Troyan V., Kiselev Y. Statistical Methods of Geophysical Data Processing
Подождите немного. Документ загружается.
Computer exercises 423
Fig. A.18 The seismogram for the observation points (km): 0.003, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35,
0.4, 0.45, 0.5.
Fig. A.19 Restoration of the longitudinal velocity : 1 it is the model of inhomogeneity; 2 it is the
restored perturbation of the velocity.
A.3.7.1 Exercises
(1) Using a script dt0mn1, to investigate an accuracy of the restoration of E by
the diffraction tomography method in dependence on the contrast and size of
inhomogeneity (for example, E = 0.05, 0.1, 0.2, 0.5 ∆ = 0.01, 0.02, 0.03, 0.04 ).
(2) To estimate accuracy of restoration in dependence on a noise value.
This page intentionally left blankThis page intentionally left blank
Appendix B
Tables
The tables which are represented below allow to implement the calculations with
Laplace function, with density functions for χ
2
distribution and Student distribu-
tion.
Table B.1 The values of the function
˜
Φ(z) = 2(1−Φ(z)) (see Sec. 1.8.1,
Eq. (1.42)).
z
˜
Φ(z) z
˜
Φ(z) z
˜
Φ(z) z
˜
Φ(z)
0,0 1,00000 1,0 0,31732 2,0 0,04550 3,0 0,00270
0,1 0,92034 1,1 0,27134 2,1 0,03572 3,1 0,00194
0,2 0,84148 1,2 0,23014 2,2 0,02780 3,2 0,00138
0,3 0,76418 1,3 0,19360 2,3 0,02144 3,3 0,00096
0,4 0,68916 1,4 0,16152 2,4 0,01640 3,4 0,00068
0,5 0,61708 1,5 0,13362 2,5 0,01242 3,5 0,00087
0,6 0,54850 1,6 0,10960 2,6 0,00932 3,6 0,00061
0,7 0,48392 1,7 0,08914 2,7 0,00694 3,7 0,00042
0,8 0,42372 1,8 0,07186 2,8 0,00512 3,8 0,00029
0,9 0,36812 1,9 0,05714 2,9 0,00374 3,9 0,00020
Table B.2 The value x
P
and probability P (x > x
P
)
with degrees of freedom n (see Sec. 1.8.4, Eq. (1.45)).
P
n 0,99 0,90 0,50 0,10 0,01
1 0,0001157 0,0158 0,455 2,706 6,635
2 0,0201 0,211 1,386 4,605 9,210
3 0,115 0,584 2,366 6,251 11,345
4 0,297 1,064 3,357 7,779 13,277
5 0,554 1,610 4,351 9,236 15,086
10 2,558 4,865 9,342 15,987 23,209
15 5,229 8,547 14,399 22,307 30,578
20 8,260 12,444 19,337 28,412 37,566
25 11,524 16,473 24,337 34,382 44,314
30 14,953 20,599 29,336 40,256 50,892
425
426 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING
Table B.3 The values t
P
and probability P of the dif-
ference t from zero average more than t
P
under a num-
ber of the degrees of freedom ν = n − 1 (see Sec. 1.8.5,
Eq. (1.47)).
P
ν 0,9 0,5 0,1 0,05 0,01
1 0,158 1,000 6,314 12,706 63,657
2 0,142 0,816 2,920 4,303 9.925
3 0,137 0,765 2,353 3,182 5,841
4 0,134 0,741 2,132 2,776 4,604
5 0,132 0,727 2,015 2,571 4,032
10 0,129 0,700 1,812 2,228 3,169
15 0,128 0,691 1,753 2,131 2,947
20 0,127 0,687 1,725 2,086 2,845
25 0,127 0,684 1,708 2,060 2,787
30 0,127 0,683 1,697 2,042 2,750
∞ 0,1257 0,6745 1,6448 1,9590 2,5758
Bibliography
Ahlberg, J., Nilson, E. and Walsh, J. (1967). The Theory of Splines and Their Applications
(Academic Press).
Akaike, H. (1974). A new look at the statistical model identification, IEEE Trans. Autom.
Control AC-19, pp. 716–723.
Aki, K. and Richards, P. (2002). Quantitative seismology. (University Science Books).
Albert, A. (1972). Regression and the Moore-Penrose Pseudoinverse (Academic Press).
Alekseev, A., Lavrentev, M. and Preobrazhenskii, N. (1985). Problems of reconstructive
tomography (in russian) (SO AN SSSR).
Alekseev, A., Mikhailenko, V. and Cheverda, V. (1983). Numerical methods of solution of
the inverse and direct problems in the theory of wave propagation (in russian) (SO
AN SSSR), pp. 165–172.
Anderson, N. (1974). On the calculation of filter coefficients for maximum entropy spectral
analysis, Geophysics 39, pp. 69–72.
Andrews, D. (1974). A robust method for multiple linear regression, Technometrics 16, 4,
pp. 523–531.
Aronov, V. (1990). Methods of creation of maps of geological and geophysical characteristics
and application of geometry for mapping of oil and gaz (in russian) (Nedra, Moscow).
Babic, V. and Buldyrev, V. (1991). Short-Wavelength Diffraction Theory. Asymptotic
Methods (Springer-Verlag).
Babic, V., Buldyrev, V. and Molotkov, I. (1999). The Space–Time Ray Method (Cambridge
University Press).
Backus, G. and Gilbert, F. (1967). Numerical applications of a formalism for geophysical
inverse problems, Geophys. J. Roy. Astr. Soc. 13, pp. 247–276.
Backus, G. and Gilbert, F. (1968). The resolving power of gross earth data, Geophys. J.
Roy Astr. Soc. 16, pp. 169–205.
Bakushinski, A. and Goncharski, A. (1994). Ill-posed problems: theory and applications
(Kluwer).
Belishev, M. (1990). Equations of the gelfand-levitan type in a multidimensional inverse
problem for the wave equation, Journal of Mathematical Sciences 50, 6, pp. 1940–
1944.
Berkhout, A. (1984.). Seismic migration — Imaging of acoustic energy by wave field ex-
trapolation (practical aspects) (Elsevier).
Berkhout, A. (1985). Seismic migration — Imaging of acoustic energy by wave field extrap-
olation (theoretical aspects) (Elsevier).
Beylkin, G. (1985). Imaging of discontinuities in the inverse scattering problem by inversion
of a causal generalized radon transform, J. Math. Phys. 26, 1, pp. 99–108.
427
428 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING
Blagoveshchenskii, A. S. (1966). About inverse problem of seismic wave propagation (in
russian), Probl. Mat. Physici , 1, pp. 68–81.
Blagoveshchenskii, A. S. (1978). Inverse problems of theory of elastic wave propagation
(in russian), Izv. AN SSSR. Fizika Zemli , 12, pp. 45–54.
Blagoveshchenskii, A. S. (2005). An inverse problem of the theory of wave propagation in
a random layered medium, Differential Equations (Pleiades Publishing) 41, 10, pp.
1442–1448.
Bleistein, N., Cohen, J. and Stockwell, J. (2000). Mathematics of Multidimensional Seismic
Imaging, Migration and Inversion (Springer).
Box, G. and Jenkins, G. (1976). Time series analysis: forecasting and control (Holden
Day).
Brandt, S. (1998). Statistical and Computational Methods in Data Analysis (Springer).
Brekhovskikh, L. (1960). Waves in layered media (Academic Press).
Brekhovskikh, L. and Godin, O. (1998). Acoustics of Layered Media I: Plane And Quasi-
Plane Waves (Springer-Verlag).
Brekhovskikh, L. and Lysanov, Y. (1991). Fundamentals of Ocean Acoustics (Springer-
Verlag).
Bukhgeim, A. (2000). Introduction to the Theory of Inverse Problems (VSP Intl Science).
Burg, J. (1972). The relationship between maximum entropy spectra and maximum like-
lihood spectra, Geophysics 37, pp. 375–376.
Claerbout, J. (1976). Fundamentals of geophysical data processing (Blackwell).
Claerbout, J. (1985). Imaging the Earth’s Interior (Blackwell).
Cramer, H. (1946). Mathematical methods of statistics (Princeton University Press).
Ermakov, S. and Mikhailov, G. (1982). Statistic simulation (in russian) (Nauka, Moscow).
Ermakov, S. and Zhigljavsky, A. (1987). Mathematical Theory of Optimal Experiment (in
russian) (Nauka, Moscow).
Faddeev, L. (1976). Inverse problems of quantum scattering theory, J. Sov. Math. , 5, pp.
334–396.
Fedorov, V. (1972). Theory of Optimal Experiments (Academic Press Inc.).
Fedotov, A. (1990). Ill-Posed Problems with Random Errors in the Data (in russian)
(Nauka, Novosibirsk).
Goldin, S. (1984). Inverse kinematical problems of reflection seismology – i. classical prob-
lems in laterally inhomogeneous media, Geophysical Journal of the Royal Astronom-
ical Society 79, 1, pp. 317–325.
Goltsman, F. (1971). Statistical models of interpretation (in russian) (Nauka, Moscow).
Helgason, S. (1999). The Radon Transform (Birkhauser).
Herman, G. (1980). Image Reconstruction from Projections (Academic).
Holland, J. (1992). Adaptation in Natural and Artificial Systems: An Introductory Analysis
with Applications to Biology, Control and Artificial Intelligence (MIT Press).
Huber, P. (2004). Robust statistics (Wiley).
Jensen, F. B., Kuperman, W. A., Porter, M. B. and Schmidt, H. (2000). Computational
ocean acoustics (Springer-Verlag).
Johnson, N. and Lion, F. (1977). Statistics and experimental design in engineering and in
the physical sciences (Wiley).
Karus, E., Kuznetsov, O. and Faizullin, I. (1986). Interhole Sonic Survey (in russian)
(Nedra, Moscow).
Kennett, T. (1983). Seismic wave propagation in stratified media (Cambridge University
Press).
Kireitov, V. and Pikalov, V. (1985). Linear and nonlinear problems of computerized to-
mography (in russian) (VTS SO AN SSSR, Novosibirsk).
Bibliography 429
Kiselev, Y. and Troyan, V. (1998). Numerical simulation in diffraction tomography (in
russian), in Voprosy geophiziki, 35 (Saint Petersburg University).
Kolmogorov, A. (1956). Foundations of the theory of probability (Chelsea Publishing Com-
pany).
Kozlov, E. (1986). Migration transform in seismic exploration (in russian) (Nedra,
Moscow).
Kravtsov, Y. (2005). Geometrical Optics in Engineering Physics (Alpha Science Interna-
tional).
Kravtsov, Y. and Apresyan, L. (1996). Radiative transfer: new aspects of the old theory,
in Progress in Optics, Vol. XXXVI (North Holland), pp. 179–244.
Kullback, S. (1978). Information theory and statistics (Dover Publications).
Lavrentev, M. M. (1967). Some improperly posed problems of mathematical physics
(Springer Tracts Nat. Philos., vol. II ) (Springer-Verlag).
Lee, Y. (1960). Statistical theory of communication (John Wiley and Sons Inc.).
Levshin, A., Yanovskaya, T., Lander, A., Bukchin, B., Barmin, M., Ratnikova, L. and
E.N., I. (1989). Seismic surface waves in a laterally inhomogeneous Earth (Kluwer
Publ.).
Marchuk, G. (1982). Methods of Numerical Mathematics (Springer-Verlag).
Meshalkin, L. and Kurochkina, A. (1981). A new approach to the parametrization of
regression dependences, Journal of Mathematical Sciences 17, 6, pp. 2269–2275.
Morozov, V. (1993). Regularization methods for ill-posed problems (CRC Press).
Newton, R. (1981). Inversion reflection data for layered media: a review of exact methods,
Geophys. J. Roy. Astr. Soc 65, 1, pp. 191–215.
Nikitin, A. (1986). Theoretical basis for processing of geophysical information (in russian)
(Nedra, Moscow).
Nolet, G. (1985). Solving or resolving inadequate and noisy tomographic systems, J. Comp.
Phys. 61, p. 463482.
Nolet, G., Trier, J. and Huisman, R. (1986). A formalism for nonlinear inversion of seismic
surface waves, Geophys. Res. Lett. 13, 1, pp. 26–29.
Penenko, V. (1981). Mathematical methods of design of experiment (in russian) (Nauka,
Novosibirsk).
Petrashen, G. (1978). Basis of the mathematical theory of elastic wave propagation, in
Voprosy dinamicheskoi teorii rasprostraneniya seismicheskikh voln (in russian), 18
(Nauka, Leningrad).
Petrashen, G. (1980). Wave propagation in anisotropic elastic media (in russian) (Nauka,
Leningrad).
Petrashen, G. and Kashtan, B. (1984). Elementary of the theory of body wave propagation
in anisotropic elastic media (in russian) (Nauka, Leningrad).
Petrashen, G., Molotkov, L. and Krauklis, P. (1985). Waves in homogeneous-layered elastic
media (in russian) (Nauka, Leningrad).
Petrashen, G. and Nakhamkin, S. (1973). Continuation of the wave fields in problems of
seismic exploration (in russian) (Nauka, Leningrad).
Pikalov, V. V. and Preobrazhenski, N. G. (1983). Computer-aided tomography and phys-
ical experiment, Sov. Phys. Usp. 26, 11, pp. 974–990.
Populis, A. and Pillai, S. (2002). Probability, random variables and stochastic processes
(Mc-Graw Hil).
Pugachev, V. S. (1984). Probability theory and mathematical statistics for engineers (Perg-
amon).
Pytev, Y. (1990). The methods of analysis and nterpretaion of the experiments (in russian)
(Vyssh. Shkola, Moscow).
430 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING
Rao, K. (1972). Linear statistical infernece and its applications (Wiley).
Romanov, V. (1974). Integral Geometry and Inverse Problems for Hyperbolic Equations.
Springers Tracts in Natural Philosophy, vol. 26 (Springer-Verlag).
Romanov, V. (1987). Inverse Problems of Mathematical Physics (VNU Science Press).
Ryzhikov, G. (1976a). About covergence of the iterative methods of the solution of ill-
posed problems (in russian), in Problemy phiziki atmosfery, 14 (Leningrad State
University).
Ryzhikov, G. (1976b). Construction of algorithm of the solution of ill-posed problemswith
the use of fructional-rational operator functions (in russian), in Problemy phiziki
atmosfery, 17 (Leningrad State University).
Ryzhikov, G. and Troyan, V. (1985). Application of the methods of computational to-
mography to interpretative seismic problems (in russian), in Vestnik Leningradskogo
Universiteta, 4 (Leningrad State University).
Ryzhikov, G. and Troyan, V. (1986a). Determination of the wave field and velocity in
the media with the perturbation method (in russian), in Vestnik Leningradskogo
Universiteta, 4 (Leningrad State University).
Ryzhikov, G. and Troyan, V. (1986b). Restoration of the velocity field in 3-d medium with
the use of the elastic wave field functionals (in russian), Izvestiya AN SSSR, Ser.
Phizika Zemli , 7, pp. 49–56.
Ryzhikov, G. and Troyan, V. (1986c). Two approaches to the problem of processing of the
sonar data (in russian), Deponirovano VINITI 12.09.86, number 6649-86.
Ryzhikov, G. and Troyan, V. (1987). Linear estimation of the functionals of the elasic
parameter fields and seismic thomographs (in russian), in Voprosy dinamicheskoi
teorii rasprostraneniya seismicheskikh voln, 27 (Nauka, Leningrad).
Ryzhikov, G. and Troyan, V. (1989). Tomographic functionals and interpretative tasks of
the elasitc wave sounding (in russian), in Voprosy dinamicheskoi teorii rasprostra-
neniya seismicheskikh voln, 28 (Nauka, Leningrad).
Ryzhikov, G. and Troyan, V. (1990). Back projection in diffraction tomography (in
russian), in Voprosy dinamicheskoi teorii rasprostraneniya seismicheskikh voln, 29
(Nauka, Leningrad).
Ryzhikov, G. and Troyan, V. (1994). Tomography and inverse sounding problems (in rus-
sian) (Saint Petersburg Unibersity).
Ryzhikov, G., Troyan, V. and Yurin, A. (1989). Regualrization methods in 3-d seismic ray
tomography problems (in russian), in Voprosy dinamicheskoi teorii rasprostraneniya
seismicheskikh voln, 28 (Nauka, Leningrad).
Schumaker, L. (2007). Spline Functions: Basic Theory (Cambridge University Press).
Sheriff, R. and Geldart, L. (1995). Exploration seismology (Cambridge University Press).
Smylie, D., Clarke, C. and Ulrych, T. (1973). Analysis of irregularities in the earth’s
rotation, in S. F. B. Adler and B. A. Bolt (eds.), Methods in Computational Physics,
Vol. 13 (Academic Press), pp. 391–430.
Sp¨ath, H. (1973). Spline-Algorithmen zur Konstruktion glatter Kurven und Fl¨achen (R.
Oldenbourg Verlag).
Stolt, R. and Weglein, A. (1985). Migration and inversion of seismic data, Geophysics 50,
8, pp. 2458–2472.
Stratonovich, R. (1975). The theory of information (in russian) (Sov. Radio, Moscow).
Sudvarg, D. (1974). About using the polinomial multisections in description of the seismic
interfaces (in russian), in Application of the matematical statistics and computational
mathematics for processing the seismic data (Nauka, Novosibirsk).
Tarantola, A. (1984). Inversion of seismic reflection data in the acoustic approximation,
Geophysics 49, 8, pp. 1259–1266.
Bibliography 431
Tikhonov, A. and Arsenin, V. (1977). Solutions of Ill-Posed Problems (Wiley).
Tikhonov, A., Arsenin, V. and Timonov, A. (1987). Mathematical Problems in Computer
Tomography (in russian) (Nauka, Moscow).
Timoshin, Y. (1978). Impulse seismic holography (in russian) (Nedra, Moscow).
Troyan, V. (1981a). Application of the spline functions for approxiamtion of geophysical
information (in russian), in Voprosy dinamicheskoi teorii rasprostraneniya seismich-
eskikh voln, 20 (Nauka, Leningrad).
Troyan, V. (1981b). Application of the spline functions for approxiamtion of the geopysical
information (in russian), in Voprosy dinamicheskoi teorii rasprostraneniya seismich-
eskikh voln, 20 (Nauka, Leningrad).
Troyan, V. (1981c). Statistacal methods for approxiamtion of the geophysical information,
in Annales Universitates Scientiarum Budapestinnensis de Rolando E¨otv¨os Nami-
natae, Sectio Geologica, XIII (Roland Eotvos University).
Troyan, V. (1982). Statistical methods of processing of seismic information in in vestiga-
tions of the complex media (in russian) (Nedra, Moscow).
Troyan, V. (1983). Solution of the interpretative seismic problems by the finite elements
method (in russian), in Vestnik Leningradskogo Universiteta, 4 (Saint Ptersburg
State University).
Troyan, V. (1984). Estimation of the seismic signal parameters by the correct moment
method (in russian), in Voprosy dinamicheskoi teorii rasprostraneniya seismicheskikh
voln, 24 (Nauka, Leningrad).
Troyan, V. (1985). The method of the local estimation of the seismic wave parameters (in
russian), in Vestnik Leningradskogo Universiteta, 2 (Saint Petersburg State Univer-
sity).
Troyan, V. and Kiselev, Y. (2000). Statistical methods of processing and interpretation of
geophysical data (in russian) (Saint Petersburg University).
Troyan, V. and Ryzhikov, G. (1994). 3-d diffraction seismic tomography. 1. tomography
functional and reconstruction algorithm (in russian), Geophisika , 6, pp. 4–12.
Troyan, V. and Ryzhikov, G. (1995). 3-d diffraction seismic tomography. 2. resolution of t
he method (in russian), Geophisika , 2, pp. 7–18.
Troyan, V. and Sokolov, Y. (1982). Approximation of the seismic horizon with taking into
account borehole data (in russian), in Voprosy dinamicheskoi teorii rasprostraneniya
seismicheskikh voln, 22 (Nauka, Leningrad).
Troyan, V. and Sokolov, Y. (1983). Determination of the bedded velocity by the area
seismic data (in russian), in Voprosy dinamicheskoi teorii rasprostraneniya seismich-
eskikh vol, 23 (Nauka, Leningrad).
Troyan, V. and Sokolov, Y. (1989). The methods of the geophysical data approximation
with the use of computer (in russian) (Nedra, Leningrad).
Troyan, V. and Windgassen, W. (1979). Approximation geophysikalischer daten unter
ber¨ucksichtigung linearer bedingungen, Geophysik und Geologie 11, 1, pp. 22–30.
Turchin, V., Kozlov, V. and Malkevich, M. (1971). The use of mathematical-statistical
methods in the solution of incorrectly possed problems, Sov. Phys. Uspechi 13, 6,
pp. 681–703.
Ursin, B. (1983). Review of elastic and electromagnetic wave propagation in horizontally
layered media, Geophysics 48, pp. 1063–1081.
Ursin, B. and Berteussen, K.-A. (1986). Comparison of some inverse methods for wave
propagation in layered media, Proc., IEEE 74, pp. 389–400.
Vasilenko, G. and Taratorin, A. (1986). Image Restoration (in russian) (Radio i Svyaz’,
Moscow).
432 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING
Voevodin, V. and Kuznetsov, Y. (1984). Matrix and calculations (in russian) (Nauka,
Moscow).
Volkov, A. (1981). Application of spline-function for geological and geophysical problems
solution (in russian) (Nauka, Moscow).
Yanovskaya, T. (1996). Ray tomography based on azimuthal anomalie, PAGEOPH 148,
1/2, pp. 319–336.
Yanovskaya, T. and Porokhova, L. (2004). Inverse Problems in. Geophysics (in russian)
(Saint Ptersburg University).
Zhdanov, M., Frenkel, M. A., Matusevich, V. I. and Urupov, A. K. (1988). Seismic and
electromagnetic migration (in russian) (Nauka, Moscow).