OPTIMIZATION MODEL OF POWER SYSTEM UNIT
COMMITMENT ALLOCATION PROBLEM
Aleffder Dedfiurk
University of Warmia and Mazury in Olsztyn
July 9, 2013
Abstract. Consider the problem of reconstruction of a small perturbation of the acoustic wave speed field
from traveltime data with linear background slowness. Mathematically, the problem is equivalent to
reconstruction of a function from the data of integrals along the circle arcs. The data are limited, in the sense
that the arc base points belong to a compact set. We propose and numerically test a new approach, based on
reduction of the problem to the inverse problem for the Radon transform. The data completion procedure is
considered as well.
Key words: Radon transform, Seismic tomography, Inverse kinematic problem, Spherical mean
transform, Interpolation of band-limited function
1. Introduction
Let H be a half-disk {x21 + x22 < 1, x2 > 0} (注:公式必须用工具编辑器编写). For a
function f , supp f ⊂ H define the arc mean transform as an integral over an arc A a,R,
centered at the daylight surface {x2 = 0}:
S p x
I p x  I p  t
 dark ,

  similar , I p  t  I p  x  I p  t
 bright ,
I p  t  I px

(1)
where a is the center, R is the radius of the arc, Fig. 1.
This transform appears in several applications. In seismic tomography, the traveltime
in ray approximation is given by an integral:
N

| I ( x)  I ( p) |  t
(2)
x ( compass ( i ))
where v = v(x) is the velocity structure of the Earth, integration is over the ray, which is
a geodesic line of the metrics dsv . Under assumption that the slowness (inverse to the
velocity) field of the media is of form 1/v(x1, x2) = (bx2 + c)−1 + εf (x1, x2), where b,
c > 0, ε ≪ 1, the mean transform M f in (1) is a linearized perturbation of the travel time
data. We refer to books [4, 19] (注: 参考文献按照时间顺序排列,并在正文中用"[]"
标注)for the practical problem of seismic waves modeling and substantiation of this
model.
In the synthetic radar aperture image processing f (x1, x2) is considered as a ground
reflectivity and measured M f (a, R) is interpreted as a mean reflectivity at distance a
around the position of the radar carrier at a time R ([2], [8])
Fig. 1.
Domain H and arc AA,R
In thermoacoustic and photoacoustic tomography ([20, 21]) f (x 1, x2) describes the
energy distribution function, and, in case of the constant wave speed, the inverse
problem can be reduced to inversion of the mean operator (1).
The inversion problem for the complete data of arcs was considered by various
authors in [4, 1, 9, 7, 6, 11, 8, 20] and some others.
We assume, that the data are known only for |a| + R < 1, i. e. Aa,R ⊂ H. Following
the paper [17], we call it the local arc problem. The local arc problem looks more
realistic from the practical point of view, at least for the seismic tomography.
In [17] the problem was reduced to the limited angle problem for the Radon
transform on the plane. In [14] the problem was reduced to other incomplete data
problem for the Radon transform on the plane. In [13] an algebraic discretization method
based on the microlocal analysis of the geometry of the arc family was proposed. This
method was implemented in [16].
In this paper we propose and numerically test a new approach, based on reduction of
the problem to the inverse problem for the Radon transform. This approach is similar to
considerations from [14].
In comparison with commonly-used algebraic discretization algorithms, the proposed
approach gives more quick algorithm. Besides, since the problem is reduced to the
standard tomographical algorithms, it is possible to apply analysis of tomographical
reconstruction algorithms to the spherical mean transform.
It is well-known (cf. [18]) that the limited angle problem for the Radon transform,
and therefore, the local arc problem, are ill-posed and the standard reconstruction algorithms work poor. To recover as much information as possible, we propose a data
completion procedure. The data completion procedure is based on the interpolation
formula from [10]. As it is shown in numerical simulation, data completion permits to
eliminate some artifacts in reconstruction.
Fig. 2.
Phantom function is equal to 1 inside ellipses and to 0 elsewhere
For the purpose of numerical simulation we consider a phantom, that consists of
characteristic function of two ellipses with semi-axes parallel to coordinate axes (see
Fig. 2). One of them is the long ellipse with the center at (−0.25, 0.33) and semi-axes of
0.1 and 0.3 respectively. The second is the wide ellipse with the center at (0.4, 0.15) and
semi-axes of 0.47 and 0.1. Both ellipses locate inside the circle {|x| < 0.9}
We will reconstruct from a discrete set of data. Fix N + 1 = 129 points on the
daylight surface:
ak = −1 + 2k/N,
k = 0, . . . , N.
enote by Ak,l the arc with the diameter [ak, al].
We suppose that the array Gk,l, 0 ≤ k < l ≤ N of data is known. Here Gk,l is the mean
transform (1) for Ak,l.
with noisy data without interpolation. So, data completion procedure seems to be stable.
(2)
6. Conclusion and future work
We proposed and numerically tested some algorithms for reconstruction of a function
from incomplete data of arc means in the seismic tomography model. Algorithms are
based on different reductions of the problem to the standard computer tomography
reconstruction. The data completion procedure is proposed as well.
Due to strong ill-posedness, reconstruction results unavoidably differ from the
original and have artifacts. However, they are still useful. And the question of stability of
reconstruction is very important. Our simulation shows that proposed algorithms are
quite stable.
Algorithms need further investigation. Specifically, it’s desirable to obtain error
estimates. These questions will be studied elsewhere.
7. Acknowledgement
Preliminary version of the work was presented at the seminar on multimedia and
com-puter graphics at the University of Warmia and Mazury in Olsztyn. The author is
grateful to professors Mokrzycki, Mąka and Tatol for their significant remarks.
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