##
Asimptotik ışın teorisi ile yapay sismogram oluşturulması

Asimptotik ışın teorisi ile yapay sismogram oluşturulması

##### Dosyalar

##### Tarih

1991

##### Yazarlar

Kaşlılar, Ayşe

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

##### Özet

îki boyutlu ortamlarda sentetik sismogramların he saplanması için Asimtotik Işm Teorisi *ne (ART) dayanan, hızlı ve etkili bir yöntem geliştirilmiştir. Ekonomik ve kullanması kolay olan bu yöntem sismik kırılma verileri nin yorumlanmasında kullanılabilecek pratik bir araçtır. Bu çalışmada kullanılan Asimtotik Işın Kuramı ile yapay sismogram üretiminde hız yapısı büyük poligonal bloklarla tanımlanmış ve her blok içinde hız gradyanı sa bit ve keyfi yönde seçilmiştir. Yansıyan ve iletilen ışın lar için genlikler sıfırıncı dereceden ART, baş dalgaları için birinci dereceden ART kullanılarak hesaplanmıştır. Yapılan çalışmada önce homojen iki tabakalı ortam incelenmiş ve bu yapı iki şekilde ele alınmıştır. îlk aşa mada yansıyan, kırılarak alt ortama iletilen dalgalarla, başdalgaları incelenmiş, ikinci aşamada bu dalgalara tek rarlı yansımalar da katılmıştır, üçüncü modelde üç tabaka lı bir ortam ele alınmış, ikinci ortam ince kalınlıklı ve düşük hızlı kabul edilmiştir. Dördüncü model için, üçüncü modeldeki üç tabakalı ortam kullanılmış fakat ikinci kat man hızı yüksek verilmiştir. Beşinci model yine üçüncü mo deldeki üç tabakalı ortam olup bu defa kalınlığı düşürül müş fakat ikinci katman hızı düşük olarak bırakılmıştır. Altıncı ve son modelde ise graben modeli incelenmiştir. Bu yöntemi kullanarak üretilen yapay sismogramlarda yansıyan, tekrarlı yansıyan ve kırılan dalgalarla birlik te başdalgaları da dikkate alınmıştır. Dönüşmüş dalgalar bu çalışmada kullanılmamıştır. Daha ileriki çalışmalarda kabuk yapısını temsil ede bilecek ilginç modeller üretilerek, gerçek verilerle kar şılaştırılmaya çalışılacaktır.

Recently, some effective and versatile methods for synthetic seismogram calculations have been developed. In this way, seismograms for complex geological structures can be calculated quite easily. Many methods have been developed for calculating synthetic seismograms. High frequency approximation methods give good results for complex structures. Standard Ray Method (SRM) and Paraxial Ray Method (PRM) for evaluating the wave field are not sufficiently accurate or even inapplicable in the so called singular regions of the ray method such as caustic regions, critical regions, transitions between illuminated and shadow regions. In the SRM, the elementary wave field can only be evaluated on rays, in the PRM on the other hand it can be evaluated not only on ray but also approximately in a vicinity of the ray. In Gaussian Beam Method (GBM), each beam is continued independently through an arbitrary inhomogeneous structure. The complete wave field at a receiver is then obtained as an integral superposition of all gaussian beams arriving in same neigbourhood of the receiver. The corresponding integral formula is valid even in various singular regions where the ray method fails. The method of the solution of the wave equation for high frequencies is called WKBJ (Wentzel, Kramers, Brillouin, Jeffreys). The solution of the wave equation can be obtained using asymptotic ray theory. The phase of this solution in slowness space is related to the spatial solution by the legendre transformation. The amplitude function is obtained by canonical transformation. Synthetic seismograms in the real spatial domain are obtained from the slowness domain via the radon transform. vi Since the solution in the slowness domain is described by asymptotic ray theory, i.e. non-dispersive propagation, the radon transform can be evaluated analytically. Synthetic seismograms calculated in this manner have been called WKBJ seismograms in laterally homogeneous media. Synthetic seismograms calculated in the same manner, but in inhomogeneous media have been called MASLOV seismograms. AMM (Alekseev - Mikhailenko Method) uses finite - difference method for calculating synthetic seismograms in inhomogeneous media. The wave number is parametrized in time-depth domain. Since the calculation is performed in time domain, no aliasing accures but the effect of the absorption can not be calculated. In the AMM, synthetic seismogram can be calculated at any point and traces can be recorded in horizantal and vertical directions. In reflectivity method, the numerical integration of the reflectivity (or plane wave reflection and transmission coefficient) of a layered medium is carried out in the horizontal wave number domain. From the displacement potentials, the displacements u(r,0,w) and w(r,0,w) (u and w are the displacement component in the frequency domain) are obtained. Inverse Fourier Transform gives u(r,0,t) and w(r,0,t) in the time domain. In ART the velocity structure is represented by large polygonal blocks, and within each blok the velocity gradient is uniform and of arbitrary orientation. Simple analytical exspressions are thus used for both ray tracing and amplitude computations. For reflected and refracted rays, amplitudes are determined by using zero-order asymptotic ray theory. Head wave amplitudes are determined using first-order asymptotic ray theory. Each blok of the model is reparametrized as a series of thin homogeneous layers perpendicular to the direction of the velocity gradient. Amplitudes are then determined by analytic expressions valid for models of homogeneous layers with plane dipping boundaries. To define the velocity structure, there are two types of boundaries, model boundaries and divider boundaries. A model boundary is a straight line of arbitrary dip, assigned a constant velocity along its length and a nonzero velocity gradient normal to its length. A divider boundary, assigned a velocity of zero, seperates a region with one velocity and gradient from a laterally adjacent region with a different velocity and gradient. Blocks may thus be defined in which the VI 1 velocity and both the magnitude and direction of the velocity gradient are arbitrary. The ray path within a given block is a circular arc, for which the travel time and distance traveled may be calculated using very simple analytical expressions (Gebrande, 1976). The source may be located along any model boundary, and rays traveling upwards or downwards from the source may be considered. If a ray is incident to boundary at an angle which is within a specified range of the critical angle, then head waves may be produced. For the precritical and multiple reflections, only the boundary or the boundaries at which reflection is desired need to be specified. At all other boundaries encountered by the ray, the behavior of the ray is controlled by the angle of incidence at the boundary. If the incident angle is less than the critical angle, then the ray refracts through the boundary; otherwise, it reflects from the boundary. Thus, if no precritical or multiple reflections are desired, then a single specification of the range of take-off angles gives all wide-angle reflections, turning rays and head waves. The corresponding travel-time curve is divided into branches such that the distance along each branch increases or decreases monotonically with distance. The family of rays associated with each travel-time branch is labeled with unique identification number, which is used in the synthetic seismogram routine for purposes of interpolation within a given ray. In a medium with an arbitrary inhomogeneous velocity distrubition, zero-order asymptotic ray theory provides a connection formula between the source at Mq and any point M for the amplitude of a reflected or refracted wave (Cerveny and Ravindra,1971). Zoeppritz amplitude cefficients Ri for transmission or reflection are taken from Cerveny and Ravindra (1971) and are calculated using a routine described by Young and Braile (1976) and Cassell (1982). The Zoeppritz algorithm assumes that the medium has a poisson ratio of 0.2 5 except if the P wave velocity is less than 1.5 km/sec, where the material is assumed to be water with a density of 1.0 x 1 03 kg/ni3 and S wave velocity of zero. The Zoeppritz routine also allows the calculation of surface reflection and surface conversion coefficients if desired. For head waves, which represent the first order coefficient in the ray expansion, a different scheme is used. The critical angle ray path toward and away from Vlli the head wave boundary is described by the Whittall and Clowes (1979) ray tracer in terms of circular arcs. Within each blok the circular ray path is divided into a large number of segments of equal length, and assumed a constant velocity along each segment. Thus the velocity model is reparametrized in terms of a series of thin homogeneous layers whose boundaries are parallel within a given block but may be nonparallel from block to block. It is applied expressions from Cerveny and Ravindra (1971) for the amplitude of head waves in a model of homogeneous layers with plane dipping interfaces. After amplitudes for the desired reflections, refractions and head wave are calculated, synthetic seismograms are generated by superimposing the displacements of all arrivals at a particular distance. The seismograms are produced at a set of equally spaced distances. Associated with each ray that riches the surface is a travel time, an epicentral distance, a complex amplitude and a travel time branch ID number. For a given branch, amplitude and travel time are linearly interpolated to the desired distance. A phase-shifted impulse is then constructed by a linear combination of a unit impulse and its Hilbert transform. The seismogram synthesis is completed by convolution with an apparent source function. In this study applications are made for several models some of may represent the crustal models in the Aegean region. In the first model the ray amplitudes are calculated for a two layered laterally homogeneous model. The first layer is 30 km thick and has 6.4 km/sec P wave velocity with 0.001 km/sec/km velocity gradient. The second layer has 8.0 km/sec velocity at the top. The velocity gradient in this layer is 0.0226 km/sec/km. This model is also used for the second model but the difference is, the second model includes multiple reflections. The third one is a thin layered model. It includes three homogeneous layers. The first layer has 4 km/sec velocity and very small velocity gradient. The second layer has 2 km/sec velocity and no velocity gradient. This is the thin and low velocity layer. The third layer has 6 km/sec velocity and 0.04 km/sec/km velocity gradient. The fourth model is same as the third model except that the thin layer has a velocity of 5 km/sec. IX In the fifth model is similar to the third model. is used. But the thickness is decreased. The thin layer has a low velocity. The sixth model is a graben model. The first layer has a velocity of 4 km/sec and a velocity gradient of the second layer has a velocity of 6.4 km/sec and a velocity gradient of In all these models several ray groups have been traced. Transmitted, reflected, multipled and head waves have been obtained. Ray trajectories, synthetic seismograms, travel time graphics and amplitude-distance graphics are given in Chapter 5. The algorithm which is used here is a fast, efficient and economical routine. It is easy to run and gives considerable results. In the later studies these models will be compared to the observed seismograms so that more realistic models of the crustal structure can be derived.

Recently, some effective and versatile methods for synthetic seismogram calculations have been developed. In this way, seismograms for complex geological structures can be calculated quite easily. Many methods have been developed for calculating synthetic seismograms. High frequency approximation methods give good results for complex structures. Standard Ray Method (SRM) and Paraxial Ray Method (PRM) for evaluating the wave field are not sufficiently accurate or even inapplicable in the so called singular regions of the ray method such as caustic regions, critical regions, transitions between illuminated and shadow regions. In the SRM, the elementary wave field can only be evaluated on rays, in the PRM on the other hand it can be evaluated not only on ray but also approximately in a vicinity of the ray. In Gaussian Beam Method (GBM), each beam is continued independently through an arbitrary inhomogeneous structure. The complete wave field at a receiver is then obtained as an integral superposition of all gaussian beams arriving in same neigbourhood of the receiver. The corresponding integral formula is valid even in various singular regions where the ray method fails. The method of the solution of the wave equation for high frequencies is called WKBJ (Wentzel, Kramers, Brillouin, Jeffreys). The solution of the wave equation can be obtained using asymptotic ray theory. The phase of this solution in slowness space is related to the spatial solution by the legendre transformation. The amplitude function is obtained by canonical transformation. Synthetic seismograms in the real spatial domain are obtained from the slowness domain via the radon transform. vi Since the solution in the slowness domain is described by asymptotic ray theory, i.e. non-dispersive propagation, the radon transform can be evaluated analytically. Synthetic seismograms calculated in this manner have been called WKBJ seismograms in laterally homogeneous media. Synthetic seismograms calculated in the same manner, but in inhomogeneous media have been called MASLOV seismograms. AMM (Alekseev - Mikhailenko Method) uses finite - difference method for calculating synthetic seismograms in inhomogeneous media. The wave number is parametrized in time-depth domain. Since the calculation is performed in time domain, no aliasing accures but the effect of the absorption can not be calculated. In the AMM, synthetic seismogram can be calculated at any point and traces can be recorded in horizantal and vertical directions. In reflectivity method, the numerical integration of the reflectivity (or plane wave reflection and transmission coefficient) of a layered medium is carried out in the horizontal wave number domain. From the displacement potentials, the displacements u(r,0,w) and w(r,0,w) (u and w are the displacement component in the frequency domain) are obtained. Inverse Fourier Transform gives u(r,0,t) and w(r,0,t) in the time domain. In ART the velocity structure is represented by large polygonal blocks, and within each blok the velocity gradient is uniform and of arbitrary orientation. Simple analytical exspressions are thus used for both ray tracing and amplitude computations. For reflected and refracted rays, amplitudes are determined by using zero-order asymptotic ray theory. Head wave amplitudes are determined using first-order asymptotic ray theory. Each blok of the model is reparametrized as a series of thin homogeneous layers perpendicular to the direction of the velocity gradient. Amplitudes are then determined by analytic expressions valid for models of homogeneous layers with plane dipping boundaries. To define the velocity structure, there are two types of boundaries, model boundaries and divider boundaries. A model boundary is a straight line of arbitrary dip, assigned a constant velocity along its length and a nonzero velocity gradient normal to its length. A divider boundary, assigned a velocity of zero, seperates a region with one velocity and gradient from a laterally adjacent region with a different velocity and gradient. Blocks may thus be defined in which the VI 1 velocity and both the magnitude and direction of the velocity gradient are arbitrary. The ray path within a given block is a circular arc, for which the travel time and distance traveled may be calculated using very simple analytical expressions (Gebrande, 1976). The source may be located along any model boundary, and rays traveling upwards or downwards from the source may be considered. If a ray is incident to boundary at an angle which is within a specified range of the critical angle, then head waves may be produced. For the precritical and multiple reflections, only the boundary or the boundaries at which reflection is desired need to be specified. At all other boundaries encountered by the ray, the behavior of the ray is controlled by the angle of incidence at the boundary. If the incident angle is less than the critical angle, then the ray refracts through the boundary; otherwise, it reflects from the boundary. Thus, if no precritical or multiple reflections are desired, then a single specification of the range of take-off angles gives all wide-angle reflections, turning rays and head waves. The corresponding travel-time curve is divided into branches such that the distance along each branch increases or decreases monotonically with distance. The family of rays associated with each travel-time branch is labeled with unique identification number, which is used in the synthetic seismogram routine for purposes of interpolation within a given ray. In a medium with an arbitrary inhomogeneous velocity distrubition, zero-order asymptotic ray theory provides a connection formula between the source at Mq and any point M for the amplitude of a reflected or refracted wave (Cerveny and Ravindra,1971). Zoeppritz amplitude cefficients Ri for transmission or reflection are taken from Cerveny and Ravindra (1971) and are calculated using a routine described by Young and Braile (1976) and Cassell (1982). The Zoeppritz algorithm assumes that the medium has a poisson ratio of 0.2 5 except if the P wave velocity is less than 1.5 km/sec, where the material is assumed to be water with a density of 1.0 x 1 03 kg/ni3 and S wave velocity of zero. The Zoeppritz routine also allows the calculation of surface reflection and surface conversion coefficients if desired. For head waves, which represent the first order coefficient in the ray expansion, a different scheme is used. The critical angle ray path toward and away from Vlli the head wave boundary is described by the Whittall and Clowes (1979) ray tracer in terms of circular arcs. Within each blok the circular ray path is divided into a large number of segments of equal length, and assumed a constant velocity along each segment. Thus the velocity model is reparametrized in terms of a series of thin homogeneous layers whose boundaries are parallel within a given block but may be nonparallel from block to block. It is applied expressions from Cerveny and Ravindra (1971) for the amplitude of head waves in a model of homogeneous layers with plane dipping interfaces. After amplitudes for the desired reflections, refractions and head wave are calculated, synthetic seismograms are generated by superimposing the displacements of all arrivals at a particular distance. The seismograms are produced at a set of equally spaced distances. Associated with each ray that riches the surface is a travel time, an epicentral distance, a complex amplitude and a travel time branch ID number. For a given branch, amplitude and travel time are linearly interpolated to the desired distance. A phase-shifted impulse is then constructed by a linear combination of a unit impulse and its Hilbert transform. The seismogram synthesis is completed by convolution with an apparent source function. In this study applications are made for several models some of may represent the crustal models in the Aegean region. In the first model the ray amplitudes are calculated for a two layered laterally homogeneous model. The first layer is 30 km thick and has 6.4 km/sec P wave velocity with 0.001 km/sec/km velocity gradient. The second layer has 8.0 km/sec velocity at the top. The velocity gradient in this layer is 0.0226 km/sec/km. This model is also used for the second model but the difference is, the second model includes multiple reflections. The third one is a thin layered model. It includes three homogeneous layers. The first layer has 4 km/sec velocity and very small velocity gradient. The second layer has 2 km/sec velocity and no velocity gradient. This is the thin and low velocity layer. The third layer has 6 km/sec velocity and 0.04 km/sec/km velocity gradient. The fourth model is same as the third model except that the thin layer has a velocity of 5 km/sec. IX In the fifth model is similar to the third model. is used. But the thickness is decreased. The thin layer has a low velocity. The sixth model is a graben model. The first layer has a velocity of 4 km/sec and a velocity gradient of the second layer has a velocity of 6.4 km/sec and a velocity gradient of In all these models several ray groups have been traced. Transmitted, reflected, multipled and head waves have been obtained. Ray trajectories, synthetic seismograms, travel time graphics and amplitude-distance graphics are given in Chapter 5. The algorithm which is used here is a fast, efficient and economical routine. It is easy to run and gives considerable results. In the later studies these models will be compared to the observed seismograms so that more realistic models of the crustal structure can be derived.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1991

##### Anahtar kelimeler

Jeofizik Mühendisliği,
Asimptotik ışın teorisi,
Yapay sismogram,
Geophysics Engineering,
Asymptotic ray theory,
Synthetic seismogram