Troyan V., Kiselev Y. Statistical Methods of Geophysical Data Processing

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Algorithms of approximation of geophysical data 243

The least squares estimate, with the use of the expansion (8.79), is written as

b = Σ

Z, where b

b = Y

ρ, Z

Z = T

If the diagonal elements σ

of the matrix Σ do not equal to zero, then the estimate

bb is

= Z

/σ

. But if some σ

are small, then such procedure is undesirable. The

values σ

should be analysed before obtaining the estimate

b. It is proved, that the

singular numbers non equal zero if and only if the columns of the matrix Ψ are linear

independent. Study of the singular numbers should implement taking into account

an accuracy of the input data and the calculation accuracy. Taking into account a

priori accuracy we choose the threshold condition σ

for the comparison of each σ

(see Sec. 6.12): if σ

≥ α, then

= Z

/σ

; if σ

< α, then

= 0. In general, at

< α the vector component b

can set an arbitrary value. Such arbitrariness is a

reason of the non-uniqueness of the least squares solution. To exclude an inﬂuence

of the non-stable components of the vector b

, in the case of σ

< a, usually, these

components put to zero.

Under linear constraints (8.75) in addition to the expansion (8.79) the expansion

a = GW F

can be used, where







0, i 6= j,

> 0, i = j, i ≤ Q

= 0, i = j, i > Q

, Q

≤ Q;

G and F are the orthogonal matrices; Q

is a number of linearly independent con-

ditions.

Let us consider the results of the model example of a determination of the

velocity taking into account the borehole data. At Fig. 8.9 the synthetic region of

observations is represented. The velocities at the proﬁles are calculated depending

on the thickness and the depth of the horizon, besides the normal noise is simulated

and the velocity gradient in x direction is introduced. At Fig. 8.10 the results

of processing with the use of four proﬁles are represented. The approximating

velocity surface is computed using the borehole data (two variants of the velocity

dependence) coincides with the velocity values in the boreholes and in other points

has the properties peculiar to the least squares method.

244 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING

Fig. 8.9 Synthetic seismic observations (a) with location of the proﬁles (I – IV), boreholes and

depth proﬁles: (b) is proﬁle I, (c) is proﬁle II, (d) is proﬁle III, e is proﬁle IV. Dash-dot lines is a

model location of the horizon, solid lines are the border of deviation area of the horizon.

Algorithms of approximation of geophysical data 245

Fig. 8.10 The results of processing of the model example from Fig. 8.9.

Lines 1 are the initial velocity data along the proﬁles; 2 is the results of the restoration of the

velocity by the seismic data without taking into account the borehole observations; 3, 4 are the

results of restoration of the velocity taking into account the borehole observations for two types

of the velocity proﬁles in the boreholes:

(a) is the proﬁle I; for borehole 1 white circle: v = 3650 m/s, ∆v = −15 m/s, dark circle:

v = 3250 m/s, ∆v = −550 m/s; for borehole 5 white circle: v = 5750 m/s, ∆v = 470 m/s, dark

circle: v = 5500 m/s, ∆v = 220 m/s;

(b) is proﬁle II; for borehole 1 white circle: v = 3650 m/s, ∆v = −150 m/s, dark circle: v = 3250

m/s, ∆v = −550 m/s; for borehole 2 white circle: v = 3375 m/s, ∆v = 125 m/s, dark circle:

v = 2750 m/s, ∆v = −500 m/s;

m/s, ∆v = −50 m/s; for borehole 3 white circle: v = 5250 m/s, ∆v = 230 m/s, dark circle:

v = 4500 m/s, ∆v = −520 m/s;

(d) is proﬁle IV; for borehole 4 white circle: v = 4350 m/s, ∆v = 190 m/s, dark circle: v = 3900

m/s, ∆v = −260 m/s.

∆v is a deviation between data, obtained by the seismic observations and in the boreholes.

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Chapter 9

Elements of applied functional analysis for

problem of estimation of the parameters

of geophysical objects

A numerical solution of the estimation problem and, generally, the methods of the

computational mathematics are based on the applied functional analysis. An opera-

tional language is very helpful for a compact representation of the algorithms of the

inverse problem solution, to estimate of the accuracy of the inverse problem solution

with taking into account the most general properties of the considered operators,

to trace the modiﬁcations of some elements of the algorithm in the dependence of

the concrete functional binding of the considered inverse problem. A solution of

the ﬁrst-kind operator equation, having the compact operator is an independent

mathematical problem: the problem of the regularization of the ill-posed tasks.

The solution of any intepretating problem, connected with the processing of the

experimental data, should contain the random component (“noise”) as a part of the

input data of the inverse problem. We should note that in this case the classical

methods of the functional analysis are inadmissible, because in general case the data

and noise do not belong to the image of the operator. This fact alone leads to using

the methods of the mathematical statistics for the construction of the algorithms

and the interpretation of the solution of the inverse problems.

The existing literature on these problems: the applied functional analysis, the

solution of the ill-posed problems and the methods of mathematical statistics are

very extensive and a presentation style usually is far from the “physical style”, when

the proof if ever produces, but without details. It is unconditionally, at such enun-

ciating the mathematical rigor is lost, but it allows to exchange the mathematical

erudition by the physical intuition, which facilitates the physical interpretation of

the solution.

9.1 Elements of Applied Functional Analysis

Let us consider the basic elements of the functional analysis.

As we set the problem of the approximate solution, for example, ﬁnding of the

function ϕ, with the deﬁnite accuracy grade satisﬁes the equation Lϕϕ = s, it is

appeared the problem of the deﬁnition of the measure of the accuracy, i.e. we

247

248 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING

should ﬁnd the measure of the functions proximity. The measure may be only a

number which corresponds to the representation of the functional space elements

ϕ ∈ Φ on the real axis R

or in other words it is necessary to deﬁne a functional

N : Φ −→ R

, and to determine the requirements, which must satisfy this functional

as the measure of the proximity of two functions e.g. ϕ

and ϕ

(1) a functions deviation is non-negative: N (ϕ

−ϕ

) ≥ 0, or N (ϕ

−ϕ

) = 0 ⇔

= ϕ

;

(2) a multiplication of the both functions on a number λ > 0 leads to a change of

the deviation in |λ| times: N (λ(ϕ

− ϕ

)) = |λ|N (ϕ

− ϕ

);

(3) by analogy with the Euclidean geometry a sum of the deviations ϕ

from ϕ

and ϕ

from ϕ

is greater than or equal to the deviation ϕ

from ϕ

: N (ϕ

−

) + N (ϕ

− ϕ

) ≥ N ((ϕ

− ϕ

) + (ϕ

− ϕ

)).

It is not diﬃcult to see that the above mentioned requirements (a, b, c) give

the deﬁnition of a norm of a function. We rewrite the conditions a)–c), replacing

(ϕ

−ϕ

) ⇒ ϕ in a) and b); ϕ

−ϕ

⇒ ψ and ϕ

−ϕ

⇒ ϕ in c) and N (ϕ) ⇒ kϕk,

in the following form:

(1) kϕk ≥ 0 , kϕk ⇔ ϕ = 0 is a non-negativity of a norm;

(2) kλϕk = |λ|kϕk is a homogeneity of a norm;

(3) kϕ + ψk ≤ kϕk + kψk is the triangle inequality.

Let’s mark, that by the genesis of the term norm is obliged to the property of

the homogeneity, which one allows at any given function ϕ by multiplying it on the

factor λ to make the magnitude of kλϕk equal to unity (to normalize function).

The conditions, deﬁned the norm, are satisﬁed for the following presentations:

(1) kϕk =

|ϕ(x)|dx. This norm deﬁnes the space of modulus integrable functions

on [a, b].

(2) kϕk = max

x∈[a,b]

|ϕ(x)|. The norm deﬁnes the space C on [a, b] of continuous

functions.

(3) kϕk =

|ϕ(x)|

1/2

. The norm deﬁnes the space L

on [a, b] of square

integrable functions.

(4) kϕk =

|ϕ(x)|

dx +

|ϕ

(x)|

1/2

. The norm in the Sobolev’s space W

on [a, b].

(5) kϕk =



i=1



1/2

. The norm in the space R

It should be noted that the norm notion coincides with the notion of a distance

metric ρ (ϕ

, ϕ

) = kϕ

− ϕ

Elements of applied functional analysis 249

A metric space (i.e. the space with the norm) is called a complete space, if any

fundamental sequence of the elements of this space converges to an element of the

space. We should remind that the fundamental sequence is such that for an arbitrary

ε > 0 is found a number n(ε), that for any elements of the sequence n

, n

> 0 are

satisﬁed the inequality ρ (ϕ

, ϕ

) < ε. The complete normalized space is called

Banach space.

By analogy with the vector analysis it is introduced a scalar product for elements

of the functional space with the next properties:

(1) S(ϕ, ϕ) ≥ 0 , S (ϕ, ϕ) = 0 ⇔ ϕ = 0 is a non-negativity of the vector modulus;

(2) S(ϕ, αψ

+ βψ

) = αS (ϕ, ψ

) + βS (ϕ, ψ

) is a linearity on the second element;

(3) S (ϕ, ψ) = S

∗

(ψ, ϕ), if ϕ , ψ are complex valued.

Hence, it follows

S(λϕ, ψ) = λ

∗

S (ϕ, ψ) .

We shall use a diﬀerent designation for the scalar product:

S(ϕ

ϕ,ψ

ψ)

∆

= (ϕ, ψ) ,

ϕ,

ψ)

∆

= hϕ|ψi

(the designation of Dirac: h

ϕ| is a bra vector,|ψi is a ket vector (“bra + ket =

bracket”: h|i). An example of a scalar product is

S(ϕ, ψ) =

∗

(x)ψ(x)dx.

The Banach space with a scalar product is called the Hilbert space H. Let us

show that the construction of a scalar product permits to introduce a concordant

norm. Actually, S(ϕ, ϕ) ≥ 0, and in this sense the ﬁrst property of a scalar product

has analogy with the property of the norm:

S(λϕ, λϕ) = λ

∗

λS(ϕ, ϕ) = |λ|

S(ϕ, ϕ) ,

i.e. in the contrast to the functional N, which is a homogeneous function of the

ﬁrst order N(λϕ) = |λ|N(ϕ), the scalar product is a homogeneous function of the

second order.

Let us show that a norm can be introduced as:

N(ϕ) = (S(ϕ, ϕ))

1/2

i.e. we will show that such functional satisﬁes the triangle inequality, and, hence,

it satisﬁes for all three properties, which have been introduced for the norm. We

write the Shwarz’s inequality:

|(ψ, ϕ)| ≤ (ϕ, ϕ)

1/2

(ψ, ψ)

1/2

0 ≤ (ϕ + αψ, ϕ + αψ) = (ϕ, ϕ) + α(ϕ, ψ) + α

∗

[(ψ, ϕ) + α(ψ, ψ)] .

250 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING

So far as a scalar product is grater than or equal zero for any α, we choose α such as

the term in square brackets was equal zero, i.e. α = −(ψ, ϕ) / (ψ, ψ) we obtain the

Schwarz’s inequality. We write down a triangle inequality ||ϕ + ψ|| ≤ ||ϕ||+ +||ψ||,

using a scalar product

(ϕ + ψ, ϕ + ψ)

1/2

≤ (ϕ, ϕ)

1/2

+ (ψ, ψ)

1/2

It is really,

(ϕ + ψ, ϕ + ψ) = (ϕ, ϕ + ψ) + (ψ, ϕ + ψ) ≤

≤ (ϕ, ϕ)

1/2

(ϕ + ψ, ϕ + ψ)

1/2

+ (ψ, ψ)

1/2

(ϕ + ψ, ϕ + ψ)

1/2

where the last inequality is the consequence of the Schwarz inequality.

We have shown that the space with the scalar product is deﬁned by the norm

(the norm in the Hilbert space)

kϕk = (ϕ, ϕ)

1/2

For instance, if the scalar product is deﬁned as

∗

(x)ψ(x) dx, so the norm of that

space reads as

kϕk =



∗

(x)ϕ(x)dx



1/2

and the corresponding Schwarz’s inequality



∗

(x)ϕ(x)dx



≤



∗

(x)ϕ(x)dx



1/2



∗

(x)ψ(x)dx



1/2

takes the name the Cauchy–Bunyakovskii inequality. We have simultaneously shown

that the functional



|ϕ(x)|



1/2

satisﬁes the triangle inequality



(ψ(x) + ϕ(x))

∗

(ψ(x) + ϕ(x) )dx



1/2

≤



|ψ(x)|



1/2



|ϕ(x)|



1/2

The scalar product is

(ϕ, ψ) =

i=1

in R

, and in the Sobolev space W

(ϕϕ, ψψ) =

∗

(x)ψ(x)dx +

0∗

(x)ψ

(x)dx .

The orthogonality condition is (ϕ, ψ) = 0.

We would remind the deﬁnition of the linear independence of elements {ϕ

}, i =

1÷n. The elements {ϕ

} are linearly independent, if from the equality

= 0

follows that α

= 0 for all i = 1 ÷ n.

Elements of applied functional analysis 251

The basis of the Hilbert space H is called the set Φ ⊆ H that for an arbitrary

element of H there is a convergent sequence, which is a linear combination of the

elements ϕ ⊆ Φ, in other words, the set Φ is the basis H, if its elements are linearly

independent and linear closure of Φ coincides with H. A basis is called orthonormal,

if all elements of the basis are mutually orthogonal and its norm is equal to 1. The

dimension of the basis is the dimension of the Hilbert space.

Let us show that in any basis the presentation ϕ =

is unique. We assume

that ϕ has an another presentation ϕ =

, then 0 = ϕ −ϕ =

(α

−β

)ϕ

But from a condition of a linear independence of the basis it follows, that α

= β

for any i.

The analogue of the vector presentation in the co-ordinate system given by

the orthonormal basis e

, ϕ

ϕ =

(

ϕ, e

, in the Hilbert space is determined by

generalized Fourier series

ϕ(x) =

(x),

where ϕ

(x) are the basis functions; α

are the coeﬃcients of the expansion,

which are equal to (ϕ, ϕ

). It is a familiar example of a countable basis in

the inﬁnite dimensional Hilbert space L

[−π, π], the basis of an trigonometrical

function: (2π)

−1/2

, (π)

−1/2

sin x, (π)

−1/2

cos x, (π)

−1/2

sin(2x), (π)

−1/2

cos(2x),

. . . , (π)

−1/2

sin(nx), (π)

−1/2

cos(nx), . . . The expansion of a function ϕ into Fourier

series in terms of trigonometrical functions:

ϕ(x) =

√

2π

−π

ϕ(x)dx +

√

∞

k=1



cos(kx)

−π

ϕ(x) cos(kx)dx

+ sin(kx)

−π

ϕ(x) sin(kx)dx



The possibility of the function presentation as a coeﬃcient set in a given basis

deﬁnes the eﬀectiveness of the introduced scalar product in a functional space.

We would remind a necessary information from the operator theory. The func-

tion L, acting from the space H

into the space H

, is called an operator:

−→ H

If H

= R

, then the operator is called a functional. The operator L is the linear

operator, if the condition

L(αϕ + βψ) = αLϕ + βLψ

is satisﬁed. The example of an integral operator in the space L

Lϕ

∆

L(x, x

)ϕ(x

)dx

252 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING

If the integral kernel L is such that

(x, x

)dxdx

< ∞, then L is the Hilbert–

Schmidt operator. The norm of the linear operator is the functional

||L|| = sup

kLϕk

kϕk

Let us compare the norm of the functional with the case of the ﬁnite dimensional

vector space, when the matrix

L transforms the vector ϕ into the vector ψψ, i.e.

Lϕ

∆

Lϕ = ψψ. In this case the norm of the operator L is a maximum possible

length |ψψ| of the vector ψψ, lying on the surface of the sphere (|ϕ| = 1). If the norm

L is less than inﬁnity, the operator is called limited. The example of an unlimited

operator is L = d

/dx

, which acting in the space of the square integrable functions

on the interval [−π, π]:



= sup

k(d

/dx

)ϕk

kϕk

Choosing as a function ϕ the orthonormal trigonometric basis, we see that

||(d

/dx

)ϕ

(x)||

||ϕ

(x)||

= k

||ϕ

(x)||

||ϕ

(x)||

k→∞

−→ ∞.

From the deﬁnition of the operator norm L it follows the inequality

kLϕk ≤ kLk · kϕk.

The operator L is called compact, if this operator transforms an each weakly conver-

gent sequence {ϕ

} from a range of its deﬁnition into a strong convergent sequence,

i.e.

L : (f, ϕ

− ϕ

)

∀f

−→ 0 ⇒ kL(ϕ

− ϕ

)k → 0.

As example of the compact operator in the space L

the integral Hilbert–Schmidt

operator

Lϕ

∆

L(x, x

)ϕ(x

)dx

can be considered. A speciﬁcity of the inﬁnite dimensional Hilbert space in compar-

ison with the ﬁnite dimensional space consists in that from the weak convergence

the strong convergence does not follow. Let’s compare with a vector space: if

∀f (f,ϕϕ

− ϕ

) ⇒ 0,

then ϕ

−→

. This statement is equivalent to the fact that in the inﬁnite-

dimensional space an unit operator is the noncompact operator. The sequence of

elements of an inﬁnite-dimensional basis {ϕ

, n = 1 ÷N } converges to zero weakly:

= (f, ϕ

)

n→∞

−→ 0,

so far as from the Parceval equality follows

= kf k

= C < ∞. The sequence

{Iϕ

= ϕ

} does not have a limit:

||I(ϕ

− ϕ

)|| = (ϕ

− ϕ

, ϕ

− ϕ

)

1/2

= [(ϕ

, ϕ

) + (ϕ

, ϕ

)]

1/2

√

2 .