Estimation of the of seismic waves parameters
We consider a problem of estimation of the parameters of a seismic signal based on real observed data. For this purpose, we propose a new mathematical model that reduces the task to a nonlinear, nonsmooth, nonconvex minimization problem. Using a special structure of the signal, we also propose a n...
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| Cite this: | Estimation of the of seismic waves parameters / V.S. Mostovoy, S.V. Mostovyi // Доповiдi Нацiональної академiї наук України. — 2014. — № 2. — С. 118-123. — Бібліогр.: 15 назв. — англ. |
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| author | Mostovoy, V.S. Mostovyi, S.V. |
| author_facet | Mostovoy, V.S. Mostovyi, S.V. |
| citation_txt | Estimation of the of seismic waves parameters / V.S. Mostovoy, S.V. Mostovyi // Доповiдi Нацiональної академiї наук України. — 2014. — № 2. — С. 118-123. — Бібліогр.: 15 назв. — англ. |
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| description | We consider a problem of estimation of the parameters of a seismic signal based on real observed
data. For this purpose, we propose a new mathematical model that reduces the task to a nonlinear, nonsmooth, nonconvex minimization problem. Using a special structure of the signal, we
also propose a numerical algorithm of finding the solution of the optimization problem. Our
method is based on the combination of the Levenberg–Marquardt algorithm and a simulated
annealing type approach. We show the convergence of the algorithm and discuss the practical
implications, which lie in a good compatibility with real seismic data, and the applicability to experiments.
Розглядається задача оцiнки параметрiв сейсмiчного сигналу, що базується на реальних
спостережених даних. Для цього пропонується нова математична модель, яка зводить
задачу до нелiнiйної негладкої задачi неопуклої мiнiмiзацiї. Пропонується чисельний алгоритм знаходження розв’язку задачi оптимизацiї, що враховує специфiку структури сигналу. Метод заснований на комбинацiї алгоритму Левенберга–Марквардта и метода симуляцiї аннiлiнга. Показано збiжнiсть алгоритму, його практичне застосування та хорошу сумiснiсть моделi з сейсмiчними экспериментальными даними.
Рассматривается задача оценки параметров сейсмического сигнала, основанная на реальных наблюденных данных. Для этого предлагается новая математическая модель, которая сводит задачу к нелинейной, негладкой задаче невыпуклой минимизации. Предлагается численный алгоритм нахождения решения задачи оптимизации, учитывающий специфику структуры сигнала. Метод основан на комбинации алгоритма Левенберга–Марквардта и метода симуляции аннилинга. Показаны сходимость алгоритма, его практическое применение и хорошая совместимость модели с сейсмическими экспериментальными данными.
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UDC 550.834
V.S. Mostovoy, S. V. Mostovyi
Estimation of the parameters of seismic waves
(Presented by Academician of the NAS of Ukraine V. I. Starostenko)
We consider a problem of estimation of the parameters of a seismic signal based on real observed
data. For this purpose, we propose a new mathematical model that reduces the task to a nonli-
near, nonsmooth, nonconvex minimization problem. Using a special structure of the signal, we
also propose a numerical algorithm of finding the solution of the optimization problem. Our
method is based on the combination of the Levenberg–Marquardt algorithm and a simulated
annealing type approach. We show the convergence of the algorithm and discuss the practical
implications, which lie in a good compatibility with real seismic data, and the applicability to
experiments.
Introduction. One of the central problems in the seismic signal processing is the determina-
tion of parameters of a signal such as the damping decrement, the principal eigenfrequency, the
moment of arrival, etc. Usually, seismic signals such as compression plane waves, distortional
waves, surface waves, Rayleigh waves, ground roll modes, and Love waves are modeled via physi-
cally realizable signals, see [2, 14, 15], that are characterized by the spectral band. It is natural to
approximate such signals by a superposition of Berlage impulses, as a generalization of the model
of simple oscillators. Such kind of modeling is widely used in seismology. With the goal to find
an optimal approach to the estimation of the “key” seismic parameters, it might be reasonable to
restrict ourself to a set of mathematical models of the seismic signals that are used in practice.
One way of doing this is to use Berlage impulses that have enough degrees of freedom for the
approximation of a wide class of seismic signal forms in comparison to other types of signals, see,
e. g., [2]. In this case, the form of a signal is defined by a 5-dimensional vector of free parameters.
In Section 2, we propose a novel approach that is based on an enlargement of the vector of free
parameters to a 6-dimensional vector. This gives more flexibility and allows one to consider dif-
ferent models in one formulation. By choosing some of the free parameters to be zero or one, we
can include almost every practically interesting case in our model and, in particular, to recover
the original 5-dimensional models. Our approach leads to a minimization problem, a solution to
which gives the optimal set of “key” parameters in the sense explained below. We also propose a
numerical algorithm for solving this optimization problem, convergence properties of which are
established and are discussed as well.
One of the merits of our method is in the balance between the ability to well approximate the
desired set of parameters of the signals (from the practical side) and the analytical tractability
(from the theoretical viewpoint). In practice, our model was successfully implemented in multiple
experiments such as the oil and gas detection and the monitoring of natural and man-made
objects, see [10, 11, 12] for details. From the theoretical point of view, first, it it proven in [12] that
the solution to our central optimization problem (5) exists. Second, since the objective function
in the minimization problem (5) is nonlinear, nonsmooth, nonconvex, and multidimensional,
the delicate optimization techniques are required to construct an approximation to the solution
to (5). Thus, we propose a numerical algorithm of solving (5), the efficiency of which is based
© V. S. Mostovoy, S.V. Mostovyi, 2014
118 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2014, №2
on local continuity, boundedness, and differentiability of the objective. Behind our numerical
method are the well-known ideas of nonlinear optimization such as the simulated annealing and
the Levenberg–Marquardt algorithm, see [7, 8, 13]. Our approach turns out to be convergent
to a true solution (in probability with respect to a certain probability measure introduced in
Section 4), as the running time increases. Convergence results were established in [12]. Finally,
it should be mentioned that our method is well-suited for parallel computing, and the running
time of numerical evaluations is of order of minutes on a standard lap top, which is convenient
in experiments.
The remainder of the paper is organized as follows: in Section 2, we describe the model of
signal, Section 3 contains the formulation of the central optimization problem (5), a numerical
algorithm of solution to which is presented in Section 4. Conclusion is stated in Section 5.
2. Mathematical model of seismic activity process. Let us consider a general represen-
tation of the impulses used to model many seismic signals, which are often used in seismology.
Free parameters of such impulses define different models. Let us define the function
S : R× [−A,A]× R+ × N
⋃
{0} × R+ × R+ × [Ω1,Ω2]× R+ → R
given by
S(t, a, α, β, ω, τ, T ) = aIτ,τ+T (t) exp{−α(t − τ)}(t − τ)βsin(ω(t− τ)), (1)
where the meaning of all the parameters is described below. From the physical point of view, all
the parameters in (1) are physically substantial ones. The meaning of the symbols in the previous
expression is the following: t stands for the time argument, the other variables are treated as the
free parameters of the model. The parameter a denotes the amplitude of the signal. It is the only
parameter that enters model (1) linearly, whereas all other parameters enter (1) nonlinearly. The
first nonlinear parameter α is the damping characteristic of oscillations. The next parameter β
is used for a correction of the impulse front. This parameter gives us the possibility to regulate
the steepness of the pulse edge. Parameter τ is used for physically realizable impulses as the
time characteristic of the signal appearance (the left endpoint of the indicator interval of the
signal). Parameter ω characterizes the angular frequency of the impulse oscillation. Parameter T
characterizes the length of the interval, where the signal exists (τ +T is the right endpoint of the
indicator interval). We restrict the admissible set of amplitudes and frequencies to the intervals
[−A,A] and [Ω1,Ω2], respectively, where A, Ω1, and Ω2 are some positive constants. In (1),
Iτ,τ+T (t) denotes the indicator function of the interval [τ, τ + T ], i. e.
Iτ,τ+T (t) =
{
1, if t ∈ [τ, τ + T ],
0, if t ∈ (−∞, τ)
⋃
(τ + T,+∞).
By changing the parameters τ and T , we have a possibility to restrict the signal to the interval
between its appearance and disappearance. We will call such impulse (1) a generalized Berlage
impulse with the linear parameter a and five nonlinear other parameters, as oppose to the Berlage
impulse defined, e. g., in [2], which is a particular case of the generalized Berlage impulse when
the parameters β = 0 and T = ∞. We also need to introduce the vector P of the free parameters
of model (1). Its representation is given by
P = {Pk}; k = 1, 6; {P1 = a, P2 = α,P3 = β, P4 = τ, P5 = ω,P6 = T}. (2)
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2014, №2 119
In view of (2), we can rewrite (1) as follows:
S(t,P) = P1IP5,P5+P6
(t) exp{−P2(t− P5)}(t− P5)
P3 sin(P4(t− P5)). (3)
Fixing certain components of the vector P of the seismic signal model, one can obtain a restriction
of the model of seismic signal. Restricted signals will be determined by a smaller number of free
parameters and will have more specific properties. In order to set up the optimization problem (5)
and to construct a numerical algorithm of its solution, it is convenient to arrange such kind of
vectors into a matrix. This matrix will determine the set of different models of seismic signals.
The index of every model will be the same as the column number of this matrix. In such a case,
we will get a possibility to model the recorded data via a superposition of signals, see (4) below.
Let us demonstrate how certain signals widely used in seismic practice can be obtained from
our model (1) by changing the parameters of the model that are encapsulated in vector P.
For example, when P3 = 1, we get a model of approximated Berlage impulse in the interval
[P5, P5 + P6]:
S(t,P) = P1IP5,P5+P6
(t) exp{−P2(t− P5)} sin(P4(t− P5)).
When P6 = ∞, the indicator of the interval IP5,P5+P6
(t) becomes the Heaviside step function
η(t − τ), where
η(x) =
{
1, if x > 0,
0, if x < 0,
and we get a Berlage impulse with the linear parameter P1 denoting its amplitude. If P3 is equal
zero and P6 = ∞, expression (3) models a fading sinusoid
S(t,P) = P1η(t− P5) exp{−P2(t− P5)} sin(P4(t− P5)),
which can be represented in terms of widely used symbols in physics as follows:
S(t, a, α, ω, τ) = aη(t− τ) exp{−α(t− τ)} sin(ω(t− τ)).
If P2 = P3 = 0 and t ∈ [P5, P5 + P6], the impulse will turn into the first Fourier harmonic on
the interval [P5, P5 + P6] with a period P6:
S(t,P) = P1IP5,P5+P6
(t) sin(P4(t− P5)).
The natural generalization, which allows us to consider more complicated signals, is based
on a superposition of signals (3). In this case, we can define a (more general) signal S(t,M) as
S(t,M) =
Q∑
q=1
S(t,Pq), (4)
where the matrix M consists of the column vectors Pq, q = 1, Q, and contains all free parameters
of our model. The signal S(t,Pq) with a specified vector Pq is called the q-th submodel. The
set of the admissible values for M is denoted by A, which is uniquely defined by the domain
of S. Finally, we are ready to state the model of signal that we used for applications and that
120 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2014, №2
is considered in the following sections. For every seismic record y(t), t ∈ [0,T ], where T is the
length of the seismic record, we suppose that y(t) is the sum of a seismic signal of the form (4)
and an additive background noise n(t),
y(t) = S(t,M) + n(t), t ∈ [0,T ].
3. Optimization problem. The mathematical problem consists of an estimation of the
matrix of free parameters M corresponding to a recorded seismogram y(t), t ∈ [0,T ], and some
statistical characteristics of the background noise n. In the simplest model of the noise n, one can
consider an additional condition that the background noise is uncorrelated, i. e. its autocorrelation
is the δ-function. We assume that the a priori distribution of M is a uniform distribution over
A and use the goodness-of-fit test for F(M) to set up an optimization problem, a solution to
which we will call the optimal value of M, where F is defined as
F(M) =
T∫
0
(y(t)− S(t,M))2dt, M ∈ A.
This leads to the optimization problem
inf
M∈A
F(M), (5)
where A is the set of admissible values of M defined above. When the objective is continuous, one
can see that problem (5) is the Mayer problem in the calculus of variations, see [3, 5]. Therefore,
variational methods are natural candidates for the analysis of (5).
Since the primal goal of this work is to construct a model and a mathematical approach for
the practical purposes, it is natural to satisfy ourself with an approximate solution to (5) with
a given tolerance ε > 0, i. e. by calculating a value M̂ such that
F(M̂)− inf
M∈A
F(M) 6 ε.
Such a value M̂ will be called a solution to (5).
One drawback of the variational approach is in the requirement of continuity (or even
smoothness) of the objective F that is often not available in practice. In particular, in our
case, impulse (1) includes the discontinuous term I corresponding to the step function, see (3).
Therefore one has to use the alternative techniques of nonlinear optimization to study (5). Addi-
tional challenges lie in the nonconvexity and the high nonlinearity of F. Of course, one can
approximate F with a smooth function, e. g. via mollifiers, see [4], and to study the correspondi-
ng smooth problem using variational techniques. However, since mollification does not remedy
nonlinearity, one still has to apply some delicate methods of nonlinear optimization, see, e. g., [1].
Solution to the optimization problem (5). The main difficulty in the construction of
an effective numerical algorithm for finding a solution to (5) is the discontinuity of the objective
F explained above. Instead of mollifying the objective, we propose a different approach that is
convergent (to the true solution of (5)) and analytically tractable in the sense explained below.
Even though our objective F is not continuous everywhere in its domain, it is (infinitely many
times) differentiable by Lebesgue almost everywhere in A, except for some set of discontinuities
of (Lebesgue) measure zero. Moreover, F does not show a pathological behavior at discontinuities,
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2014, №2 121
outside of which it stays locally smooth and uniformly bounded on compact sets. Therefore, we
can use optimization techniques to find the local minima. Numerically, we use the Levenberg–
Marquardt algorithm for this purpose, see [7, 8, 13]. Having the procedure of finding the local
minima, we can hope to recover the global minimum by repeating the Levenberg–Marquardt
procedure many times starting from different (randomly chosen) initial points. The reader can see
the intimate relation to the simulated annealing and Metropolis–Hastings algorithms, see [6, 9].
In our settings, the problem is studied in [12], and the convergence in probability (for a given
tolerance, with respect to a probability measure on A that is used for the generation of starting
points) is proven there. Moreover, we can even show the almost sure convergence if the set A
is assumed to be compact (this is a reasonable and non-restrictive assumption that often holds
in practice, if we have some a priori information about the possible ranges of the components
of M). Thus, if we run our algorithm for a sufficiently long time, the estimate very close to the
true value of a minimizer to (5) will be found. The description of the applications of our method
to various practical problems is contained in [12].
To guarantee the convergence of the algorithm in probability, any probability distribution
over A, such that the Lebesgue measure on A is absolutely continuous in the probability measure,
i. e. every subset of A with a positive Lebesgue measure has positive probability, could be used
for the generation of starting points. From the practical point of view, our algorithm is tailor
made for parallel computing. It converges fast enough to be used on a usual computer, since
the Levenberg–Marquardt procedure is extremely efficient for minimizing twice differentiable
functions, which is almost our case, except for the fact that F is differentiable (infinitely many
times) locally, not globally.
Conclusion. We proposed a new model of seismic data and an algorithm of finding the
optimal values of the seismic parameters of special interest, such as the damping decrement,
the principal eigenfrequency, and the moment of arrival. Our model is a generalization of the
classical seismic models in [2]. The determination of the optimal values of free parameters of the
model is based on a criterion, which we aim to minimize. This leads to a nontrivial optimization
problem (5), a solution to which can be obtained via a numerical algorithm we constructed
as well. In turn, our numerical algorithm is studied both theoretically (here, the existence of
a solution and the convergence of the algorithm are proven) and practically in the numerical
experiments (see examples in [10]).
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122 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2014, №2
11. Mostovyi S. V., Starostenko V. I. Interpretation of geophysical data under uncertainty information // Izv.
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Received 19.08.2013Institute of Geophysics of the NAS of Ukraine, Kiev
National Aviation University, Kiev
В.С. Мостовий, С.В. Мостовий
Оцiнка параметрiв сейсмiчних хвиль
Розглядається задача оцiнки параметрiв сейсмiчного сигналу, що базується на реальних
спостережених даних. Для цього пропонується нова математична модель, яка зводить
задачу до нелiнiйної негладкої задачi неопуклої мiнiмiзацiї. Пропонується чисельний алго-
ритм знаходження розв’язку задачi оптимизацiї, що враховує специфiку структури сигна-
лу. Метод заснований на комбинацiї алгоритму Левенберга–Марквардта и метода симу-
ляцiї аннiлiнга. Показано збiжнiсть алгоритму, його практичне застосування та хорошу
сумiснiсть моделi з сейсмiчними экспериментальными даними.
В.С. Мостовой, С.В. Мостовой
Оценка параметров сейсмических волн
Рассматривается задача оценки параметров сейсмического сигнала, основанная на реаль-
ных наблюденных данных. Для этого предлагается новая математическая модель, кото-
рая сводит задачу к нелинейной, негладкой задаче невыпуклой минимизации. Предлагается
численный алгоритм нахождения решения задачи оптимизации, учитывающий специфи-
ку структуры сигнала. Метод основан на комбинации алгоритма Левенберга–Марквардта
и метода симуляции аннилинга. Показаны сходимость алгоритма, его практическое приме-
нение и хорошая совместимость модели с сейсмическими экспериментальными данными.
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2014, №2 123
|
| id | nasplib_isofts_kiev_ua-123456789-86980 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1025-6415 |
| language | English |
| last_indexed | 2025-11-24T15:13:06Z |
| publishDate | 2014 |
| publisher | Видавничий дім "Академперіодика" НАН України |
| record_format | dspace |
| spelling | Mostovoy, V.S. Mostovyi, S.V. 2015-10-07T19:21:46Z 2015-10-07T19:21:46Z 2014 Estimation of the of seismic waves parameters / V.S. Mostovoy, S.V. Mostovyi // Доповiдi Нацiональної академiї наук України. — 2014. — № 2. — С. 118-123. — Бібліогр.: 15 назв. — англ. 1025-6415 https://nasplib.isofts.kiev.ua/handle/123456789/86980 550.834 We consider a problem of estimation of the parameters of a seismic signal based on real observed data. For this purpose, we propose a new mathematical model that reduces the task to a nonlinear, nonsmooth, nonconvex minimization problem. Using a special structure of the signal, we also propose a numerical algorithm of finding the solution of the optimization problem. Our method is based on the combination of the Levenberg–Marquardt algorithm and a simulated annealing type approach. We show the convergence of the algorithm and discuss the practical implications, which lie in a good compatibility with real seismic data, and the applicability to experiments. Розглядається задача оцiнки параметрiв сейсмiчного сигналу, що базується на реальних спостережених даних. Для цього пропонується нова математична модель, яка зводить задачу до нелiнiйної негладкої задачi неопуклої мiнiмiзацiї. Пропонується чисельний алгоритм знаходження розв’язку задачi оптимизацiї, що враховує специфiку структури сигналу. Метод заснований на комбинацiї алгоритму Левенберга–Марквардта и метода симуляцiї аннiлiнга. Показано збiжнiсть алгоритму, його практичне застосування та хорошу сумiснiсть моделi з сейсмiчними экспериментальными даними. Рассматривается задача оценки параметров сейсмического сигнала, основанная на реальных наблюденных данных. Для этого предлагается новая математическая модель, которая сводит задачу к нелинейной, негладкой задаче невыпуклой минимизации. Предлагается численный алгоритм нахождения решения задачи оптимизации, учитывающий специфику структуры сигнала. Метод основан на комбинации алгоритма Левенберга–Марквардта и метода симуляции аннилинга. Показаны сходимость алгоритма, его практическое применение и хорошая совместимость модели с сейсмическими экспериментальными данными. en Видавничий дім "Академперіодика" НАН України Доповіді НАН України Науки про Землю Estimation of the of seismic waves parameters Оцiнка параметрiв сейсмiчних хвиль Оценка параметров сейсмических волн Article published earlier |
| spellingShingle | Estimation of the of seismic waves parameters Mostovoy, V.S. Mostovyi, S.V. Науки про Землю |
| title | Estimation of the of seismic waves parameters |
| title_alt | Оцiнка параметрiв сейсмiчних хвиль Оценка параметров сейсмических волн |
| title_full | Estimation of the of seismic waves parameters |
| title_fullStr | Estimation of the of seismic waves parameters |
| title_full_unstemmed | Estimation of the of seismic waves parameters |
| title_short | Estimation of the of seismic waves parameters |
| title_sort | estimation of the of seismic waves parameters |
| topic | Науки про Землю |
| topic_facet | Науки про Землю |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/86980 |
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