR (statement Ia), we compute the gradients of t = τR(ξ∼) and t = τDR(ξ∼

, r∼) with respect

to ξ∼. We start with ∇

∼ξτR(ξ∼), the jth component of which is given by

∂τR(ξ∼)

∂ξj=

∂

∂ξjτDR(ξ∼

, r∼R(ξ∼

))

=∂τDR(ξ∼

, r∼)

∂ξj+

∂τDR(ξ∼, r∼)

∂rk

∂ r∼Rk

∂ξj, (A-1)

in which the summation convention has been used. We now invoke Fermat’s principle, which

states that a reflection ray is stationary among all rays that join a fixed source-receiver pair

to all points in the vicinity of the specular reflection point. In our case, this simply means

24

that∂τDR(ξ∼

, r∼)

∂rkr∼R

= 0 , (A-2)

Note that the stationarity condition (A-2) actually defines the coordinate vector r∼R = r

∼R(ξ∼)

of the reflection point MR as a function of the given vector parameter ξ∼. In our case, this

given parameter is ξ∼R so that r

∼R = r∼R(ξ∼R). Substituting equation (A-2) into equation (A-1)

yields∂τR(ξ∼

)

∂ξj=

∂τDR(ξ∼, r∼)

∂ξj=

∂τD(ξ∼, r∼, z)

∂ξj, (A-3)

which, of course, holds at ξ∼

= ξ∼R and r

∼= r

∼R. For the right equality, we have used the fact

that the partial ξ∼

dependencies of τDR(ξ∼, r∼) and τD(ξ∼

;MR) are identical. Now, equation

(A-3) is our desired result as it proves the tangency of the reflection and diffraction traveltime

surfaces t = τR(ξ∼) and τD(ξ∼

;MR) at NR. This concludes the proof of statement Ia.

Proof of statement Ib

Consider the isochrone z = ζI(r∼;NR) for the given point NR on the reflection-time

surface ΓR that corresponds to the target reflector ΣR. By definition, this isochrone contains

all depth points MI = MI(r∼;NR) for which the diffraction traveltime τD(ξ∼

;MI) equals the

traveltime value t = τR(ξ∼R). Hence, the point MR, dual to NR, belongs to the isochrone

defined by NR as well as to the reflector ΣR. To prove the tangency of both surfaces at MR,

we consider the gradients of the traveltime functions τDR(ξ∼, r∼) and τDI(ξ∼

, t, r∼) with respect

to r∼, the jth components of which are given by

∂τDR(ξ∼, r∼)

∂rj=

∂τD(ξ∼;M)

∂rj+

∂τD(ξ∼R, r∼R, z)

∂zz = ζR(r∼R)

∂ζR(r∼)

∂rj(A-4a)

and

∂τDI(ξ∼, t, r

∼)

∂rj=

∂τD(ξ∼;M)

∂rj+

∂τD(ξ∼R, r∼R, z)

∂zz = ζR(r∼R)

∂ζI(r∼;N)

∂rj, (A-4b)

25

respectively. The left side of equation (A-4b) vanishes identically for all ξ∼

in A because of the

isochrone definition [see equation (5)]. Moreover, at ξ∼

= ξ∼R and r

∼= r

∼R, also the left side

of equation (A-4a) equals zero because of the stationarity condition (A-2). It follows from the

above considerations that∂ζI(r∼

;NR)

∂rj=

∂ζR(r∼)

∂rj(A-5)

at r∼

= r∼R. This expresses the fact that isochrone and reflector are tangent at MR and proves

statement Ib.

Proof of statement Ic

In order to find the relationship between the vertical stretch factors (∂τD/∂z)(ξ∼R;MR)

and (∂ζI/∂t)(r∼R;NR), we differentiate the traveltime function (5) with respect to t. We readily

find∂τDI(ξ∼

, t, r∼)

∂t=

∂τD(ξ∼;M)

∂z

∂ζI(r∼;N)

∂t= 1 (A-6)

at r∼

= r∼R, from which we conclude that

∂τD∂z

(ξ∼R;MR) =

[

ζI∂t

(r∼R;NR)

]

−1

. (A-7)

In other words, the vertical stretch factors of the Huygens surface for MR at NR and of the

isochrone for NR at MR are reciprocal. This concludes the proof of statement Ic.

Second duality theorem

Proof of statement IIa

In order to prove equation (13a), we differentiate the left and right sides of equation

(A-3) with respect to ξi. Taking into account the functional dependency r∼R = r

∼R(ξ∼) which

stems from the stationarity condition (A-2) and applying the chain rule, we have in matrix

26

notation

∂2τR(ξ∼)

∂ξi∂ξj

≡ H≈R = H

≈D + Λ≈DRX≈R , (A-8)

where we have used equations (6) and (9). Moreover, we have introduced the matrix

X≈R =

∂rRi(ξ∼)

∂ξj

. (A-9)

evaluated at ξ∼

= ξ∼R. To get rid of the auxiliary matrix X

≈R in equation (A-8), we take the

derivative with respect to ξi of equation (A-2), which identically vanishes for all ξ∼

in A. We

have

0 =∂2τDR(ξ∼

, r∼)

∂ξi∂rj+

∂2τDR(ξ∼, r∼)

∂rk∂rj

∂rRk

∂ξi, (A-10)

which, after evaluation at ξ∼

= ξ∼R and r

∼= r

∼R and after use of equations (8), (9), and (A-9),

can be recast in matrix form as

0≈

= Λ≈

TR + H

≈DRX≈R . (A-11)

As long as H≈DR is nonsingular, equation (A-11) can be solved for X

≈R and inserted into

expression (A-8). This yields immediately equation (13a) and completes the proof of statement

IIa.

Proof of statement IIb

We now only have to compute the difference between the Hessian matrices H≈DR of

τDR(ξ∼, r∼) and H

≈DI of τDI(ξ∼, t, r

∼) [equation (5)] keeping in mind that the latter is identical

to zero for all ξ∼

in A. The desired Hessian matrices can be computed by taking the derivatives

of equations (A-4) with respect to ri. Appropriate use of the chain rule yields

∂2τDR(ξ∼, r∼)

∂ri∂rj=

∂2τDR(ξ∼, r∼)

∂ri∂rj−

∂2τDI(ξ∼, t, r

∼)

∂ri∂rj

=∂mR(r∼

)

∂rj

[

∂ζR(r∼)

∂ri−

∂ζI(r∼;N)

∂ri

]

27

+∂mR(r∼

)

∂ri

[

∂ζR(r∼)

∂rj−

∂ζI(r∼;N)

∂rj

]

+∂2τD(ξ∼

;M)

∂z2

[

∂ζR(r∼)

∂ri

∂ζR(r∼)

∂rj−

∂ζI(r∼;N)

∂ri

∂ζI(r∼;N)

∂rj

]

+ mR(r∼)

∂2ζR(r∼)

∂ri∂rj−

∂2ζI(r∼;N)

∂ri∂rj

. (A-12)

We now evaluate the above equation at ξ∼

= ξ∼R and r

∼= r

∼R. Using the tangency of

the isochrone and the reflector expressed by equation (A-5), and expressing the result in

matrix notation by means of equations (7) and (8), we directly arrive at equation (13b). This

completes the proof of statement IIb.

28

APPENDIX B

Curvature matrices

In this Appendix, we derive the relationship between a Hessian matrix of second deriva-

tives of a given surface in arbitrary Cartesian coordinates and the curvature matrix of that

surface. Let

z = ζ(r∼) (B-1)

denote a surface defined in an arbitrary Cartesian coordinate system (r1, r2, r3 = z) with

r∼

= (r1, r2)T . For a given point P on that surface, the Hessian matrix is

Z≈

=

∂2ζ

∂r12∂2ζ

∂r1∂r2

∂2ζ

∂r2∂r1

∂2ζ

∂r22

. (B-2)

Moreover, consider the 3-D Cartesian coordinate system (r′1, r′

2, r′

3 = z′) for which the r′1r′

2-

plane is tangent to the surface at P . The orientation of the r′1- and r′2-axes may be arbitrary

within the tangent plane. Then, a curvature matrix for that surfaces at P is defined as the

Hessian matrix

K≈

=

∂2ζ ′

∂r′1

2

∂2ζ ′

∂r′1∂r′

2

∂2ζ ′

∂r′2∂r′

1

∂2ζ ′

∂r′2

2

. (B-3)

where

z′ = ζ ′(r∼

′) (B-4)

is the same surface, now represented in the new coordinate system with r∼

′ = (r′1, r′

2)T . Note

that curvature matrices are unique up to a rotation of the r∼

′ coordinates on the tangent plane

to the surface at P .

29

In order to find a representation of K≈

in terms of Z≈, we first have to establish the

relationship between ζ(r∼) and ζ ′(r

∼′). For that purpose, we choose the new primed coordinate

system to be constructed from the unprimed one by a single rotation of the z-axis onto the

surface normal at P . Let the rotation angle (i.e., the angle between the z-axis and the surface

normal) be denoted by βR.

Moreover, without loss of generality, we may suppose that the r2-axis of the old system

and the r′2-axis of the new system coincide. In fact, this is achieved by a further (horizontal)

rotation of the new system in the r′1r′

2-plane (tangent to the surface at P ) which does not

affect the definition of K≈. Under the above conditions, the transformations between the two

coordinate systems can be written as

r′1 = r1 cosβR − z sinβR , (B-5a)

r′2 = r2 , (B-5b)

z′ = r1 sinβR + z cosβR . (B-5c)

We now insert equations (B-1) and (B-4) into the transformation formula (B-5c) to

obtain the relationship between ζ and ζ ′ as

ζ ′(r∼

′) = r1 sinβR + ζ(r∼) cosβR . (B-6)

The desired relationship between matrices Z≈

and K≈

can now be derived from the derivatives

of equation (B-6) with respect to primed coordinates. We have

∂ζ ′

∂r′1

= sinβR∂r1∂r′

1

+ cosβR∂ζ

∂rk

∂rk∂r′

1

(B-7a)

and

∂ζ ′

∂r′2

= cosβR∂ζ

∂rk

∂rk∂r′

2

. (B-7b)

As βR is a constant angle, the second derivatives are

∂2ζ ′

∂r′i∂r′j= cosβR

∂2ζ

∂rk∂rl

∂rk∂r′i

∂rl∂r′j

(B-8)

30

or in matrix notation

K≈

= cosβR P≈

T Z≈P≈

, (B-9)

where P≈

is the projection matrix

P≈

=

∂r1∂r′

1

∂r2∂r′

1

∂r1∂r′

2

∂r2∂r′

2

=

cosβR 0

0 1

. (B-10)

Equation (B-9) is our desired relationship that expresses the curvature matrix in terms

of the Hessian matrix in arbitrary Cartesian coordinates. Moreover, from equation (B-9), we

observe that the relationships between the determinants and signatures of the matrices Z≈

and

K≈, that are needed in the text, are given by

detK≈

= cos4 βR detZ≈

(B-11a)

and

SgnK≈

= SgnZ≈

. (B-11b)

The additional rotation of the r2-axis onto the r′2 axis, say by an angle φ, that is in

general necessary leads to a modified projection matrix

P≈

=

cosβR 0

0 1

cosφ − sinφ

sinφ cosφ

(B-12)

in equation (B-9). Equations (B-11) remain unaffected.

31

APPENDIX C

Relationship between Λ≈DR and hB

In this Appendix, we prove the relationship between the determinant of Λ≈DR and the

Beylkin determinant hB given in equation (23).

We start from equation (3), the derivative of which with respect to rj can be rewritten

as∂τDR(ξ∼

, r∼)

∂rj=

∂τD(ξ∼, r∼, z)

∂rj+

∂τD(ξ∼, r∼, z)

∂zζR(r∼

)

∂ζR(r∼)

∂rj. (C-1)

Form equation (A-2) we know that at r∼

= r∼R this expression vanishes. We conclude that

∂ζR(r∼)

∂rjr∼R

= −1

mD

∂τD(ξ∼, r∼, z)

∂rjr∼R

, (C-2)

where we have inserted formula (10a).

The second mixed derivative of τDR(ξ∼, r∼) with respect to ξi and rj can be written

accordingly as

∂2τDR(ξ∼, r∼)

∂ξi∂rj=

∂2τD(ξ∼, r∼, z)

∂ξi∂rj+

∂2τD(ξ∼, r∼, z)

∂ξi∂z

∂ζR(r∼)

∂rj. (C-3)

We take equation (C-3) at r∼R, insert equation (C-2) and take 1/mD out of the brackets to

arrive at

∂2τDR(ξ∼, r∼)

∂ξi∂rjr∼R

= −1

mD

(∂τD(ξ∼, r∼, z)

∂rjr∼R

∂2τD(ξ∼, r∼, z)

∂ξi∂zr∼R

−

−∂τD(ξ∼

, r∼, z)

∂zr∼R

∂2τD(ξ∼, r∼, z)

∂ξi∂rjr∼R

)

32

= −1

mDdet

∂τD(ξ∼, r∼, z)

∂rj

∂τD(ξ∼, r∼, z)

∂z

∂2τD(ξ∼, r∼, z)

∂ξi∂rj

∂2τD(ξ∼, r∼, z)

∂ξi∂z

r∼R

. (C-4)

Inserting this result into

detΛ≈DR =

∂2τDR(ξ∼, r∼)

∂ξ1∂r1r∼R

∂2τDR(ξ∼, r∼)

∂ξ2∂r2r∼R

−∂2τDR(ξ∼

, r∼)

∂ξ1∂r2r∼R

∂2τDR(ξ∼, r∼)

∂ξ2∂r1r∼R

(C-5)

yields after some tedious by straightforward algebra

detΛ≈DR =

1

mD

∂τD(ξ∼, r∼, z)

∂r1det

∂2τD(ξ∼, r∼, z)

∂ξ1∂r2

∂2τD(ξ∼, r∼, z)

∂ξ1∂z

∂2τD(ξ∼, r∼, z)

∂ξ2∂r2

∂2τD(ξ∼, r∼, z)

∂ξ2∂z

−

−∂τD(ξ∼

, r∼, z)

∂r2det

∂2τD(ξ∼, r∼, z)

∂ξ1∂r1

∂2τD(ξ∼, r∼, z)

∂ξ1∂z

∂2τD(ξ∼, r∼, z)

∂ξ2∂r1

∂2τD(ξ∼, r∼, z)

∂ξ2∂z

+

+∂τD(ξ∼

, r∼, z)

∂zdet

∂2τD(ξ∼, r∼, z)

∂ξ1∂r1

∂2τD(ξ∼, r∼, z)

∂ξ1∂r2

∂2τD(ξ∼, r∼, z)

∂ξ2∂r1

∂2τD(ξ∼, r∼, z)

∂ξ2∂r2

r∼R

. (C-6)

Introducing the vector

∇∼τD(ξ∼

, r∼, z) =

∂τD(ξ∼, r∼, z)

∂r1,∂τD(ξ∼

, r∼, z)

∂r2,∂τD(ξ∼

, r∼, z)

∂z

, (C-7)

33

we may write

detΛ≈DR =

1

mDdet

∇∼τD(ξ∼

, r∼, z)

∂

∂ξ1∇∼τD(ξ∼

, r∼, z)

∂

∂ξ2∇∼τD(ξ∼

, r∼, z)

r∼R

=hBmD

, (C-8)

where we have recognized the well-known Beylkin determinant hB defined by Beylkin (1985)

and given in equation (22).

2-D relationship

In 2-D, the quantity ΛDR is a scalar given by

ΛDR =∂2τDR

∂ξ∂rξR, rR

. (C-9)

Correspondingly to equation (C-2), we have from the very same stationarity condition (A-2)

in two dimensions

∂ζR∂r

= −1

mD

∂τD∂r

, (C-10)

where the derivatives are all taken at ξR and rR. The second derivative of τDR with respect

to ξ is then found to be

∂2τDR

∂ξ∂r=

∂2τD∂ξ∂r

−1

mD

∂2τD∂ξ∂z

∂τD∂r

, (C-11)

where equation (C-10) has already been inserted. In the same way as before for the 3-D case,

we take 1/mD out of the brackets and arrive at

ΛDR = −1

mD

(

∂τD∂r

∂2τD∂ξ∂z

−∂τD∂z

∂2τD∂ξ∂r

)

ξR, rR

34

= −1

mDdet

∂τD∂r

∂τD∂z

∂2τD∂ξ∂r

∂2τD∂ξ∂z

ξR, rR

= −hBmD

, (C-12)

where we have introduced the 2-D Beylkin hB determinant according to equation (29) in full

correspondence to the well-known 3-D Beylkin determinant (22).

35

Figure Captions

Fig. 1: (a) Schematic 2-D sketch of a generally 3-D seismic record section. From all seismic

traces that define the reflection-signal strip, only the one at ξ∼

= ξ∼R is depicted. At NR,

the Huygens surface ΓM computed for the depth pointMR [see part (b)] is tangent to the

reflection time surface ΓR. (b) Schematic 2-D sketch of a generally 3-D seismic model.

The depth-migrated strip attached to the reflector ΣR results from a depth migration

of the reflection-signal strip [see part (a)]. At point MR, the isochrone ΣN computed

for the time point NR [see part (a)] is tangent to the reflector ΣR. The points MR

and NR are thus dual to each other as explained in the text. Also shown is the ray

S(ξ∼R)MR(r∼R)G(ξ

∼R) that uniquely defines the dual pair (MR, NR). The angles αR and

βR denote the reflection angle and the local dip angle at MR.

36

reflection-signal strip

R

t

R

R( )

RD( ; M )

z

r

MR

M

NR

Rr

depth-migrated strip

RR

(a)

(b)

(r; N )RIR

(r)R

R

M

R

N

S( )R

G( )R

Fig. 1.

37