Seismic Imaging

Kyle de Mos

August 4, 2018

Bachelor thesis Mathematics and Physics & Astronomy

Supervisor: dr. Chris Stolk, dr. Rudolf Sprik

Institute of Physics

Korteweg-de Vries Institute for Mathematics

Faculty of Sciences

University of Amsterdam

Abstract

In this thesis seismic imaging is studied. Seismic imaging is done by sending in wavesinto the subsurface and then measuring the reflections. Waves in the subsurface aremathematically described by the wave equation. Because of the varying wave speedin the subsurface of the earth, this wave equation has varying coefficients. In thisthesis existence and uniqueness of solutions of such partial differential equations arediscussed and continuous dependence of the solutions on those coefficients. After that alinearization is used to find approximate solutions given a function for the wave speed.This is called Born forward modelling. This Born model is then used to obtain migrationformulas which construct images of the subsurface, given measurements of reflectedwaves. A specific example of a migration formula we look at is Reverse Time Migration orRTM. Finally some MATLAB programmes are examined which illustrate the principlesof Born forward modeling and RTM.

Title: Seismic ImagingAuthors: Kyle de Mos, kyle.demos@uva.nl, 10800603Supervisors: dr. Chris Stolk, dr. Rudolf SprikSecond grader: prof. dr. Daan Crommelin, dr. Tom HijmansEnd date: August 4, 2018

Institute of PhysicsUniversity of AmsterdamScience Park 904, 1098 XH Amsterdamhttp://www.iop.uva.nl

Korteweg-de Vries Institute for MathematicsUniversity of AmsterdamScience Park 904, 1098 XH Amsterdamhttp://www.kdvi.uva.nl

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Contents

1 Introduction 4

2 Mathemathical preliminaries 72.1 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 The Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 More on the Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Uniqueness and existence of solutions in case of variable coefficients 113.1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Rewriting the PDE as an ODE with values in a Hilbert space . . . . . . . 133.3 Existence and uniqueness of solutions . . . . . . . . . . . . . . . . . . . . 15

4 Born forward modeling 194.1 In time domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 In frequency domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 Comparison of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Migration 235.1 Derivation of migration formula from Born forward modeling . . . . . . . 235.2 MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2.1 Finite difference method . . . . . . . . . . . . . . . . . . . . . . . . 265.2.2 Born modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2.3 RTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2.4 Adjoint test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.3 Comments on practicality . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6 Conclusion 32

Populaire samenvatting 33

Bibliografie 35

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1 Introduction

The goal in seismic imaging is to find information about the earth’s substructure bysending in waves and measuring the reflections [5]. This is used mostly by the oil andgas industry to find petroleum deposits, but also by geophysicists for academic use[5].

The result we are looking for is an image such as in figure 1.1.

Figure 1.1: Example of a seismic image from (http://www.crs4.it/research/energy-and-environment/imaging-and-numerical-geophysics/seismic-depth-imaging/)

The experiment executed to obtain the data to create such images consists of so-called sources, which create seismic waves, and receivers which measure vibrations onthe surface. This experiment can be done on land or on water. Figure 1.2 illustratesthe experiment on land: a vibrator truck acts as source and creates seismic waves whichreflect on a layer of the earth. These reflected waves are then measured by so-called geo-phones, which act as receivers in this experiment. Figure 1.3 illustrates the experimenton water. Here airguns act as sources and the receivers are hydrophones.

After collecting the data it needs to be interpreted. This means translating the datato an ”image” of the subsurface. Wave propagation in the subsurface is described by thewave equation. So the data can be viewed as samples of solutions to the wave equation.In this case the wave equation has variable coefficients, specifically the wave speed, whichvaries in different materials in the subsurface. Finding the solutions when we know thecoefficients is the forward problem for our partial differential equation. We are interestedin the inverse problem: finding the coefficients when we know the wavefield at certainpoints.

In practice we look for approximate solutions. Geophysicists have created methods to

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Figure 1.2: Illustration of the seismic imaginging experiment on land from the website(http://www.innoseis.com/seismic-surveying/)

Figure 1.3: Illustration of the seismic imaginging experiment above waterfrom the website (http://clubofmozambique.com/news/oil-and-gas-exploration-seismic-survey-launched-in-nampula-and-zambezia/)

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find such approximate solutions, named migration schemes [5]. However these were notoriginally created as approximate solutions to the inverse problem of the wave equation,but as methods to obtain images of the subsurface [5].

The aim of this thesis is to critically evaluate the mathemathics behind imaging asused in seismology and give a description of how commonly used migration formulasfollow from the wave equation.

To obtain migration formulas from the wave equation a couple of steps need to betaken. First we find approximate solutions to the wave equation by linearizing it, fac-tually taking into account only single scattering. This gives us an expression for theapproximate solution which can be viewed as a linear operator working on certain pa-rameters from the wave equation. Since we are interested in finding these parameterswe then try to find something approximating the inverse of this operator, so that if weapply this to the measured wavefield, it yields back the parameters. It turns out that theadjoint of the mentioned operator is a good choice. An important result, which requirestoo complicated mathematics to be proven in this thesis, is that of Beylkin [9]. He provesthat the discontinuities in the result gained by applying the adjoint to the data are asubset of the dicontinuities of the actual parameterfunction. When we think of a layeredsubsurface, these discontinuities lie at the transitions between layers, which is exactlywhat we would like to map. This also motivates a somewhat more precise definitionof an image: any function which has the same dicontinuities as the parameterfunctionsdescribing the system, albeit perhaps with different amplitudes.

The structure of this thesis is as follows: In chapter 2 we will give some mathemath-ical preliminaries which are necessary for later chapters. In chapter 3 we will discussuniqueness, existence and continuous dependence on the parameters of the wave equa-tion with variable parameters. In chapter 4 we will discuss the Born approximation;which is the above mentioned linearization of the wave equation and will give us ap-proximate solutions if we know the parameter functions. In chapter 5 we first discussmigration formulas and how they follow from the Born approximation. After that weexamine some MATLAB programmes from [6] which illustrate the Born approximationand migration, in particular Reversed Time Migration (RTM).

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2 Mathemathical preliminaries

In this chapter we will introduce some mathematics that is necessary for the followingchapters. In particular we will discuss the wave equation, with constant and varyingcoefficients, the fourier transform, the Helmholtz equation and the concept of a Green’sfunction.

2.1 The wave equation

The governing equation is the scalar wave equation:(1

c(x)2

∂2

∂t2−4

)u(x, t) = f(x, t), (2.1)

where the wave speed c(x) is seen as a function c : R3 → R≥0 of position and f(x, t) isa forcing term. The requirements for existence and uniqueness of solutions of (2.1) willbe discussed in chapter 3.

We assume zero initial conditions: u(x, 0) = ut(x, 0) = 0. We construct solutions byusing a Green’s function, i.e. a solution g(x, t, xs, ts) to:(

1

c(x)2

∂2

∂t2−4

)g = δ(x− xs)δ(t− ts). (2.2)

A solution for equation (2.1) with a general forcing term f(x, t) is then given by corre-lating f with the Green’s function [2]:

u(x, t) =

∫ t+

0

∫R3

g(x, t, xs, ts)f(xs, ts)dxsdts (2.3)

For constant c the Green’s function is given by [2]:

g(x, t, xs, ts) =δ(t− ts − |x− xs|/c)

4π|x− xs|. (2.4)

In which case 2.3 simplifies to:

u(x, t) =

∫R3

f(xs, t− |x− xs|/c)4π|x− xs|

dxs. (2.5)

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2.2 Fourier Transformation

We define the fourier transform of a function f as follows:

F(f)(ω) = f(ω) =

∫ ∞−∞

f(t)eiωtdt, (2.6)

and the inverse Fourier transform of a function g as:

F−1(g)(t) = g(t) =1

2π

∫ ∞−∞

g(w)e−iωtdω. (2.7)

A direct calculation shows:

F(f ′)(ω) = −iωFf(ω). (2.8)

And if we define the convolution of two functions f and g as

(f ∗ g)(x) =

∫ ∞−∞

f(x− y)g(y)dy,

we have the following result:

F(f ∗ g)(ω) = f(ω)g(ω). (2.9)

which we call the convolution theorem. More specifically if we fill in t = 0 we find:∫ ∞−∞

f(t)g(−t)dt = (f ∗ g)(0) = F−1(f g)(0) (2.10)

=1

2π

∫ ∞−∞

e−iω0f(ω)g(ω)dω =1

2π

∫ ∞−∞

f(ω)g(ω)dω. (2.11)

Another result about Fourier transforms we’ll need relates F−1(f)

to f :

F−1(f(ω)

)=

1

2π

∫ ∞−∞

e−iωtf(ω)dω =1

2π

∫ ∞−∞

eiωtf(ω)dω (2.12)

=1

2π

∫ ∞−∞

e−iω(−t)f(ω)dω = f(−t) = f(−t). (2.13)

If we combine this with our previous result we find:∫ ∞−∞

f(t)g(t)dt =1

2π

∫ ∞−∞

f(ω)g(ω)dω. (2.14)

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2.3 The Helmholtz Equation

If we apply a Fourier Transform with respect to time to the wave equation we obtain:

(4+ω2

c2)u(x, ω) = −f(x, ω). (2.15)

If we then define the wavenumber k as k = ωc , write U for u and h for f we obtain the

Helmholtz equation:

(4+ k2)U = −h. (2.16)

This problem can again be solved by means of a Green’s function. To make a distinctionwith the Green’s function for the wave equation we will call it G(x, xs). This Green’sfunction is the solution to:

(4+ k(x)2)G(x, xs) = −δ(x− xs). (2.17)

The solution to equation (2.16) then becomes [2]:

U(x) =

∫R3

h(xs)G(xs)dxs (2.18)

When k is a constant we can write down an explicit form for G(x, xs) [2]:

G(x, xs) =eik|x−xs|

4π|x− xs|. (2.19)

2.4 More on the Green’s functions

In this section we discuss some more properties of the Green’s functions described above.First we would like to note something about the Green’s function in the time domain.Because there are no constants varying with time in equation (2.1) we know that g mustbe of the form g(x, xs, t, ts) = ψ(x, xs, t − ts) for some ψ : R3 × R3 × R≥0 → R. Thephysical idea is that it should not matter whether we send a pulse now and measure it in10 seconds or send the pulse at some other time and measure it 10 seconds after havingsent it out then.

Now we would like to mention something about the relation between the Green’sfunctions in the time domain and in the Fourier domain and causality. If we Fouriertransform equation (2.2) and compare it to equation (2.17) we find a relation betweenour two Green’s functions: F(g)(x, xs, ω) = G(x, xs)e

iωts . Note that if G is a solution to(2.17) then so is G. By using equation (2.13) we see that the inverse Fourier transformof G(x, xs) is g(x, xs,−t, 0), which is also a solution to equation (2.2) with ts = 0. Herewe need to make a distinction between the causal and the the acausal Green’s function.If g(x, xx, t, ts) is the causal Green’s function, which propogates forward in time, theng(x, xs, t, ts) := g(x, xs,−t,−ts) is the acausal Green’s function which propogates back-ward in time. If we have a constant wave velocity these are δ(t − ts − |x − xs|/c) and

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δ(t− ts + |x− xs|/c), see figure 2.1. It is illustrated how the causal Green’s function isa cone in positive time and the acausal Green’s function is a cone in negative time. Tosolve the wave equation we usually use the causal Green’s function because we do notallow the effect to come before its cause. We also call G(x, xs) and G(x, xs) the causaland the acausal Green’s functions for the Helmholtz equation.

Figure 2.1: Figure from [1] illustrating the causal and acausal Green’s functions.

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3 Uniqueness and existence of solutionsin case of variable coefficients

In this chapter we will investigate for which functions c(x) equation (2.1) has a uniquesolution. We will also look at continuous dependence on c of such solutions. This isrelevant, because in reality we never exactly know its values and if a small difference inc gives an entirely different solution, we have little hope of finding good approximations.The theory discussed is based on [4]. However as this is a dissertation the level exceedsthat of a bachelor thesis. Therefore we will not derive all that is done in the chapter weare interested in, but rather highlight some proofs and ideas and discuss these in somemore detail, while referring to [4] for more difficult but important results.

3.1 Sobolev Spaces

In this section we will give a discussion of Sobolev spaces. These function spaces are notonly relevant for the generalized wave equation we are discussing, but in many partialdifferential equation problems [8]. The definitions and theorems discussed here are from[8].

For convenience we first introduce the so-called multi-index notation. Let x = (x1, . . . , xn)be a point in Rn. We call the vector α = (α1, . . . , αn) a multi-index of order |α| =α1 + · · · + αn. We then define xα as xα = xα1

1 . . . xαn

n . If we have a function f : Rn ⊃Ω → R which is Ck(Ω) we use multi-index notation to denote its partial derivatives:

Dαf = ∂|α|f∂xα11 ...∂xαnn

. Having this concept we can define so-called weak derivatives:

Definition 3.1.1. Let u, v ∈ L1loc(Ω). We say v is the αth weak partial derivative of u

if for all φ ∈ C∞0 (Ω): ∫ΩuDαφdx = (−1)|α|

∫Ωvφdx, (3.1)

in which case we write Dαu = v.

Using these weak derivaties we define Sobolev spaces as follows:

Definition 3.1.2. The Sobolev space W k,p(Ω) consists of all u ∈ L1loc(Ω) such that for

each multi-index α with |α| ≤ k, Dαu exists in the weak sense and belongs to Lp(Ω). Ifp = 2 we usually write Hk(Ω).

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Sobolev spaces are Banach spaces if we equip them with the following norm [8]:

||u||Wk,p(Ω) =∑|α|≤k

||Dαu||Lp(Ω) =∑|α|≤k

(∫Ω|Dαu(x)|pdx

) 1p

. (3.2)

For p = 2 the Sobolev spaces are even Hibert spaces if we equip them with the followinginner product [8]:

〈u, v〉 =

∫ ∑|α|≤m

(Dαu) (Dαv) dx. (3.3)

In case p = 2 and Ω = Rn we can give an alternative definition of Sobolev spaces usingthe Fourier transform. So far we have only defined the Fourier transform for functionsof one variable. For functions of several variables we define the Fourier transform asfollows:

Definition 3.1.3. If u ∈ L1(Rn) we define its Fourier transform as follows:

u(ξ) = F(u)(ξ) =

∫Rnu(x)eiξ·xdx, (3.4)

where ξ · x = ξ1x1 + · · ·+ ξnxn.

With this definition we have the following:

F(Dαu) = (−iξ)αF(u). (3.5)

Using Plancherel’s formula we then have:∫Rn|Dαu(x)|2dx =

1

(2π)n

∫Rn|(−iξ)αu(ξ)|2dξ =

1

(2π)n

∫Rnξ2α|u(ξ)|2dξ. (3.6)

From this it follows:

||u||2Hk(Rn) =1

(2π)n

∫Rn

∑|α|≤k

ξ2α

|u(ξ)|2dξ. (3.7)

Now there is a constant C such that (1 + |ξ|2)k ≤∑|α|≤k ξ

2α ≤ C(1 + |ξ|2)k. Integratingthis gives us:∫

Rn(1 + |ξ|k)2|u(ξ)|2dξ ≤

∫Rn

∑|α|≤k

ξ2α

|u(ξ)|2dξ ≤ C∫Rn

(1 + |ξ|k)2|u(ξ)|2dξ. (3.8)

Combining this with the previous equality gives us that ||(1 + |ξ|2)k/2u||L2(Rn) is anequivalent norm with ||u||Hk(Rn). This means that we could also have defined Sobolev

spaces for Ω = Rn by saying u is in Hk(Rn) if ||(1 + |ξ|2)k/2u||L2(Rn) < ∞. This nowallows us to define Sobolev spaces with non-integer order as follows:

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Definition 3.1.4. Let u ∈ L2(Rn). Then u ∈ Hk(Rn) if (1 + |ξ|k)u ∈ L2(Rn).

We would like to note that there are also ways to define Sobolev Spaces of fractionalorder when the underlying space is not all of Rn, this is discussed in [10].

An important theorem states that if a function has enough weak derivatives, it has acertain degree of smoothness. To explain this we first define Holder spaces:

Definition 3.1.5. A function u : Ω → R is Holder continous with exponent γ if thereis c > 0 such that |u(x)− u(y)| < c||x− y||γ for all x, y ∈ Ω. The space Ck,γ(Ω) consistsof all functions whose partial derivatives up to order k exist(in the classical sense) andwhose kth partial derivatives are Holder continuous with exponent γ.

With this definition we can now state the so-called Sobolev embedding theorem:

Theorem 3.1.6. Let Ω be a bounded open subset of Rn, with a C1 boundary. Assumeu ∈W k,p(Ω).

1. if k < n/p,Then u ∈ Lq(Ω) where 1/q = 1/p+ 1/n.

2. if k > n/p,

Then u ∈ Ck−[np

]−1, γ, where γ = [n/p] + 1 − n/p if n/p is not an integer and γ

being any number lesser than 1 is n/p is an integer.

Here [x] means x rounded to an integer, where 1/2 is rounded to 0. So for example if

u ∈ H1(Ω) for Ω ∈ R, then u also lies in C0, 12 .

3.2 Rewriting the PDE as an ODE with values in aHilbert space

We are looking to solve the following generalized wave equation

−∑i

∂

∂xi

1

ρ(x)

∂

∂xiu(x, t) + κ(x)

∂2

∂t2u(x, t) = f(x, t), (3.9)

where u is the pressure, ρ(x) the mass density and κ(x) the compressibility. Often wetry to model a system of layers of different material in the earth’s subsurface, whichimplies that our coefficients ρ(x),κ(x) can vary discontinuously. Because of this we willinterpret equation (3.9) in a distributional sense, that is:

∫ (−∑i

v(x, t)∂

∂xi

1

ρ(x)

∂

∂xiu(x, t) + v(x, t)κ(x)

∂2

∂t2u(x, t)

)dxdt =

∫v(x, t)f(x, t)dxdt,

(3.10)

for all test functions v ∈ C∞0 (Rn×[0, T ],C). If we apply partial integration this becomes:

13

∫ (∑i

1

ρ(x)

∂u

∂xi(x, t)

∂v

∂xi(x, t)− v(x, t)κ(x)

∂u

∂t(x, t)

∂v

∂t(x, t)

)dxdt =

∫v(x, t)f(x, t)dxdt.

(3.11)

for all test functions v ∈ C∞0 (Rn × [0, T ],C). Now comes the point at which we need tochoose our space of solutions. Equation (3.11) does not require u or f to be smooth.We will choose our solutions u ∈ H1(Rn × [0, T ],C) and our forcing term f ∈ L2(Rn ×[0, T ],C). These choices turn out to give us uniqueness and existence of solutions to(3.11).

We choose initial conditions:

u(x, 0) = u0(x), (3.12)

∂u

∂t(x, 0) = u1(x), (3.13)

with u0 ∈ H(1)(Rn,C) and u1 ∈ L2(Rn,C). A priori we do not know whether u(x, 0)

and ∂u∂t (x, 0) are well defined, because general L2-functions cannot be restricted to a

hyperplane. In our case they can be; this will be discussed below equation (3.19).We are going to view u(x, t) as a function t 7→ u(t) with values u(t) : x 7→ u(x, t) in a

Hilbert space of functions on Rn. Fubini’s theorem says that if a function h : X×Y → Cis absolutely integrable, we can calculate the integral over X ×Y as a repeated integral,that is: ∫

X×Yhd(x, y) =

∫X

∫Yh(x, y)dydx. (3.14)

In particular this means we can identify the space L2(X × Y,C) with L2(X,L2(Y,C)).If we apply this to the situation as described above we obtain u ∈ H1(Rn × (0, T ),C)if and only if u ∈ L2((0, T ), H1(Rn,C)) and u′ ∈ L2((0, T ), L2(Rn,C)). For conveniencewe will define V = H1(Rn,C) and H = L2(Rn,C).

Let the operators A,C be defined as:

A = −∑i

∂

∂xi

1

ρ(x)

∂

∂xi, (3.15)

C = κ(x). (3.16)

Then we have certain symmetric forms associated with A and C:

a(u, v) =

∫ ∑i

1

ρ(x)

∂u

∂xi(x)

∂v

∂xi(x)dx (3.17)

c(u, v) =

∫κ(x)u(x)v(x)dx (3.18)

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and with these definitions our differential equation becomes:∫ T

0[a(u(t), v(t))− c(u(t), v(t))]dt =

∫ T

0〈f(t), v(t)〉dt. (3.19)

for all v ∈ C∞0 ([0, T ], V ), which can be explained as the weak form of:

Au(t) + C∂2u

∂t2= f(t). (3.20)

The initial conditions above can then be written as:

u(0) = u0, (3.21)

u′(0) = u1, (3.22)

with u0 ∈ V , u1 ∈ H. Again, since t 7→ u(t) is only L2, u(0) is not defined a priori.But because u′ ∈ L2((0, T ), H) we have u lying in a version of a Sobolev space, onlywith values in a Hilbert space in stead of R. If we accept that such Sobolev spaceshave the same Sobolev embedding theorems, we find that u is Holder continuous withexponent 1

2 , and we can restrict a continuous function to a hyperplane. Also, since usatisfies the above differential equation, we know that u′′ ∈ L2(0, T ), V ′). So by the sameargument we know that u′ is Holder continuous with exponent 1

2 and can be restrictedto a hyperplane.

We will require 1ρ(x) and κ(x) to be measurable and bounded away from zero, in

particular we require there to be constants α, β ≥ 0 such that

1

ρ(x)≥ α, (3.23)

κ(x) ≥ β. (3.24)

We would like to mention that since the wavespeed is given by√κ/ρ, [5], this corresponds

to c being measurable and bounded away from zero.This gives us for λ > α:

c(u, u) ≥ β||u||2H , (3.25)

a(u, u) + λ||u||2H ≥ α||u||2V . (3.26)

For convenience we will from now on write | · | for || · ||H and || · || for || · ||V .

3.3 Existence and uniqueness of solutions

Now that we have our ordinary differential equation we will show uniqueness of solutionsand continuous dependence on parameters. To do this we define the energy of the systemas follows:

E(t) =1

2(a(u(t), u(t)) + c(u′(t), u′(t)) + λ|u(t)|2). (3.27)

Now we can prove a lemma about our energy:

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Lemma 3.3.1. Suppose a,c,u,f are as above and u satisfies (3.19). Then there is asubset I0 of (0, T ) that differs from (0, T ) by a set of measure zero, such that the energyas defined above is Holder continuous of order 1

2 on I0 and it obeys

E(t) ≤ C(E(s) +

∫ T

0|f(σ)|2dσ

), (3.28)

for all s, t ∈ I0, C = C(α, β, λ, T ).

Proof. In order to proof the theorem we will use a regularizing sequence, that is, asequence (ρm) of functions in C∞0 ((0, T )), which satisfies

ρm(t) ≥ 0, (3.29)∫ρm(t)dt = 1, (3.30)

suppρm → 0. (3.31)

Now if h ∈ Lp((0, T )), then h ∗ ρm(s) ∈ C∞0 ((0, T )) with nth derivative h ∗ ρ(n)m (s)

and we have h ∗ ρm(s) → h(s) in L2-norm if m → ∞ [3]. If we now define v(t) =ρm(s− t)(ρ′m ∗ u)(s), then v ∈ C∞0 ([0, T ], V ). Because of this u satisfies equation (3.19)

with this choice of v. However∫ T

0 ρm(s− t)u(t)dt equals ρm ∗ u(s), thus:

a(ρm ∗ u(s), ρ′m ∗ u(s)) + c(ρ′′m ∗ u(s), ρ′m ∗ u(s)) = 〈ρm ∗ f(s), ρ′m ∗ u(s)〉.

Next we define:

um(s) = ρ′m ∗ u(s), (3.32)

fm(s) = ρm ∗ f(s), (3.33)

Em(s) =1

2(a(um(s), um(s)) + c(u′m(s), u′m(s)) + λ|u(t)|2). (3.34)

Now, we want to differentiate Em to find bounds for Em(t) and then let m → ∞ toobtain the statement in the lemma. The derivative of the inner product of two functionscan be found as follows

d

ds〈u(s), v(s)〉 = lim

h→0

〈u(s+ h), v(s+ h)〉 − 〈u(s), v(s)〉h

(3.35)

= limh→0〈u(s+ h),

v(s+ h)− v(s)

h〉+ 〈u(s+ h)− u(s)

h, v(s)〉 (3.36)

= 〈u(s), v′(s)〉+ 〈u′(s), v(s)〉, (3.37)

where we used the continuity of the inner product to interchange the innerproduct andthe limit in the last step. If we apply this to u = v we find:

d

ds〈u(s), u(s)〉 = 〈u(s), u′(s)〉+ 〈u′(s), u(s)〉 = 2 Re〈u(s), u′(s)〉. (3.38)

16

Using this relation we can differentiate Em(s) to obtain:

dEmds

= Re(a(um(s), u′m(s)) + c(u′′m(s), u′m(s)) + λ〈um(s), u′m(s)〉) (3.39)

= Re(〈fm(s), u′m(s)〉+ λ〈um(s), u′m(s)〉, (3.40)

where we used equation (3.3) in the second step. If we send m→∞ um, u′m and fm go

to u, u′ and f respectively in the appropriate L2-norms, therefore all the inner productsand the energy Em go to their unconvolved version in L1([0, T ]). So if we let m → ∞equation (3.40) becomes:

dE

dt= Re(〈f, u〉+ λ〈u,u′〉+ λ〈u, u′〉. (3.41)

This means dEdt ∈ L

1((0, T )) and thus t 7→ E(t) is continous and bounded. Because of(3.25), (3.26) it follows that ||u(t)||, |u′(t)| are bounded. It follows:

dE

dt≤ |f(t)||u(t)|+ λ|u(t)||u′(t)| ≤ C1|f(t)|+ C2. (3.42)

Integrating this we obtain:

|E(t)− E(s)| ≤∫ t

s(C1|f(σ)|+ C2dσ (3.43)

= C1

∫ T

01[s,t]|f(σ)dσ + C2(t− s) (3.44)

= C1

√t− s

[∫ T

0|f(σ)|dσ

]1/2

+ C2(t− s). (3.45)

It follows that E(t) is Holder continuous of order 12 on I0.

For any inner product the following inequalities hold:

〈a− b, a− b〉 = |a|2 − 2 Re(〈a, b〉) + |b|2 ≥ 0 (3.46)

Re(〈a, b〉) ≤ 1

2(|a|2 + |b|2) (3.47)

If we integrate equation (3.40) and apply the above inequality we find:

Em(t) ≤ Em(s) +

∫ t

s

1

2

(|fm(σ)|2 + (1 + λ)|u′m(σ)|2 + λ|um(σ)|2

). (3.48)

Using the boundedness of 1ρ(x) and κ(x) we find:

Em(t) ≤ Em(s) +

∫ T

0|12f(σ)|2dσ + C

∫ t

sEm(σ)dσ. (3.49)

17

Applying Gronwall’s lemma we obtain:

Em(t) ≤ C(Em(s) +

∫ T

0|12f(σ)|2dσ

), (3.50)

with C depending on α,β, λ, T . If we now take the limit m→∞ we obtain (3.3.1).

In [4] they now continue to proof that u and u′ are continuous after modifications ona set of measure zero. We will omit these proofs and refer to [4] for them as they do notfall within the scope of this bachelor thesis.

We will now state the final result on existence and uniqueness of solutions to theproblem (3.11),(3.12),(3.13). The proof for existence again falls outside the scope of thisbachelor thesis, but we will give the proof of the uniqueness of the solutions and theircontinuous dependence on parameters.

Theorem 3.3.2. The problem as described above has a solution u ∈ C([0, T ], V ) ∩C1([0, T ], H). The solution u is unique and depends continuously on u0, u1 and f .

Proof. UniquenessSuppose u, u are both solutions to the problem. Then u = u − u is a solution to thedifferential equation with f = 0, u0 = 0 and u1 = 0. Since u(t) is continuous in V andu′(t) in H [4], we can use estimate (3.3.1) and find:

E(t) ≤ C(E(0) +

∫ T

0|f(σ)|2dσ

)= C

(||u0||2 + |u1|2 +

∫ T

0|f(σ)|2dσ

)= 0.

Thus by (3.25), (3.26) u = 0 and u = uContinuous dependence on parameters

Now suppose we have two problems, one with u0,u1,f and one with u0,u1,f . Then u0

solves the differential equation with u0 = u0 − u0, u1 = u1 − u1, f = f − f . Equation(3.3.1) gives us:

E(t) ≤ C(||u0 − u0||2 + |u1 − u1|2 +

∫|f(σ)|2dσ

),

which implies continuity of the problem with respect to u0, u1 and f .

18

4 Born forward modeling

In this chapter we construct approximate solutions to the wave equation given the wavespeed function. We do this by linearizing the wave equation. This is called Born modelingor the first order perturbation approximation. We will perform the same linearization forthe Helmholtz equation to obtain approximate solutions in the Fourier domain. Finallywe will show how the results from the time domain and frequency domain agree. Theresults of this chapter will be the starting point for obtaining migration formulas in thenext chapter.

4.1 In time domain

Let s(x) be the slowness field defined by s(x) = 1/c(x). Then the wave equation (2.1)can be recast as: (

s(x)2 ∂2

∂t2−4

)u(x, t) = f(x, t). (4.1)

The idea is to write the slowness field s(x) as the sum of a ”slowly varying”, smoothbackground field s0(x) and a ”rapidly varying”, perhaps even discontinous, perturbationfield δs(x). We also decompose our solution u(x, t) into a background solution u0(x, t),which solves equation (4.1) with s(x) = s0(x), and a perturbation δu(x, t). Furthermore,we let g0(x, xs, t, ts) be the background Green’s function which solves equation (4.1) withs(x) = s0(x) and f(x, t) = δ(x− xs)δ(t− ts), so that we can write u0(x, t) as:

u0(x, t) =

∫ ∫g(x, xs, t, ts)f(xs, ts)dtsdxs. (4.2)

If we now plug in u(x, t) = u0(x, t) + δu(x, t) and s(x) = s0(x) + δs(x) into equation(4.1) and ignore second order terms(and thereby linearize the problem) we obtain:(

s20

∂2

∂t2+ 2s0δs

∂2

∂t2−4

)(u0 + δu) = f, (4.3)(

s20

∂2

∂t2−4

)δu+ 2s0δs

∂2

∂t2u0 = 0, (4.4)(

s20

∂2

∂t2−4

)δu = −2s0δs

∂2

∂t2u0. (4.5)

Notice that this is the wave equation for δu with forcing term −2s0δs∂2

∂t2u0. We interpret

2s0δs as an approximate reflection distribution. Then equation (4.5) can be interpreted

19

as follows: whenever the background wave is at a reflector(that is: a place where 2s0δsis not zero) it serves as a source for the perturbation field. In this sense the perturba-tion solution δu can be viewed as a reflected wave, taking into account only first orderreflections.

Because equation (4.5) is just the wave equation with forcing term −2s0δs∂2

∂t2u0 we can

write its solution using the background Green’s function. Since the operators s20∂2

∂t2−4

and − ∂2

∂t2commute we can also solve the problem with a forcing term 2s0δsu0 and then

apply − ∂2

∂t2. This gives:

δu(x, t) = − ∂2

∂t2

∫ ∫2s0(x′)δs(x′)u0(x′, t′)g0(x, t, x′, t′)dx′dt′. (4.6)

A case of special interest is f(x, t) = δ(x− xs)δ(t− ts), so that u0(x, t) = g0(x, xs, t, ts)and u(x, t) = g(x, xs, t, ts), in which case δu(x, t) can be viewed as a perturbation on thesystem’s Green’s function, we will call it δg. Plugging this in gives:

δg(x, xs, t, ts) = − ∂2

∂t2

∫ ∫2s0(x′)δs(x′)g0(x′, t′, xs, ts)g0(x, t, x′, t′)dx′dt′. (4.7)

If we take the background slowness field s0 to be constant, we know the backgoundGreen’s function and can therefore give a more explicit formula for δg:

δg(x, xs, t, ts) = − ∂2

∂t2

∫ ∫2s0δs(x

′)δ(t− t′ − |x− x′|/c)

4π|x− x′|δ(t′ − ts − |x′ − xs|/c)

4π|x′ − xs|dx′dt′.

(4.8)

If we evaluate the t′-integral this further simplifies to:

δg = − ∂2

∂t2

∫R3+

2s0δs(x′)δ(t− ts − |x− x′|/c− |x′ − xs|/c)

16π2|x′ − xs||x− x′|.dx′. (4.9)

This formula can be interpreted as follows: for a fixed location x and time t the per-turbed Green’s function δg(x, t, xs, ts) is given by weightedly integrating the reflectiondistribution over the ellips |x′ − xs|+ |x− x′| = c(t− ts), see figure 4.1.

Figure 4.1: Illustration of the integration domain for equation (4.9).

20

4.2 In frequency domain

Here we will repeat the process from the last section, but then in the frequency domain.That is, we will linearize the Helmholtz equation in a similar way as we did for the waveequation. We let k(x) = ωs(x) and again write s(x) = s0(x) + δs(x) where s0(x) is thebackground slowness field and δs(x) a perturbation on the slowness field. Furthermorewe let U0 be the solution to the Helmholtz equation with k0(x) = ωs0(x) and G0(x, xs)the corresponding Green’s function. We write U(x) = U0(x)+δU(x) and δk(x) = ωδs(x).Plugging this all into the Helmholtz equation and ignoring second order terms we obtain:

(4+ k20 + 2k0δk(x))(U0 + δU) = −h(x), (4.10)

(4+ k20)δU = −2k0δkU0 = −2ω2s0δsU0. (4.11)

Using the background Green’s function we obtain a solution:

δU(x) = ω2

∫2s0(x′)δs(x′)U0(x′)G0(x, x′)dx′. (4.12)

If the source is f(x) = δ(x− xs) then U(x) = G(x, xs) = G0(x, xs) + δG(x, xs) and:

δG(x, xs) = ω2

∫2s0(x′)δs(x′)G0(x′, xs)G0(x, x′)dx′. (4.13)

And if we take the background slowness to be constant this gives us:

δG(x, xs) = ω2

∫2s0δs(x

′)eik0|x

′−xs|

4π|x′ − xs|eik0|x−x

′|

4π|x− x′|dx′. (4.14)

4.3 Comparison of results

In this section we will show that the results of the Born approximation in time domainand in frequency domain agree. First we will show that equations (4.14) is the Fouriertransform of equation (4.8). Since the amplitude factors agree I will focus on the partconcerning time dependence. Let f1(t) = δ(t− |x− x′|/c), f2(t) = δ(t− ts − |x− xs|/c).Then we find:

(f1 ∗ f2)(t) =

∫δ(t− t′ − |x− x′|/c)δ(t′ − ts − |x− xs|/c)dt′, (4.15)

which we recognize as the time dependent part of equation (4.8). The Fourier transformsof f1 and f2 are f1(ω) = eiω|x−x

′|/c and f2(ω) = eiω|x−xs|/ceiωts . If we now apply theconvolution theorem (2.9) we find:

f1 ∗ f2(ω) = f1(ω)f2(ω) = eiω|x−x′|/ceiω|x−xs|/ceiωts . (4.16)

The factor eiωts is a phase shift we don’t have in the frequency domain. However if weFourier transform equation (2.2) we would also get this term, which would then carry

21

to the solution. The second time derivative in equation (4.8) will turn to a factor −ω2

in the frequency domain. If we plug in k = ωc and ignore the phase term(that is, assume

the pulse was sent at t = 0) we find that the Fourier-transform with respect to time of(4.8) is (4.14).

We will now show a more general version of this. That is, we will show that equation(4.12) is the Fourier transform of (4.6) if h = f . In that case we have u0 = U0 andg0 = G0e

iωts . In general we know that the time dependence of the Green’s function forthe wave equation with parameters independent of time to be of the form g(x, t, xs, ts) =ψ(x, x′, t − ts), see section 2.4. If we then again look at the time-dependent part ofequation (4.6) we can define f1(t) = g0(x, x′, t, 0) = ψ(x, x′, t) and f2(t) = u0(x′, t) andfind:

(f1 ∗ f2)(t) =

∫f1(t− t′)f2(t′)dt′ =

∫ψ(x, x′, t− t′)u0(x′, t′)dt′ (4.17)

=

∫g0(x, x′, t, t′)u0(x′, t′)dt′, (4.18)

which we recognize as the time dependent part of equation (4.6). If we now applya Fourier transform in the time domain and use the convolution theorem we obtain(f1 ∗ f2)(ω) = f1(ω)f2(ω) = G0(x, x′)U0(x′) which is exactly what we find in equation

(4.12). The − ∂2

∂t2in (4.6) becomes the ω2 in (4.12) and so we see that equation (4.12) is

indeed the Fourier transform of equation (4.6).

22

5 Migration

This chapter consists of three parts. In the first part we give an argument based on[1] why we should apply the adjoint of the operator that turns the reflectivity functioninto the approximate solution to the data to obtain an image of the subsurface. A muchstronger result was proven by Beylkin in [9]; he proved that the resulting adjoint operatorgives back the discontinuities of the wave speed function, but to prove this we wouldneed quite some mathemathics that is far beyond the scope of this bachelor thesis. Wealso give some interpretation of the resulting formulas. In the second part we illustratethe ideas of Born forward modeling and migration with a MATLAB programme from[6]. In the third part we discuss some practical issues of migration.

5.1 Derivation of migration formula from Born forwardmodeling

In this section we will use the Born Forward Model derived in the last chapter to obtaina method for imaging. This is based on [1].

We will, for now, work in the frequency domain. The experiment consists of sources swhich cause waves in the subsurface, which are then measured by geophones g. We willbe interested in the wavefield due to a specific source and will hence use the followingnotation: U0(x|s) will denote the background wavefield due to source s and δU(x|s) willdenote the perturbation wavefield due to source s. In the time domain we will use thenotation u0(x, t|s) and δu(x, t|s). Using this notation we write the approximation wefound in the previous chapter in the following form:

δU(x|s) = ω2

∫G0(x, x′)m(x′)U0(x′|s)dx′, (5.1)

where m(x) = 2s0(x)δs(x) is the reflection distribution [1]. We now regard U0(x|s) asa ”downgoing wave” caused by the source and δU(x|s) as the reflected ”upgoing wave”.We therefore expect to measure δU(x|s) at the location of the geophones. Thus if wecall D(g|s) the data measured at geophone g due to a signal sent out by source s we canre-interpret the equation as:

D(g|s) = ω2

∫G0(x, x′)m(x′)U0(x′|s)dx′. (5.2)

Since the right side is linear in the function m we may write this as a linear equation:

d = Lm. (5.3)

23

Where d stands for the data D(g|s), m for the reflection distribution function and L forthe linear operator as in equation (5.2). When actually doing calculations the problemwill always be discretized and we may approximate equation (5.3) by a matrix equation:

di =N∑j=1

Lijmj , (5.4)

where we divided the region under interest into a regular grid with N cells, di is thedata vector for source-geophone pair i (i ∈ 1, . . . ,M), mj = m(xj) with xj the centerof the jth cell, and Lij = ω2G0(g, xj)U0(xj |s) where U0(xj |s) is the background wavecaused by a source at location s.

In general this system of equations is overdetermined and should therefore be inter-preted in a least squares sense, that is we try to find m such that ||d−Lm||2 ≤ ||d−Lm||2for all m ∈ RN . Note that Lm|m ∈ RN = Col(L). So we are trying to findy = Lm ∈ Col(L) that minimizes ||y − d||. But this is just finding the best approx-imation of d in Col(L) and we know this is the orthogonal projection of d on Col(Y ),which is characterized by y − d ⊥ Col(Y ). Being orthogonal to Col(L) means beingorthogonal to the columns of L. Thus y is the best approximation of d in Col(L) ifL∗(y − d) = 0. If we plug in y = Lm we obtain the so-called normal equations:

L∗Lm = L∗d. (5.5)

If L∗L is invertible the solution is:

m = (L∗L)−1L∗d. (5.6)

This can be approximated by:

mi =(L∗d)i(L∗L)ii

, (5.7)

when L∗L is ”diagonally dominant” [1]. We can then also normalize L∗L so the diagonalelements are normalized to unity. We then obtain the compact formula

m ≈ L∗d, (5.8)

which says that the model can be obtained by applying the adjoint of the forwardmodelling operator on the data. If we write this out in a matrix equation it reads

mj =

M∑i=1

L∗jidi =

M∑i=1

ω2G0(g(i)|xj)diU0(xj |s(i)). (5.9)

If we remember wehere this equation came from we can see that this is an approximateform of:

m(x) ≈∫ ∞−∞

∫g

∫sω2G0(g, x)D(g|s)U0(x|s)dgdsdω. (5.10)

24

where we integrate over all frequencies to stack together the images for all frequencies.Note that is we choose suitable function spaces for m(x) and D(g|s) with inner products〈f(x), g(x)〉1 =

∫f(x)g(x)dx and 〈A(g|s), B(g|s)〉2 =

∫∞−∞

∫ ∫A(g|s)B(g|s)dgdsdω it

would also be the adjoint in that sense.Equation (5.10) is called the migration formula. We will now give an interpretation.

Let us write equation (5.10) in a more suggestive form:

m(x) ≈∫s

∫ ∞−∞

[∫gG0(g, x)D(g|s)dg

] [ω2U0(x|s)

]dωds. (5.11)

If we remember that G0(x, xs) is the acausal Green’s function we may interpret∫g G0(g, x)D(g|s)dg as a backpropagated version of the wavefield measured by the geo-

phones. If we now apply formula (2.14) from section 2.3 to equation (5.10) we see thatit can be written in the following form

m(x) ≈ − 1

2π

∫s

∫ ∞−∞F−1

[∫gG0(g, x)D(g|s)dg

]∂2

∂t2u0(x, t|s)dtds, (5.12)

which means we are multiplying the backpropagated upward going reflected wave withthe second time derivative of the forward propogated wave at time t and location x overall time to give us our reflection distribution at location x. Intuitively this makes sensebecause when the upgoing wave and downgoing wave are at the same location at thesame time we expect there to be a reflector at that location. This method of migrationis called Reverse Time Migration or RTM for short.

We could also directly use an adjoint for equation (4.8). This is called Kirchhoffmigration and is described in [11].

5.2 MATLAB

Here we will discuss a series of MATLAB files from [6], which illustrate the born modelingand migration which we have discussed in the previous sections. This is done for a twodimensional situation. The code can be applied to different velocity models. We willuse a three layer system to describe the different steps. We divide the area in a grid ofdx × dx sized cells. A velocity model is defined in the form of a matrix whose (j, k)-thentry represents the wave velocity in the (j, k)-th cell, where j represents the x-coordinateand k the z-coordinate. The slowness field is then calculated as the reciprocal of thevelocity field. Then it is smoothed to obtain a background slowness field. The differencebetween the original slowness field and the smoothed slowness field then becomes δs.The smoothing is done by taking for every entry of the matrix the average of a smallbox of the matrix centred at that entry. If we do this for our three layer system weobtain the fields as in figure 5.1.

25

Figure 5.1: Velocity model used for Born forward modeling and RTM. On the left the”true” velocity field is shown. In the middle the smoothed velocity field,which is used as the background velocity. On the right the relflection distri-bution derived from the background and true velocity field is shown.

A source and geophone geography is defined. Then born modeling is done numericallyto generate data. Afterwards this data is used to perform RTM and produce an image.Then a so-called adjoint test is done to see in how far RTM is the adjoint process ofborn modeling and if the code functions properly.

5.2.1 Finite difference method

To approximate the wave equation the MATLAB-programmes use the so-called FiniteDifference method. In this method derivatives are replaced by finite differences. Becauseof this we can express the field at a next time step using the current field and the onefrom the previous time step. More specifically we use a second-order approximation forthe second order time derivative:

(∂2u

∂t2

)nj,k

≈un+1j.k − 2unj,k + un−1

j,k

dt2, (5.13)

and an eighth order approximation for the second spatial derivate in the x-direction :(∂2u

∂x2

)nj,k

≈ 1

dx2

(c1u

nj,k +

5∑i=2

ci[unj+i−1,k + unj−i+1,k

]). (5.14)

Where the constants ci are respectively -205/7, 8/5, -1/5, 8/315 and -1/560. Wherethese come from is discussed in [7]. The second spatial derivative in the y-direction isapproximated in the same manner as the one in the x-direction. If we substitute thisinto the wave equation (2.1) we obtain:

1

v2j,k

un+1j,k − 2unj,k + un−1

j,k

dt2=

1

dx2

(2c1u

nj,k +

5∑i=2

ci[unj+i−1,k + unj−i+1,k

](5.15)

+5∑i=2

[unj,k+i−1 + unj,k−i+1

])+ fnj,k,

26

where unj,k is the wavefield at the (j, k)-th x,z location and at the nth time step, vj,kis the velocity field at the (j, k)-th position and fnj,k represents the forcing term. We

now define αj,k = v2j,kdt

2/dx2. We can rewrite the equation to calculate the wavefield attimestep n+ 1 from the wavefield at the previous two timesteps:

un+1j,k = (2 + 2c1)unj,k − un−1

j,k

+αj,k

(2c1u

nj,k +

n∑i=2

ci[unj+i−1,k + unj−i+1,k + unj,k+i−1 + unj,k−i+1

])+ βj,kf

nj,k. (5.16)

An important thing to note is that we use a finite matrix to describe our systemwhen computing. If we used normal boundaries for our region of interest we would getreflections that we do not want. A solution to this would be to use a much larger matrixthan our region of interest, so that the waves do not have the time to reflect and comeback to our region of interest, but this is computatively very expensive. To deal with thisso called absorbing boundary conditions are used: a layer is created around our regionof interest in which the solutions decay. A detailed description of this falls outside thescope of this thesis, but for more information see [12].

5.2.2 Born modeling

In the programme doing the Born-modeling a background wavefield P and a perturbationwavefield Q are defined. The background wavefield is updated at each time step byequation (5.16) with fnj,k a predetermined sourcefunction s which only acts at the sourcelocation. For the three layer system a so-called Ricker wavelet was used. See figure 5.2.

Figure 5.2: The ricker wavelet which is used as input signal.

To update the perturbed field we first need a Finite Difference version of equation(4.6). For this we need an approximation of m∂2u

∂t2. Note that if there is no source at a

location then the background wavefield satisfies ∂2u∂t2

= v24u. If we use a second order

27

Finite Difference scheme for this we obtain:(m∂2u

∂t2

)nj,k

≈ mj,kvj,k(unj+1,k + unj−1,k + unj,k+1 + unj,k−1 − 4unj,k). (5.17)

So this term is used as the forcing term fnj,k for the perturbation field. In figure 5.3 wesee a screenshot of the background and perturbation wavefield obtained with the bornmodeling code.

Figure 5.3: Snapshot of the Born modeling done in [6]. On the left the backgroundwavefield is shown and on the right the perturbation wavefield.

Something worth noting is that the background field is not purely ”downgoing”, butalso contains(weaker) reflections.

5.2.3 RTM

The RTM-function takes a velocity model and seismogram as input. It first uses forwardmodeling as in equation (5.16) to calculate the background wavefield at the final twotimesteps. Notice now that equation (5.15) is symmetric in time. That is, we can alsouse it to calculate un−1

j,k from the next two timesteps:

un−1j,k = (2 + 2c1)unj,k − un+1

j,k

+αj,k

(2c1u

nj,k +

n∑i=2

ci[unj+i−1,k + unj−i+1,k + unj,k+i−1 + unj,k−i+1

])+ βj,kf

nj,k. (5.18)

Using this the forward propogated downgoing wave is backpropogated in time again.At the same time the upward going wave is approximated by using the seismogramdataas sources. That is, each geophone acts a source for the backward propogation of theupgoing wave, in accordance with the Huygens’ principle. The image is created bymultiplying the velocity squared times the laplacian of downgoing wavefield with the

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upgoing wavefield at each timestep and summing, in accordance with equation (5.12).For our three layer system data was generated by born modeling and then used a inputfor the RTM function. The resulting image is shown in figure 5.4.

Figure 5.4: RTM image made using code from [6], based on the velocity model in 5.1and a synthetic seismogram made with Born modeling.

With some imagination, we can see the two lines of reflection we started out with.There is also quite some noise. A part of this can be explained by the fact that the”downgoing” background wavefield also has upgoing components which will be multipliedwith the backpropogated perturbation field to add to the image. Also we should keepin mind that the RTM is only an adjoint process and not an inverse process. We arealso only using a finite amount of geophones. The lines are also deformed at the ends,causing them to be ”smiles”. This can be explained by the absence of measured datafrom the reflections at the end of the lines, because there are no geophones more to theright or to the left. Another important cause of artefacts relates to aperture: we cannotmeasure all directions and therefore miss k-vectors in our spectrum. Also the rickerwavelet misses the lower frequencies. For more information on this, see [11].

In figure 5.5 we see an image created by using a constant background velocity and a”single-point” reflectivity function. The same process of using Born modeling to obtaina seismogram and then using RTM to make an image was used. We see that instead ofjust getting back the point, we get a ”cross” around the point. This can be explainedby a lack of frequencies in the input signal. Again, for more information, see [11].

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Figure 5.5: ”Migration Green’s function”, obtained by using a constant background ve-locity and a single-point reflectivity for Born modeling and RTM, with codefrom [6].

5.2.4 Adjoint test

In section 5.1 we described RTM to be the adjoint of Born modeling. Remember thatif T : X → Y is a linear operator between two Hilbert spaces we call S : Y → X itsadjoint if:

〈Tx, y〉Y = 〈x, Sy〉X , (5.19)

for all x ∈ X and y ∈ Y . Applying that to our situation we check whether the RTMfunction indeed acts as the adjoint of Born modeling. In the space where the reflectivitydistribution lies we use: ∑

j,k

m2j,k (5.20)

as inner product and for our data(seismogram) we use the following inner product:∑g,n

(unjg ,kg

)2. (5.21)

This corresponds with the integral inner products in section 5.1 up to scaling(but thesame scaling for both inner products). To test whether or not two operators are eachother’s adjoints one can take some elements x ∈ X and y ∈ Y see if 〈Tx, y〉 = 〈x, Sy〉. Inthe case of the three layer model this was done with m and Lm as elements and L andR as operators, where m is the reflectivity distribution, L is born forward modeling and

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R is RTM. The results were 〈Lm,Lm〉 = 957.0849, 〈m,RLm〉 = 957.0667. By using theobtained image from RTM as reflectivity for born modeling it was also possible to calcu-late 〈Lm,LRLm〉 = 13430328.0751 and compare it with 〈RLm,RLm〉 = 13412715.6765.These values support the idea that Born modeling and RTM are adjoint operations. Theadjoint test also serves as a test for the functioning of the code.

5.3 Comments on practicality

We’d now like to make a short comment on the matrix mentioned in section 5.1. Forour three layer system we have 201× 81 = 16281 locations and 100 geophones. So, withour one source, that would mean that the matrix from section 5.1 would have 1628100entries. And that is for just one time step. We also have 2001 timesteps. This is justfor our simple small 2D system. In real situations these numbers grow quickly as thereare many more geophones, more sources and a larger area under interest, which is also3D. In practice the matrix is therefore not calculated explicitly. But even then hugeamounts of data are necessary for the imaging process.

Also we need a backgorund velocity to do RTM in a real situation. Finding a suitablebackground velocityfield is called seismic velocity estimation. To go into detail aboutthis falls outside the scope of this thesis, but for more information how it is done andcan be described mathematically, see [5].

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6 Conclusion

We have discussed the wave equation with variable constants and it was shown that it hasunique solutions in an appropriate Sobolev space if the wave speed c was measurable andbounded away from zero. It was also shown that the solution then depends continuouslyon c. This is important because if we do not have this continuity we cannot hope to findgood approximations if we do not precisely know c. After this we linearized the waveequation and obtained the Born forward modeling formula. This was done in both thetime and frequency domain and it was shown both results agree. This Born formulathen lead to a migration formula, which consisted of applying the adjoint of the Bornoperator to the data to obtain an image. The procedures of Born forward modeling andmigration were then illustrated using MATLAB code from [6].

Further research could be done to look at microlocal analysis and the results fromBeylkin [9], which mathematically justifies all migration algorithms by proving thatusing the adjoint of the modeling operator gives back the right discontinuities of theparameterfunctions describing the subsurface. It would also be worthwile to take acloser look at the concept of aperture and how this effects the images obtained by usingmigration methods. One could for example look at [11] for this. Another point of interestcould be to take a better look at the numerical methods used to execute migrationalgorithms, such as Finite Differenence methods and Absorbing boundary conditions.

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Populaire samenvatting

Vaak wilt men in de seismologie een ”afbeelding” van de ondergrond maken, bijvoorbeeldvoor olie- of gasbedrijven zodat ze de locatie van olie en gas in de ondergrond kunnenvaststellen, maar het kan ook voor wetenschappelijke redenen interessant zijn. De manierwaarop zo’n afbeelding wordt gemaakt is met golven. In het algemeen worden veel af-beeldingen gemaakt met behulp van golven. Denk hierbij aan optische experimenten,waarbij we objecten bekijken met behulp van lichtgolven, maar ook aan ultrasound echo-scopie voor medische toepassingen of radar. Bij die laatste twee zijn het geluidsgolvendie gebruikt worden om een afbeelding te maken. Dat is in ons geval ook zo. Er wordtdus een experiment met geluidsgolven uitgevoerd om informatie over de ondergrond tevinden. Als het experiment op land wordt uitgevoerd worden grote vibrerende trucksof explosieven gebruikt om die geluidsgolven te creeren. Die geluidsgolven weerkaatsendan in de ondergrond en worden op het oppervlak gemeten met een soort microfoons,zie figuur 6.1. Zo’n experiment kan ook op het water worden uitgevoerd. Dan wordenzogenaamde ”airguns” gebruikt om geluidsgolven te creeren en lijnen vol met microfoonsop het wateroppervlak om de gereflecteerde golven te meten.

Figure 6.1: Illustration of the seismic imaginging experiment on land from the website(http://www.innoseis.com/seismic-surveying/)

Als de golven gemeten zijn aan het oppervlak, moet aan de hand van die data eenafbeelding gemaakt worden van de ondergrond. Daar gaat deze scriptie over. Wiskundiggezien worden golven beschreven door de golfvergelijking. De oplossing van de golfvergeli-

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jking is een functie die voor elk punt en voor elk tijdstip zegt hoe de golf eruitziet. Degolfsnelheid speelt een rol in de golfvergelijking. Maar de golfsnelheid verschilt in ver-schillende materialen en moet zelf dus ook beschreven worden door een functie van deplaats. Het ”exact” oplossen van de golfvergelijking met variabele golfsnelheid blijkt erglastig te zijn. Maar door wat versimpelingen te maken kunnen we wel een benaderendeoplossing vinden van de golfvergelijking met variabele golfsnelheid. Dit geeft ons dan een”operator” die de golfsnelheid vertaalt naar de golffunctie. Eigenlijk zijn we natuurlijkgeınteresseerd in het omgekeerde proces: gegeven de goffunctie op bepaalde plaatsen(diewe gemeten hebben), willen we juist de golfsnelheidsfunctie kunnen reconstrueren, wantdie vertelt ons waar wat voor materialen in de ondergrond zitten. We willen wiskundiggezegd eigenlijk een ”inverse” van de net genoemde operator. Dit blijkt ook heel moeilijkte zijn, maar we kunnen wel iets doen wat er op lijkt. Deze ”benaderende inverse” geeftons dan de afbeelding waar we naar op zoek zijn.

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[3] Hormander, L. (1983). The Analysis of Linear Partial Differential Operators (2nded.). Berlin, Germany: Springer-Verlag.

[4] Stolk, C. C. (2000). On the modeling and inversion of seismic data.

[5] Symes, W.W. (2009) Inverse Problems 25 123008

[6] CSIM Wave-equation Series Lab. (z.d.). Retrieved July 11 2018, fromhttps://csim.kaust.edu.sa/files/ErSE328.2013/LAB/WE LABS/lab born.html

[7] Fornberg, B. (1988). Generation of Finite Difference Formulas on Arbitrary Grids.Mathemathics of Computation, 51, 699-706.

[8] C Evans, L. (2010). Partial Differential Equations (2e ed.). New York, United States:American Mathemathical Society.

[9] Beylkin, G. (1985). Imaging of discontinuities in the inverse scattering problem byinverse of a causal generalized Radon transform. Journal of Mathemathical Physics,26, 99-108. doi:10.1063/1.526755

[10] Lions, J. L., & Magenes, E. (1972). Non-Homogeneous Boundary Value Problemsand Applications. New York, United States: Springer-Verlag.

[11] Bleistein, N., Cohen, J. K., & Stockwell, J. W. (2001). Mathemathics of Multi-dimensional Seismic Imaging, Migration and Inversion. New York, United States:Springer.

[12] Johnson, S. G. (2007, August). Notes on PerfectlyMatched Layers [Notes]. Retrieved August 3, 2018, fromhttps://ia801007.us.archive.org/32/items/flooved979/flooved979.pdf

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