Rastgele ortamlarda dalga yayılımının modellenmesi
Başlık çevirisi mevcut değil.
- Tez No: 39458
- Danışmanlar: PROF.DR. NEZİHİ CANITEZ
- Tez Türü: Doktora
- Konular: Jeofizik Mühendisliği, Geophysics Engineering
- Anahtar Kelimeler: Dalga yayılması, Rastgele ortamlar, Sismik dalgalar, Wave propagation, Random mediums, Seismic waves
- Yıl: 1993
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET İnhomojen ortamlardaki dalga yayılımı rastgele ortam tanımı kullanarak incelenebilir. Uzaysal hızların ve yoğunlukların dağılımının stokastik süreçler ile incelenebildiği, istatistiksel momentler ile tanımlanan rastgele ortam kullanımı ile oluşturulan sismogramlar, gerçek bir sismogramda olduğu gibi, yayılma zamanlarındaki ve genlikteki sapmaları, soğrulmayı, ikincil dalgalan ve koda dalgalarının oluşumunu içermektedir. Sismogramlardaki bu tür etkileri kullanarak, ortamdaki inhomojenliklerin büyüklüklerinin kestirilmesi tezin ana fikrini oluşturmaktadır. Rastgele ortam olarak Gaussian, von Karman ve genelleştirilmiş Gaussian ortamlar ile çalışılmıştır. Gaussian ortam belirli ölçekten değişimlere izin verdiğinden, ancak yavaş değişim içeren ortamların modellenmesinde kullanılmıştır, von Karman ortam ise, her türlü ölçekten değişim içeren ortamların modellenmesi amacıyla kullanılmıştır. Sürekli ortamlar için tanımlı olan bu yaklaşım ile sonlu ve ayrık değerli gözlem uzayında çalışmak mümkün değildir. Bundan dolayı, fiziksel problemlerin incelenebilmesi amacıyla bu tezde, genelleştirilmiş Gaussian adını verdiğimiz yeni bir ortam tanımlaması oluşturulmuştur. Rastgele ortamlardaki dalga yayılımı, yeni bir yöntem olan ve bu tezde tanımlanan Hartley yöntemi ile modellenmiştir. Oluşturulan sismogramlara ait kodalarm spektrumları modellenerek, rastgele ortamı tanımlayan istatistiksel büyüklüklerin kestirimi yapılmaya çalışılmıştır. Modellemede kullanılan bağıntılar, Born yaklaşımı kullanılarak elde edilmiştir. Gaussian ortam için bulunan sonuçlar ile ortamı tanımlayan istatistiksel büyüklüklerin kestirimi tam doğrulukla yapılmıştır. Sürekli ortamlar için tanımlı olan von Karman ortam için elde edilen sonuçların kestirimi ise yetersiz duyarlıkta elde edilmiştir. Bu tezde tanımlanan ve fraktal özellik gösteren von Karman ortama yaklaşım sağlayan genelleştirilmiş Gaussian ortamı tanımlayan büyüklüklerin kestirimi tam olarak yapılabilmiştir. Oluşturulan bu yaklaşım TUBITAK-Gebze sayısal deprem istasyonunun kaydettiği patlatma verilerinden seçilen bir veri setine uygulanmıştır. Aynı kaynak-alıcı konumu için bulunan koda spektrumları benzer özelliklere sahip olarak elde edilmiş ve teori ile uyumlu sonuçlar bulunmuştur. Sonuç olarak, farklı ölçekten değişim içeren İnhomojen ortamlardaki istatistiksel büyüklüklerin eldesi için yeni bir yöntem tanımlanmıştır. Oluşturulan bu yöntem, benzeşimler ve gerçek veri ile uyumlu sonuçlar vermiştir. Ek olarak, iki yeni yaklaşım tanımlanmıştır. İlk yaklaşım, her türlü ortamdaki dalga yayılımınm modellenmesini sağlayan Hartley yönteminden oluşmaktadır, ikinci yaklaşım ise, farklı ölçekten değişimleri içeren rastgele ortamı tanımlayan genelleştirilmiş Gaussian ortam tanımını içermektedir. vi
Özet (Çeviri)
asymmetric form. The lower limit approximately equals the sampling distance and agrees with our theoretical results. Also, the correlation function of het erogeneous medium is derived. The agreement between the simulated and theoretical correlations is excellent. In the last part of this thesis, we apply these concepts on a real data set. This data set was from quarry blast records of the TUBITAK-Gebze digital seismic station. All of the records have the same source-receiver path. Each coda power spectrum shows a band limited form approximately in the [1-21^2] interval. Using the backscattering coefficient which is defined for generalized Gaussian media, the lower and upper limit of the correlation function is es timated. The lower limit is found to be 910m and the upper limit 1200m. This results have confidence in this interval according to the source-receiver distance and the spatial sampling. However, we can not correlated with other data sets which has different source-receiver path; because, thus far we are not be able to collect additional data suitable for this purpose. In conclusion, a method for the determination of statistical parameters of a multi-scale heterogenous medium has been developed. This method appears to give very satisfactory results for the simulated and real data sets. Also, this method does not need any special tools such as spectral estimation techniques. In addition, two new ideas are demonstrated. First, the Hartley method is introduced for modeling wave propagation in general media. Next, generalized Gaussian random media, for defining multi-scale random media, is defined and demonstrated. xivrandom media. They simulated wave propagation with the 2- D finite difference method. Their results showed that the backscattering coefficient can be suc cessfully predicted by the codas power spectrum for Gaussian random media. However, they could not invert the backscattering coefficient for exponential random media, successfully. Roth and Korn (1993) investigated the validity of single scattering theory in weakly inhomogeneous 2-D random media using the finite difference approximation. They used different autocorrelation functions from Frankel and Clayton (1986). Their results showed that single scattering theory can be used to determine the energy loss of a plane wave due to scat tering in both 1-D and 2-D random media and agree with a analytical result of Sato (1984). Finite difference modeling has also been applied to the elastic wave equation; both to study the relationship between the medium and the ob served scattered field (Frankel and Clayton, 1986; Mclaughlin and Anderson, 1987; Dougherty and Stephens, 1988; Charrette, 1990; Coates and Charrette, 1993) and to study the response of typical seismic processing streams used on data collected in highly heterogeneous regions (Gibson and Levander, 1988; 1990). In the scattering literature, highly heterogenous media are often approx imated by random fields. The advantage of this approach is that a complex, multi-dimensional velocity function can be expressed in terms of a few simple statistical parameters. In Chapter 2, the conditions for statistical characteri zation is described. One statistical parameter which can be used to describe the variability of a velocity field is the autocorrelation function. The prop erties of three commonly used autocorrelation functions, the Gaussian, expo nential, and von Karman are investigated, and their applicability to earth is discussed. All three spectra are nearly flat at low wavenumbers, but at higher wavenumbers the Gaussian falls off exponentially, while the exponential and von Karman fall off with a power law dependence. The fall off rate controls the roughness of the medium. Those characterized by the Gaussian autocorre lation are smoothly varying, while the exponential and von Karman functions are more highly textured. These functions are isotropic and inhomogeneities do not have any preferred orientation. It is probably realistic to consider, for the entire crust, that small scale inhomogeneities do not have a preferred orientation. However, in seismic exploration, we are interested in inhomo geneities in a particular rock or in small region of the crust where anisotropy gains importance. One example might be the deposition of overlapping lenses with different lithologies. The lens shape suggests that the correlation length of these features might be different in the horizontal and vertical direction. Although each lens may have isotropic elastic moduli, the composite medium may display an effective or apparent anisotropy. The preferred orientation of the fluctuations should be reflected in the autocorrelation function. For sim plicity, it will be assumed that all azimuthal variation in the autocorrelation function can be explained through the dimensionless ellipsoidal norm. The interesting aspect of ellipsoidal autocorrelation functions is that they allow us to describe media in which the inhomogeneities are isotropic, elongated in a particular direction, or even flattened.random media. They simulated wave propagation with the 2- D finite difference method. Their results showed that the backscattering coefficient can be suc cessfully predicted by the codas power spectrum for Gaussian random media. However, they could not invert the backscattering coefficient for exponential random media, successfully. Roth and Korn (1993) investigated the validity of single scattering theory in weakly inhomogeneous 2-D random media using the finite difference approximation. They used different autocorrelation functions from Frankel and Clayton (1986). Their results showed that single scattering theory can be used to determine the energy loss of a plane wave due to scat tering in both 1-D and 2-D random media and agree with a analytical result of Sato (1984). Finite difference modeling has also been applied to the elastic wave equation; both to study the relationship between the medium and the ob served scattered field (Frankel and Clayton, 1986; Mclaughlin and Anderson, 1987; Dougherty and Stephens, 1988; Charrette, 1990; Coates and Charrette, 1993) and to study the response of typical seismic processing streams used on data collected in highly heterogeneous regions (Gibson and Levander, 1988; 1990). In the scattering literature, highly heterogenous media are often approx imated by random fields. The advantage of this approach is that a complex, multi-dimensional velocity function can be expressed in terms of a few simple statistical parameters. In Chapter 2, the conditions for statistical characteri zation is described. One statistical parameter which can be used to describe the variability of a velocity field is the autocorrelation function. The prop erties of three commonly used autocorrelation functions, the Gaussian, expo nential, and von Karman are investigated, and their applicability to earth is discussed. All three spectra are nearly flat at low wavenumbers, but at higher wavenumbers the Gaussian falls off exponentially, while the exponential and von Karman fall off with a power law dependence. The fall off rate controls the roughness of the medium. Those characterized by the Gaussian autocorre lation are smoothly varying, while the exponential and von Karman functions are more highly textured. These functions are isotropic and inhomogeneities do not have any preferred orientation. It is probably realistic to consider, for the entire crust, that small scale inhomogeneities do not have a preferred orientation. However, in seismic exploration, we are interested in inhomo geneities in a particular rock or in small region of the crust where anisotropy gains importance. One example might be the deposition of overlapping lenses with different lithologies. The lens shape suggests that the correlation length of these features might be different in the horizontal and vertical direction. Although each lens may have isotropic elastic moduli, the composite medium may display an effective or apparent anisotropy. The preferred orientation of the fluctuations should be reflected in the autocorrelation function. For sim plicity, it will be assumed that all azimuthal variation in the autocorrelation function can be explained through the dimensionless ellipsoidal norm. The interesting aspect of ellipsoidal autocorrelation functions is that they allow us to describe media in which the inhomogeneities are isotropic, elongated in a particular direction, or even flattened.a background part and perturbative part (Born approximation or first Born approximation). Chernov (1960) investigated the applicability of the Born ap proximation for scattering in random acoustic media. The generality of his analysis lead to an overly strict validity criterion which is only valid when the scattered field depends linearly on the incident field. This condition is verified if kD <(6c/c0)2> <1 where D is the diameter of the scattering object, cq is the mean value of the velocity distribution, 8c is the fluctuations in the velocity field and k is the largest wavenumber contributing to the solution. Much of the early work on coda characterization on seismology utilized a random media model and the Born approximation (Aki, 1969, 1973; Aki and Chouet, 1975; Dainty and Toksöz, 1977 Sato, 1977a,b; Wu, 1982a,b; Dainty, 1984; Malin and Phinney, 1985, Charrette, 1990; Korn, 1993). More recently this approach has been used in exploration seismology as well (O'Doherty ve Anstey, 1971; Schoenberger and Levin, 1974, 1978; Richards and Menke, 1983; Burridge and et al., 1987, 1988; Burridge and et al., 1989; Asch and et al., 1990; Gibson and Levander, 1988, 1990; Kemer, 1992; Tran and Bamberger, 1993; Emmerich and et al., 1993). The Born approximation assumes that the incident wave is unchanged during propagation through the heterogeneous region. As a result, energy scattered from the incident wave is not subtracted form the background field and the total energy in the medium increases with time. The validity con ditions at the Born approximation for various single scattering theories have been studied (Chernov, 1960; Hudson and KnopofF, 1966; Kennett, 1972; Hud son and Heritage, 1981; Beydoun and Tarantola, 1988, Coates and Charrette, 1993). Whereas most analytic solutions for scattered waves are valid only when scattering is weak, it may be possible to solve the problem exactly via nu merical methods. The first numerical simulations of seismic wave propagation in stochastic media were accomplished using a two-dimensional ray tracing technique (Mereu and Ojo, 1981). In that study, it was found that variations in travel time and amplitude of the incident wave are controlled mainly by the long wavelength variations in the medium. Frankel and Clayton (1984) used the finite difference technique to model acoustic waves in random media and were able to produce coda waves and study apparent attenuation. For the three random media that they studied (Gaussian, exponential, and von Karman), they found that apparent attenuation increased with frequency un til the correlation length of the scatterers was comparable to the wavelength. At higher frequencies, there was no decrease in apparent attenuation in the exponential and von Karman media, but there was a noticeable decrease in the Gaussian media. These results agree well with analytic solutions (Dainty, 1984). Jannaud et al. (1991) checked the proportionality between the spectral power of codas and the backscattering coefficient for Gaussian and exponential ixrandom media. They simulated wave propagation with the 2- D finite difference method. Their results showed that the backscattering coefficient can be suc cessfully predicted by the codas power spectrum for Gaussian random media. However, they could not invert the backscattering coefficient for exponential random media, successfully. Roth and Korn (1993) investigated the validity of single scattering theory in weakly inhomogeneous 2-D random media using the finite difference approximation. They used different autocorrelation functions from Frankel and Clayton (1986). Their results showed that single scattering theory can be used to determine the energy loss of a plane wave due to scat tering in both 1-D and 2-D random media and agree with a analytical result of Sato (1984). Finite difference modeling has also been applied to the elastic wave equation; both to study the relationship between the medium and the ob served scattered field (Frankel and Clayton, 1986; Mclaughlin and Anderson, 1987; Dougherty and Stephens, 1988; Charrette, 1990; Coates and Charrette, 1993) and to study the response of typical seismic processing streams used on data collected in highly heterogeneous regions (Gibson and Levander, 1988; 1990). In the scattering literature, highly heterogenous media are often approx imated by random fields. The advantage of this approach is that a complex, multi-dimensional velocity function can be expressed in terms of a few simple statistical parameters. In Chapter 2, the conditions for statistical characteri zation is described. One statistical parameter which can be used to describe the variability of a velocity field is the autocorrelation function. The prop erties of three commonly used autocorrelation functions, the Gaussian, expo nential, and von Karman are investigated, and their applicability to earth is discussed. All three spectra are nearly flat at low wavenumbers, but at higher wavenumbers the Gaussian falls off exponentially, while the exponential and von Karman fall off with a power law dependence. The fall off rate controls the roughness of the medium. Those characterized by the Gaussian autocorre lation are smoothly varying, while the exponential and von Karman functions are more highly textured. These functions are isotropic and inhomogeneities do not have any preferred orientation. It is probably realistic to consider, for the entire crust, that small scale inhomogeneities do not have a preferred orientation. However, in seismic exploration, we are interested in inhomo geneities in a particular rock or in small region of the crust where anisotropy gains importance. One example might be the deposition of overlapping lenses with different lithologies. The lens shape suggests that the correlation length of these features might be different in the horizontal and vertical direction. Although each lens may have isotropic elastic moduli, the composite medium may display an effective or apparent anisotropy. The preferred orientation of the fluctuations should be reflected in the autocorrelation function. For sim plicity, it will be assumed that all azimuthal variation in the autocorrelation function can be explained through the dimensionless ellipsoidal norm. The interesting aspect of ellipsoidal autocorrelation functions is that they allow us to describe media in which the inhomogeneities are isotropic, elongated in a particular direction, or even flattened.this form of heterogeneities. A Gaussian correlation function characterizes a single-scale smoothly heterogeneous media, while real heterogeneities in the earth are often multi-scale. So, we have used the zero order von Karman func tion which has a self similar form. This type of media represents one of the fractal geometries described by Mandelbrot (1977). The essential property of fractal media is that the irregularities in all length scales can be represented. However, self similar media has continuous form and has some disadvantages in band limited discrete medium. In this case, there must be a lower limit for scales which depends on the sampling distance. Briefly, observational data sets have disagreement at higher wavenumbers with von Karman modeling. In this thesis, we define a new medium which gives a good approxima tion to von Karman medium. It is called the generalized Gaussian medium. The main idea is stacking different Gaussian media with different correlation length. Maximum correlation length equals the normal correlation length sim ilar to von Karman media. Minimum correlation length can be taken to be the sampling distance. The result of the stacking process must be normalized according to the difference between the maximum and minimum correlation lengths. Also, this scaling factor equals the variance of the random field. In order to achieve direct compatibility between smaller and larger scales than sampling rate, we applied a local averaging (smoothing) and resampling (thin ning) operator (Jones and Thomas, 1985). Variation of correlation length can be chosen using the fractional Brownian motion. The linear variation gives an approximation to the von Karman media. According to the variation rule, other types of media such as Brownian can be defined. The generalized Gaussian medium is very convenient for modeling discrete medium in scattering problem, because, it is to supply a transition between Gaussian and von Karman medium and uses the simple and elementary func tion like Gaussian. The spectral power of codas normalized by the one of the first arrival is averaged over several seismograms. There are two equivalent possibilities to perform this statistical analysis. The average can be performed either over several receivers located in the same medium or over several media with the same statistical properties and one receiver in each medium. We merged the two approaches due to the limited computer facilities. As a result, the average is taken using many media with several receivers. For a Gaussian medium, the average codas power spectrum is indeed proportional to the backscattering coefficient as predicted by linear theory. But, in von Karman media it appears that there is a disagreement between the theoretical and that of inversion solution. As it is pointed out, this medium has continuous form, therefore, it has a lower limit when discretized. The generalized Gaussian medium shows a good agreement with theo retical results. It has a lower and upper limit for correlation length and an Xlllrandom media. They simulated wave propagation with the 2- D finite difference method. Their results showed that the backscattering coefficient can be suc cessfully predicted by the codas power spectrum for Gaussian random media. However, they could not invert the backscattering coefficient for exponential random media, successfully. Roth and Korn (1993) investigated the validity of single scattering theory in weakly inhomogeneous 2-D random media using the finite difference approximation. They used different autocorrelation functions from Frankel and Clayton (1986). Their results showed that single scattering theory can be used to determine the energy loss of a plane wave due to scat tering in both 1-D and 2-D random media and agree with a analytical result of Sato (1984). Finite difference modeling has also been applied to the elastic wave equation; both to study the relationship between the medium and the ob served scattered field (Frankel and Clayton, 1986; Mclaughlin and Anderson, 1987; Dougherty and Stephens, 1988; Charrette, 1990; Coates and Charrette, 1993) and to study the response of typical seismic processing streams used on data collected in highly heterogeneous regions (Gibson and Levander, 1988; 1990). In the scattering literature, highly heterogenous media are often approx imated by random fields. The advantage of this approach is that a complex, multi-dimensional velocity function can be expressed in terms of a few simple statistical parameters. In Chapter 2, the conditions for statistical characteri zation is described. One statistical parameter which can be used to describe the variability of a velocity field is the autocorrelation function. The prop erties of three commonly used autocorrelation functions, the Gaussian, expo nential, and von Karman are investigated, and their applicability to earth is discussed. All three spectra are nearly flat at low wavenumbers, but at higher wavenumbers the Gaussian falls off exponentially, while the exponential and von Karman fall off with a power law dependence. The fall off rate controls the roughness of the medium. Those characterized by the Gaussian autocorre lation are smoothly varying, while the exponential and von Karman functions are more highly textured. These functions are isotropic and inhomogeneities do not have any preferred orientation. It is probably realistic to consider, for the entire crust, that small scale inhomogeneities do not have a preferred orientation. However, in seismic exploration, we are interested in inhomo geneities in a particular rock or in small region of the crust where anisotropy gains importance. One example might be the deposition of overlapping lenses with different lithologies. The lens shape suggests that the correlation length of these features might be different in the horizontal and vertical direction. Although each lens may have isotropic elastic moduli, the composite medium may display an effective or apparent anisotropy. The preferred orientation of the fluctuations should be reflected in the autocorrelation function. For sim plicity, it will be assumed that all azimuthal variation in the autocorrelation function can be explained through the dimensionless ellipsoidal norm. The interesting aspect of ellipsoidal autocorrelation functions is that they allow us to describe media in which the inhomogeneities are isotropic, elongated in a particular direction, or even flattened.asymmetric form. The lower limit approximately equals the sampling distance and agrees with our theoretical results. Also, the correlation function of het erogeneous medium is derived. The agreement between the simulated and theoretical correlations is excellent. In the last part of this thesis, we apply these concepts on a real data set. This data set was from quarry blast records of the TUBITAK-Gebze digital seismic station. All of the records have the same source-receiver path. Each coda power spectrum shows a band limited form approximately in the [1-21^2] interval. Using the backscattering coefficient which is defined for generalized Gaussian media, the lower and upper limit of the correlation function is es timated. The lower limit is found to be 910m and the upper limit 1200m. This results have confidence in this interval according to the source-receiver distance and the spatial sampling. However, we can not correlated with other data sets which has different source-receiver path; because, thus far we are not be able to collect additional data suitable for this purpose. In conclusion, a method for the determination of statistical parameters of a multi-scale heterogenous medium has been developed. This method appears to give very satisfactory results for the simulated and real data sets. Also, this method does not need any special tools such as spectral estimation techniques. In addition, two new ideas are demonstrated. First, the Hartley method is introduced for modeling wave propagation in general media. Next, generalized Gaussian random media, for defining multi-scale random media, is defined and demonstrated. xiv
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