In geophysical exploration different types of measurements are used to probe the same subsurface ... more In geophysical exploration different types of measurements are used to probe the same subsurface region. In this paper we show that the wavelet transform can aid the process of linking different data types. The continuous wavelet transform, and in particular the analysis of amplitudes along wavelet transform modulus maxima lines, is a powerful tool to analyze the characteristic properties of local variations in a signal. The amplitude-versus-scale curve of a particular transition in a signal can be seen as its fingerprint. Hence, local variations in different data types can be linked by comparing their fingerprints in the wavelet transform domain. Insight in the physics underlying the different types of measurements is required to 'tune' the different wavelet transforms in such a way that a particular geological transition in the Earth's subsurface leaves the same fingerprint in the wavelet transform of each data type. We discuss the wavelet transform as a tool for geophysical data integration for three situations. First we discuss how one can link the scale-dependent properties of outliers in borehole data to those of reflection events in surface seismic data. We use wave theory to derive relations between the two data types in the wavelet transform domain. Next we analyze the relation between the wavelet transforms of detailed geological models and (simulated) migrated seismic data, with the aim of improving the geological interpretation. A spatial resolution function provides the link between the wavelet transforms of the geological model and the migrated seismic data. Finally we consider the integration of geotechnical (cone penetration test) data with shallow shear wave seismic data. We illustrate with a real data example that specific geological features of the shallow subsurface can be identified in the wavelet transforms of both data types. We conclude that the wavelet transform can be used as a tool that aids the integration of different types of data.
We formulate reciprocity theorems for time-lapse seismic methods, based on the full and the onc w... more We formulate reciprocity theorems for time-lapse seismic methods, based on the full and the onc way wave equations. The latter form allows a straighlforward physical interpretation of the various contributing terms. Ur ike difference data taken at the acquisition surface, the boundary integral in the one-way reciprocity theorem Íepresents the true timelapse changes of the reflectivity of the top reflectoÍ of a reservoir below the boundary at which Ihis integral is evaluated. Evalualron of the boundary integÍal yields therefore suited inpuÍ foÍ timelapse AVO arnlysis-KEY WORDS: recipÍ(rciry. r ime-lapse. one-wa) wave equalion.
Subsurface Sensing Technologies and Applications, 2003
We discuss a time-lapse technique using a full nonlinear inversion scheme for subsurface monitori... more We discuss a time-lapse technique using a full nonlinear inversion scheme for subsurface monitoring purposes. In this technique we relate two measurements of the same nature but at different time instants to characterize the change of the subsurface during this time of evolution. This technique is very popular in seismic geophysical explorations. In this paper we investigate the possibility of
employing a causality based imaging condition. Jounal oÍ Seismic Explorution. l1t 19-5L. We formu... more employing a causality based imaging condition. Jounal oÍ Seismic Explorution. l1t 19-5L. We formulate the theory for a direct nonlmear seismic inversion method in the acouslic approximation. It is a completely data-driven method, aiming at the determination of subsurface properties direclly from the data. The theory is presented for the full three-dimensional, larerally varying case. For this situation we derive a layer replacement method based on the reciprociry lheorem and we de ve an imaging condition based on causality. Next we sjmplify the theory for lhe one-dimensional case and give some synthetic results for this case. We explain how the limited bardwidth of seismic data influences our method. We propose a solution method ro deal with the lack of low frequencies in the data. This method uses the absolute value of the data. Finally we prese a synthetic inversion example for a laterally varying earth model using commoÍ-midpoinl techniques.
Reciprocity theorems play an important role in formulating true amplitude operations on seismic w... more Reciprocity theorems play an important role in formulating true amplitude operations on seismic wave fields, such as multiple elimination, migration and characterization. In general, a reciprocity theorem interrelates the quantities that characterize two admissible physical states that could occur in one and the same domain (de Hoop, 1988). One state is identified with an actual measurement,
Ultrasonic experimcÍrts carried out on Rotliegend reservoir sandstone samples havc shown a specif... more Ultrasonic experimcÍrts carried out on Rotliegend reservoir sandstone samples havc shown a specific stress-dependent behaviour of the transmission response. Apart from the well-known velocity increase as ambient stress iÍEreases, the amplitude and the time are scaled when the stress is changed from onc value to another. Our hypothesis is that whcn stess changes, some mineralogical constinlents of the rock may change dleir acoustic prop€rties differently from other constitucnts. As a consequenc€, different scattering attenuation effects take place within the rock. The observed stress-dependent scaling bchaviour can be a consequence of the latter phenomenon. In order to quantiry th € scaling behaviour, two approaches arc used. First, a heuristically derived model from thc experimental data is tested on numerically simulated data. Next, an analytically derived model from a modified version of the O'Doherty-Anstey expression for the transmission response througb finely layercd medi...
... Data for Object Detection and Characterization Evert C. Slob,* J. Groenenboom,† and Jacob T. ... more ... Data for Object Detection and Characterization Evert C. Slob,* J. Groenenboom,† and Jacob T. Fokkema ... This is a first step towards true amplitude processing of GPR data, which is necessary for a quantitative reconstruction of the 3D electromagnetic contrast distribution. ...
SEG Technical Program Expanded Abstracts 2001, 2001
Ultrasonic experiments carried out on several reservoir sandstone samples have shown a speci c pr... more Ultrasonic experiments carried out on several reservoir sandstone samples have shown a speci c pressuredependent behaviour of the transmission response. Apart from the well-known velocity increase as ambient pressure increases, the amplitude and the time axis are scaled as a function of the pressure. In order to quantify the scaling phenomenon two approaches are used. First, a heuristically derived model from the experimental data is tested on numerically simulated data. On the other hand, an analytically derived model from a modi ed version of the O'Doherty-Anstey expression for the transmission response through nely layered media is also analyzed and tested on numerically simulated data. Both scaling models present t w o scalar parameters that relate a wavelet recorded at a higher ambient pressure with another recorded at a relatively lower one. Estimating these parameters from measurements for a range of di erent a m bient pressures gives valuable information about the pressure-dependent behaviour of the reservoir rock.
3D imaging of ground penetrating radar data is a challenging task. Most imaging procedures are ba... more 3D imaging of ground penetrating radar data is a challenging task. Most imaging procedures are based on scalar seismic imaging algorithms. However, for electromagnetic imaging the vectorial character of the emitted eld and the radiation characteristics of the source and the receiver play an important role. In this paper a new imaging algorithm is presented, which is dedicated to the electromagnetic case. The vectorial character of the electromagnetic waves is taken into account for a monostatic xed o set ground penetrating radar gpr survey, when the source and receiver are present on a dielectric medium. The forward model is described by the 3D wave eld extrapolator which takes into account the vectorial character of the electromagnetic eld. The imaging algorithm is based on an approximate inverse of this 3D wave eld extrapolator, which is determined in the spatial Fourier domain. For a homogeneous space and a zero-o set con guration it is feasible to carry out an analytical approximation of the inverse operator. The approximate inverse of a scalar 3D wave eld extrapolator is not stable, which motivates the use of a multi-component imaging algorithm. For two homogeneous halfspaces it is not feasible to carry out a similar analytical approach. However, in numerical sense the same procedure can be carried out, which has the important bene t that also the o set between the source and receiver can be taken into account. Numerical results are presented for a point scatterer in a homogeneous space, which indicates the spatial resolution which can be obtained. Imaging results of experiments are presented, which take into account the vectorial character of the measured electric eld, the oset between the source and receiver and the presence of two homogeneous halfspaces.
SEG Technical Program Expanded Abstracts 2008, 2008
This abstract discusses a receiver deghosting method for seismic pressure wave-fields that have b... more This abstract discusses a receiver deghosting method for seismic pressure wave-fields that have been recorded at arbitrary depth. For horizontal streamers, the deghosted pressure wavefield can be directly constructed from the scattered pressure wave-field in the spectral domain. For non-horizontal streamers, a system of equations is solved using a preconditioned conjugate gradient method. The mathematical derivation of the method is shown, and implementation issues are discussed. The effectiveness of the method is demonstrated using two synthetic examples.
In this paper we discuss a new type of regularization technique for the nonlinear inverse scatter... more In this paper we discuss a new type of regularization technique for the nonlinear inverse scattering problem, namely the multiplicative technique. The main advantage is that we do not have to determine the regularization parameter before the inversion process is started. We consider different norms of the total variation as regularization factor. Specifically, we investigate a weighted L 2-norm, and by using an appropriate updating scheme we show that this multiplicative regularization factor does not increase the nonlinearity of the inversion problem. Numerical examples using synthetic and experimental data demonstrate the robustness of the presented method.
Subsurface Sensing Technologies and Applications, 2002
Electric dipoles radiate an electromagnetic field into their surroundings. The electric and magne... more Electric dipoles radiate an electromagnetic field into their surroundings. The electric and magnetic fields from a point source or from linear dipoles have components parallel to a spherical surface but also a radial component. In most investigations on radiated fields, only the part of the electric field that is parallel to a spherical surface is taken into account. The argument is that only that component contributes to the radiated energy. Here we show that the phase front in the lower half space is not a spherical surface and hence, the radial components contribute to radiated energy. We show differences in radiation patterns for point electric dipoles and linear dipoles, either modeled as perfectly conducting wires or as resistively loaded wires. Our primary interest lies with improving image resolution through processing of Ground-Penetrating Radar data. Here the emphasis lies with understanding the radiation characteristics of linear dipoles that can be incorporated with imag...
Subsurface Sensing Technologies and Applications, 2003
One-way wave equations play a fundamental role in the formulation of imaging schemes for reflecti... more One-way wave equations play a fundamental role in the formulation of imaging schemes for reflection measurements, as in seismic exploration and ground-penetrating radar. In this paper we review the theory for electromagnetic one-way wave fields, including dissipation, and we derive reciprocity theorems for these one-way wave fields. First we consider the special situation of a 2-D medium in which the medium parameters vary only in the lateral direction. Next we generalize the theory for 2-D inhomogeneous media. For the latter situation we derive one-way reciprocity theorems of the convolution type and of the correlation type. These reciprocity theorems find their applications in forward and inverse problems, respectively.
Numerical modeling of thin-wire antennas is interesting to people working on antenna design and t... more Numerical modeling of thin-wire antennas is interesting to people working on antenna design and to people working on the processing and interpretation of radar data. They have different demands regarding the output of the model code. Our primary interest lies with improving image resolution through processing of ground-penetrating radar data. For this goal the effects of two antennas placed close together on the Earth's surface are studied. To study the antenna to antenna coupling and antenna to ground coupling, a model for linear dipoles is used. The model uses a thin-wire approximation for the transmit and receive antennas. The antennas are placed just above the air-Earth interface. The Earth is modeled as a horizontal layer stack. Using this model, we study the coupling effects between two parallel antennas, either in each other's E-plane or H-plane. For ordinary antenna distances, the coupling between two parallel antennas when they are resistively loaded is on the order of À70 dB, which is not negligible for GPR applications. The influence of the lower halfspace on antenna behavior is strong but limited to a fraction of the wavelength in depth.
Reciprocity theorems have proven their usefulness in the study of forward and inverse scattering ... more Reciprocity theorems have proven their usefulness in the study of forward and inverse scattering problems. Most reciprocity theorems in the literature apply to the total wave field and are thus not compatible with one-way wave theory, which is often applied in situations in which there is a clear preferred direction of propagation, like in electromagnetic or acoustic wave guides and in seismic exploration. In this paper we review the theory for one-way wave fields (or bidirectional beams), and we extensively discuss the symmetry properties of the square root operator appearing in the one-way wave equation. Using these symmetry properties, it appears to be possible to derive reciprocity theorems of the convolution type and of the correlation type for electromagnetic or acoustic one-way wave fields in dissipative inhomogeneous media along the same lines as the usual derivation of the reciprocity theorems for the total wave field. The one-way reciprocity theorem of the convolution type provides a basis for representations of scattered one-way wave fields in terms of generalized Bremmer series expansions or generalized primaries. The one-way reciprocity theorem of the correlation type finds its application in reflection imaging based on inverse one-way wave field propagators.
Journal of Geophysical Research: Solid Earth, 1992
The plane wave scattering in a two-dimensional medium consisting of two quarter spaces below a ha... more The plane wave scattering in a two-dimensional medium consisting of two quarter spaces below a haft-space, also known as the vertical fault model, is the simplest two-dimensional model applicable to geophysical exploration. The wave field obeys the two-dimensional Helmholtz equation. Its formulation is equivalent to the electromagnetic case when the fundamental unknown is the horizontally polarized electric field vector and to the acoustic case when the unknown is the pressure. In the electromagnetic case the medium parameters are the electrical conductivity and the magnetic permeability, and in the acoustic case they are the compressibility and the density. The formulation of the scattered field in all space is defined by an integral representation composed of two one-sided Fourier transforms in each medium. It represents an extension of Weaver's formulation. The spectral constituents of the Fourier integrals are determined by means of the Neumann's series, as a solution to the set of integral equations obtained by application of the boundary conditions. The numerical results show that the series converge and that the field values have a precision always better than 5%. The same formulation can be extended to solve the problem of scattering in more complex two-dimensional or in three-dimensional medium constituted of perpendicular blocks. Fields and spectral constituents may be used in the inversion of geophysical data.
Journal of Geophysical Research: Solid Earth, 2004
The interaction of elastodynamic waves with imperfect interfaces is usually described by the so-c... more The interaction of elastodynamic waves with imperfect interfaces is usually described by the so-called linear slip model. In this model it is assumed that the particle displacement of an elastic wave at an interface jumps by a finite amount, linearly proportional to the stress at the interface. In this paper we postulate general boundary conditions at arbitrarily shaped imperfect interfaces for acoustic waves in fluids, elastodynamic waves in solids, electromagnetic waves in matter, poroelastic waves in porous solids, and seismoelectric waves in porous solids in such a way that they cover the linear slip model and other existing models for imperfect interfaces. These boundary conditions are expressed by a single matrix-vector equation in the space-frequency domain. Using this equation, we extend two unified reciprocity theorems (one of the convolution type and one of the correlation type) for the various wave phenomena, with an extra integral over the imperfect interfaces. By considering two special cases of these reciprocity theorems, we observe that (1) source-receiver reciprocity remains valid when the source and receiver are separated by imperfect interfaces and that (2) the extra integral in the correlation-type reciprocity theorem quantifies the power dissipated by the imperfect interfaces.
In geophysical exploration different types of measurements are used to probe the same subsurface ... more In geophysical exploration different types of measurements are used to probe the same subsurface region. In this paper we show that the wavelet transform can aid the process of linking different data types. The continuous wavelet transform, and in particular the analysis of amplitudes along wavelet transform modulus maxima lines, is a powerful tool to analyze the characteristic properties of local variations in a signal. The amplitude-versus-scale curve of a particular transition in a signal can be seen as its fingerprint. Hence, local variations in different data types can be linked by comparing their fingerprints in the wavelet transform domain. Insight in the physics underlying the different types of measurements is required to 'tune' the different wavelet transforms in such a way that a particular geological transition in the Earth's subsurface leaves the same fingerprint in the wavelet transform of each data type. We discuss the wavelet transform as a tool for geophysical data integration for three situations. First we discuss how one can link the scale-dependent properties of outliers in borehole data to those of reflection events in surface seismic data. We use wave theory to derive relations between the two data types in the wavelet transform domain. Next we analyze the relation between the wavelet transforms of detailed geological models and (simulated) migrated seismic data, with the aim of improving the geological interpretation. A spatial resolution function provides the link between the wavelet transforms of the geological model and the migrated seismic data. Finally we consider the integration of geotechnical (cone penetration test) data with shallow shear wave seismic data. We illustrate with a real data example that specific geological features of the shallow subsurface can be identified in the wavelet transforms of both data types. We conclude that the wavelet transform can be used as a tool that aids the integration of different types of data.
We formulate reciprocity theorems for time-lapse seismic methods, based on the full and the onc w... more We formulate reciprocity theorems for time-lapse seismic methods, based on the full and the onc way wave equations. The latter form allows a straighlforward physical interpretation of the various contributing terms. Ur ike difference data taken at the acquisition surface, the boundary integral in the one-way reciprocity theorem Íepresents the true timelapse changes of the reflectivity of the top reflectoÍ of a reservoir below the boundary at which Ihis integral is evaluated. Evalualron of the boundary integÍal yields therefore suited inpuÍ foÍ timelapse AVO arnlysis-KEY WORDS: recipÍ(rciry. r ime-lapse. one-wa) wave equalion.
Subsurface Sensing Technologies and Applications, 2003
We discuss a time-lapse technique using a full nonlinear inversion scheme for subsurface monitori... more We discuss a time-lapse technique using a full nonlinear inversion scheme for subsurface monitoring purposes. In this technique we relate two measurements of the same nature but at different time instants to characterize the change of the subsurface during this time of evolution. This technique is very popular in seismic geophysical explorations. In this paper we investigate the possibility of
employing a causality based imaging condition. Jounal oÍ Seismic Explorution. l1t 19-5L. We formu... more employing a causality based imaging condition. Jounal oÍ Seismic Explorution. l1t 19-5L. We formulate the theory for a direct nonlmear seismic inversion method in the acouslic approximation. It is a completely data-driven method, aiming at the determination of subsurface properties direclly from the data. The theory is presented for the full three-dimensional, larerally varying case. For this situation we derive a layer replacement method based on the reciprociry lheorem and we de ve an imaging condition based on causality. Next we sjmplify the theory for lhe one-dimensional case and give some synthetic results for this case. We explain how the limited bardwidth of seismic data influences our method. We propose a solution method ro deal with the lack of low frequencies in the data. This method uses the absolute value of the data. Finally we prese a synthetic inversion example for a laterally varying earth model using commoÍ-midpoinl techniques.
Reciprocity theorems play an important role in formulating true amplitude operations on seismic w... more Reciprocity theorems play an important role in formulating true amplitude operations on seismic wave fields, such as multiple elimination, migration and characterization. In general, a reciprocity theorem interrelates the quantities that characterize two admissible physical states that could occur in one and the same domain (de Hoop, 1988). One state is identified with an actual measurement,
Ultrasonic experimcÍrts carried out on Rotliegend reservoir sandstone samples havc shown a specif... more Ultrasonic experimcÍrts carried out on Rotliegend reservoir sandstone samples havc shown a specific stress-dependent behaviour of the transmission response. Apart from the well-known velocity increase as ambient stress iÍEreases, the amplitude and the time are scaled when the stress is changed from onc value to another. Our hypothesis is that whcn stess changes, some mineralogical constinlents of the rock may change dleir acoustic prop€rties differently from other constitucnts. As a consequenc€, different scattering attenuation effects take place within the rock. The observed stress-dependent scaling bchaviour can be a consequence of the latter phenomenon. In order to quantiry th € scaling behaviour, two approaches arc used. First, a heuristically derived model from thc experimental data is tested on numerically simulated data. Next, an analytically derived model from a modified version of the O'Doherty-Anstey expression for the transmission response througb finely layercd medi...
... Data for Object Detection and Characterization Evert C. Slob,* J. Groenenboom,† and Jacob T. ... more ... Data for Object Detection and Characterization Evert C. Slob,* J. Groenenboom,† and Jacob T. Fokkema ... This is a first step towards true amplitude processing of GPR data, which is necessary for a quantitative reconstruction of the 3D electromagnetic contrast distribution. ...
SEG Technical Program Expanded Abstracts 2001, 2001
Ultrasonic experiments carried out on several reservoir sandstone samples have shown a speci c pr... more Ultrasonic experiments carried out on several reservoir sandstone samples have shown a speci c pressuredependent behaviour of the transmission response. Apart from the well-known velocity increase as ambient pressure increases, the amplitude and the time axis are scaled as a function of the pressure. In order to quantify the scaling phenomenon two approaches are used. First, a heuristically derived model from the experimental data is tested on numerically simulated data. On the other hand, an analytically derived model from a modi ed version of the O'Doherty-Anstey expression for the transmission response through nely layered media is also analyzed and tested on numerically simulated data. Both scaling models present t w o scalar parameters that relate a wavelet recorded at a higher ambient pressure with another recorded at a relatively lower one. Estimating these parameters from measurements for a range of di erent a m bient pressures gives valuable information about the pressure-dependent behaviour of the reservoir rock.
3D imaging of ground penetrating radar data is a challenging task. Most imaging procedures are ba... more 3D imaging of ground penetrating radar data is a challenging task. Most imaging procedures are based on scalar seismic imaging algorithms. However, for electromagnetic imaging the vectorial character of the emitted eld and the radiation characteristics of the source and the receiver play an important role. In this paper a new imaging algorithm is presented, which is dedicated to the electromagnetic case. The vectorial character of the electromagnetic waves is taken into account for a monostatic xed o set ground penetrating radar gpr survey, when the source and receiver are present on a dielectric medium. The forward model is described by the 3D wave eld extrapolator which takes into account the vectorial character of the electromagnetic eld. The imaging algorithm is based on an approximate inverse of this 3D wave eld extrapolator, which is determined in the spatial Fourier domain. For a homogeneous space and a zero-o set con guration it is feasible to carry out an analytical approximation of the inverse operator. The approximate inverse of a scalar 3D wave eld extrapolator is not stable, which motivates the use of a multi-component imaging algorithm. For two homogeneous halfspaces it is not feasible to carry out a similar analytical approach. However, in numerical sense the same procedure can be carried out, which has the important bene t that also the o set between the source and receiver can be taken into account. Numerical results are presented for a point scatterer in a homogeneous space, which indicates the spatial resolution which can be obtained. Imaging results of experiments are presented, which take into account the vectorial character of the measured electric eld, the oset between the source and receiver and the presence of two homogeneous halfspaces.
SEG Technical Program Expanded Abstracts 2008, 2008
This abstract discusses a receiver deghosting method for seismic pressure wave-fields that have b... more This abstract discusses a receiver deghosting method for seismic pressure wave-fields that have been recorded at arbitrary depth. For horizontal streamers, the deghosted pressure wavefield can be directly constructed from the scattered pressure wave-field in the spectral domain. For non-horizontal streamers, a system of equations is solved using a preconditioned conjugate gradient method. The mathematical derivation of the method is shown, and implementation issues are discussed. The effectiveness of the method is demonstrated using two synthetic examples.
In this paper we discuss a new type of regularization technique for the nonlinear inverse scatter... more In this paper we discuss a new type of regularization technique for the nonlinear inverse scattering problem, namely the multiplicative technique. The main advantage is that we do not have to determine the regularization parameter before the inversion process is started. We consider different norms of the total variation as regularization factor. Specifically, we investigate a weighted L 2-norm, and by using an appropriate updating scheme we show that this multiplicative regularization factor does not increase the nonlinearity of the inversion problem. Numerical examples using synthetic and experimental data demonstrate the robustness of the presented method.
Subsurface Sensing Technologies and Applications, 2002
Electric dipoles radiate an electromagnetic field into their surroundings. The electric and magne... more Electric dipoles radiate an electromagnetic field into their surroundings. The electric and magnetic fields from a point source or from linear dipoles have components parallel to a spherical surface but also a radial component. In most investigations on radiated fields, only the part of the electric field that is parallel to a spherical surface is taken into account. The argument is that only that component contributes to the radiated energy. Here we show that the phase front in the lower half space is not a spherical surface and hence, the radial components contribute to radiated energy. We show differences in radiation patterns for point electric dipoles and linear dipoles, either modeled as perfectly conducting wires or as resistively loaded wires. Our primary interest lies with improving image resolution through processing of Ground-Penetrating Radar data. Here the emphasis lies with understanding the radiation characteristics of linear dipoles that can be incorporated with imag...
Subsurface Sensing Technologies and Applications, 2003
One-way wave equations play a fundamental role in the formulation of imaging schemes for reflecti... more One-way wave equations play a fundamental role in the formulation of imaging schemes for reflection measurements, as in seismic exploration and ground-penetrating radar. In this paper we review the theory for electromagnetic one-way wave fields, including dissipation, and we derive reciprocity theorems for these one-way wave fields. First we consider the special situation of a 2-D medium in which the medium parameters vary only in the lateral direction. Next we generalize the theory for 2-D inhomogeneous media. For the latter situation we derive one-way reciprocity theorems of the convolution type and of the correlation type. These reciprocity theorems find their applications in forward and inverse problems, respectively.
Numerical modeling of thin-wire antennas is interesting to people working on antenna design and t... more Numerical modeling of thin-wire antennas is interesting to people working on antenna design and to people working on the processing and interpretation of radar data. They have different demands regarding the output of the model code. Our primary interest lies with improving image resolution through processing of ground-penetrating radar data. For this goal the effects of two antennas placed close together on the Earth's surface are studied. To study the antenna to antenna coupling and antenna to ground coupling, a model for linear dipoles is used. The model uses a thin-wire approximation for the transmit and receive antennas. The antennas are placed just above the air-Earth interface. The Earth is modeled as a horizontal layer stack. Using this model, we study the coupling effects between two parallel antennas, either in each other's E-plane or H-plane. For ordinary antenna distances, the coupling between two parallel antennas when they are resistively loaded is on the order of À70 dB, which is not negligible for GPR applications. The influence of the lower halfspace on antenna behavior is strong but limited to a fraction of the wavelength in depth.
Reciprocity theorems have proven their usefulness in the study of forward and inverse scattering ... more Reciprocity theorems have proven their usefulness in the study of forward and inverse scattering problems. Most reciprocity theorems in the literature apply to the total wave field and are thus not compatible with one-way wave theory, which is often applied in situations in which there is a clear preferred direction of propagation, like in electromagnetic or acoustic wave guides and in seismic exploration. In this paper we review the theory for one-way wave fields (or bidirectional beams), and we extensively discuss the symmetry properties of the square root operator appearing in the one-way wave equation. Using these symmetry properties, it appears to be possible to derive reciprocity theorems of the convolution type and of the correlation type for electromagnetic or acoustic one-way wave fields in dissipative inhomogeneous media along the same lines as the usual derivation of the reciprocity theorems for the total wave field. The one-way reciprocity theorem of the convolution type provides a basis for representations of scattered one-way wave fields in terms of generalized Bremmer series expansions or generalized primaries. The one-way reciprocity theorem of the correlation type finds its application in reflection imaging based on inverse one-way wave field propagators.
Journal of Geophysical Research: Solid Earth, 1992
The plane wave scattering in a two-dimensional medium consisting of two quarter spaces below a ha... more The plane wave scattering in a two-dimensional medium consisting of two quarter spaces below a haft-space, also known as the vertical fault model, is the simplest two-dimensional model applicable to geophysical exploration. The wave field obeys the two-dimensional Helmholtz equation. Its formulation is equivalent to the electromagnetic case when the fundamental unknown is the horizontally polarized electric field vector and to the acoustic case when the unknown is the pressure. In the electromagnetic case the medium parameters are the electrical conductivity and the magnetic permeability, and in the acoustic case they are the compressibility and the density. The formulation of the scattered field in all space is defined by an integral representation composed of two one-sided Fourier transforms in each medium. It represents an extension of Weaver's formulation. The spectral constituents of the Fourier integrals are determined by means of the Neumann's series, as a solution to the set of integral equations obtained by application of the boundary conditions. The numerical results show that the series converge and that the field values have a precision always better than 5%. The same formulation can be extended to solve the problem of scattering in more complex two-dimensional or in three-dimensional medium constituted of perpendicular blocks. Fields and spectral constituents may be used in the inversion of geophysical data.
Journal of Geophysical Research: Solid Earth, 2004
The interaction of elastodynamic waves with imperfect interfaces is usually described by the so-c... more The interaction of elastodynamic waves with imperfect interfaces is usually described by the so-called linear slip model. In this model it is assumed that the particle displacement of an elastic wave at an interface jumps by a finite amount, linearly proportional to the stress at the interface. In this paper we postulate general boundary conditions at arbitrarily shaped imperfect interfaces for acoustic waves in fluids, elastodynamic waves in solids, electromagnetic waves in matter, poroelastic waves in porous solids, and seismoelectric waves in porous solids in such a way that they cover the linear slip model and other existing models for imperfect interfaces. These boundary conditions are expressed by a single matrix-vector equation in the space-frequency domain. Using this equation, we extend two unified reciprocity theorems (one of the convolution type and one of the correlation type) for the various wave phenomena, with an extra integral over the imperfect interfaces. By considering two special cases of these reciprocity theorems, we observe that (1) source-receiver reciprocity remains valid when the source and receiver are separated by imperfect interfaces and that (2) the extra integral in the correlation-type reciprocity theorem quantifies the power dissipated by the imperfect interfaces.
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Papers by Jacob Fokkema