NATO Science for Peace and Security Series C: Environmental Security, 2009
This work presents the results of gravity, TEM and geological surveys conducted in the area of Ka... more This work presents the results of gravity, TEM and geological surveys conducted in the area of Kato Souli (Greece), as study of its hydro-geological characteristics and mapping through GIS. The gravity survey offered a rather detailed image of the alpine basement and together with surface geological observations, insight into the post-alpine tectonic processes that have controlled the development of the area. The TEM survey produced detailed three-dimensional images of the aquifer systems and salination conditions. The results have shown that (a) the alpine basement is located much deeper than previously thought and, (b) that the sea water intrusion takes place both near sea level and at depth. The depth and morphology of the alpine basement are believed to have been fashioned by faults that either have not been active during the Quaternary, or are buried under thick terrestrial and alluvial deposits. Sea water intrusion forms at least two distinct salination horizons, presumably as...
The accumulation of stress and strain is known to induce changes in the electrical properties of ... more The accumulation of stress and strain is known to induce changes in the electrical properties of rocks, which can be monitored for signs of earthquake prepara-tion processes. To this effect, the Magnetotelluric sounding method presents some unique advantages. However, single-site MT data are notoriously susceptible to natural or anthropogenic time-varying coherent noise, which may severely bias the response function estimators and degrade their repeatability, unless treated with dedicated processing techniques. Such a technique is presented herein, involving the W-estimator with random error weighting, followed by an iterative robustification scheme based on an influence function approach. The algorithm is demonstrated on a set of severely distorted data exhibiting a marginal distribution of outliers and is shown to effectively reduce the bias errors and the variance. It is also applied to the long-term monitoring of crustal resistivity with MT response functions at a noisy site loc...
We present a model of spontaneous electric current generation (electrification) involving the mot... more We present a model of spontaneous electric current generation (electrification) involving the motion of charged edge dislocation arrays, during crack formation and propagation (microfracturing) in rocks under stress. Inasmuch as the seismogenic zone comprises a rock volume filled with cracks, massive pre-seismic crack propagation may produce macroscopic electrical earthquake precursors by the superposition of the electric fields of propagating cracks, varying proportionally to the changes in dislocation /crack density and rock resistivity. The motion of dislocations occurs parallel to the applied shear stress, generating a dipole electric field parallel to their slip vector, hence quasi-parallel to the slip vector of shear cracks and by the self-similarity of fragmentation processes, to the slip vector of the upcoming earthquake. We simulate the evolution of crack populations using an ad hoc kinetic theory based on Maxwell-Boltzman statistics. Our results indicate that crack propaga...
We outline a theoretical framework to derive the time-to-failure power-law from basic principles.... more We outline a theoretical framework to derive the time-to-failure power-law from basic principles. We use energy conservation in a faulted crustal volume undergoing stress changes and we assume that the fault system obeys a fractal / hierarchical distribution law. Furthermore, we assume that the precursory seismic activation extends over a broad area around the impending failure and rapidly converges to the rupture zone as a function of the time-to-failure. By considering the analytic conditions near the time of failure, we derive the time-to-failure power-law and show that the critical exponent m is a function of the fractal dimension and that the cumulative precursory crustal deformation (A) is a function of the energy supplied to the system and the size of the rupture. The fractallity of the fault system is a necessary condition for the appearance of power-law acceleration in the seismic release rates. From a theoretical point of view things appear to be quite illuminated but real...
Surface and groundwater resources are important assets for the economic and so-cial welfare of th... more Surface and groundwater resources are important assets for the economic and so-cial welfare of the population of a given area; they are renewable and with proper management, they contribute substantially to sustained development. Unfortunately, the expansion of industrial and agricultural activities lead to increasing withdrawal of wa-ter from the aquifers, excessive use of pesticides and fertilizers and (frequently illicit) disposal of industrial waste. This results in rapid reduction in the quantity and quality of water reserves. There's a compelling call for the development of advanced and inex-pensive means to monitor the quantitative and qualitative characteristics of water re-sources. This paper introduces a multi-channel system, developed in response to the re-quirements specified by Directive 2000/60 of the European Commission, to measure up to 15 physical and chemical parameters (pollutant concentration) facilitating the evalu-ation of water quality in real time. The sy...
On the assumption of causality, it is shown that for general (three-dimensional) conduc- tivity d... more On the assumption of causality, it is shown that for general (three-dimensional) conduc- tivity distributions integrable over any cuboid region of the Earth, the magnetotelluric field possesses very restrictive analytic properties: the singularities of the electric and magnetic field components are all simple zeros confined on the positive imaginary axis of the complex frequency plane. This means that transfer functions comprising simple ratios of orthogonal electric and magnetic field components should also have simple poles and zeros located on the positive imaginary axis. Three-dimensional impedance tensors can be reduced to diagonal or anti-diagonal forms with elements comprising simple ratios of orthogonal field components, using such methods, as the Canonical Decomposition or the SVD, which can be shown to constitute 3-D rotations. Then, it can be shown that the Schmucker Response function derived from the characteristic (singular) values of the impedance tensor can be cast in...
In a prequel to this paper, it was shown that Schmucker response function derived from the charac... more In a prequel to this paper, it was shown that Schmucker response function derived from the characteristic values of the impedance tensor can be cast into a simple Cauer form (expansion). This representation epitomizes the properties of the magnetotelluric responses, which are a direct consequence of its sensu stricto causality. It may also form the basis for a practical means to test a measured magnetotelluric response function for realizability, i.e. physical validity and origin in a real and recoverable Earth structure. Moreover, owing to its analyticity, the Cauer representation may also be used to interpolate (reconstruct) distorted portions of the observed response functions. A procedure and algorithm to realize these objectives is presented herein and its effectiveness is demonstrated with a number of applications to synthetic and measured MT data.
The study of vertical and horizontal crustal movements at the Santorini Volcanic Complex (SVC), a... more The study of vertical and horizontal crustal movements at the Santorini Volcanic Complex (SVC), as deduced by Differential GPS measurements revealed that an intricate pattern of five distinct domains with different horizontal kinematics: The West SVC (Akrotiri peninsula and Therassia) with very significant NNW-ward motion, North Thera with rather significant NW-ward motion, East Thera (Monolithos), with significant SE-ward motion, South Thera with significant NW-ward motion and, finally, central Thera with small westward motion. An apparently dextral, NNW-SSE oblique-to-strike-slip fault emerges as a prominent tectonic structure, separating the West SVC from the rest of the complex; this is the "Santorini Fault Zone". Additional insight is afforded by the results of MT and GDS surveys: a significant NNW-SSE conductive zone was detected, which is collocated with the purported NNW-SSE fault zone indicated by DGPS analysis and may be explained as an epiphenomenal conductivity...
ABSTRACT The possibility of electrical earthquake precursors (EEP) has long been appreciated, but... more ABSTRACT The possibility of electrical earthquake precursors (EEP) has long been appreciated, but it has proven difficult to construct a solid theory to describe their generation and expected characteristics, or proven techniques to identify and discriminate true pre-cursors from noise. To this end, a large number of laboratory experiments have been conducted, which have demonstrated the generation of transient electric potential prior to rupture in both dry and wet rock specimens. The principal mechanisms proposed to explain these observations are the piezo-electric and electrokinetic effects. The piezo-electric effect, often quoted in the literature as the principal generator of EEP, cannot explain why non-piezoelectric rock specimens also generate precursory electric phe-nomena. Streaming potential can be generated in many kinds of saturated rocks, but the fracturing of dry rocks also produces transient electric effects. Therefore, these two mechanisms may not be the basic contributors to the precursory phenomena ob-served in the laboratory (and to possible EEP signals). Herein we present a series of laboratory experiments on the microfracturing electrification of dry marble samples under stress, and discuss their possible relationship to field observations of purported EEP. The marble samples were subjected to uniaxial compression,at constant and variable stress-rates, in both the elastic and the plastic domains. During the experiments, pres-sure stimulated currents (PSC) were observed. The PSC was linearly related to the stress rate, so long as the stressed material deformed elastically. Deviation from lin-earity arose when the specimen was driven into the plastic deformation range; this effect has been counterparted with the dependence of the PSC on the stress rate and, ultimately, with the inverse of the changing (decreasing) Young's modulus. The emit-ted current appears very intense and non-linear just prior to failure. The dependence of the emitted pressure stimulated currents on the stress rate and the Young modulus of the material was thus demonstrated. Repeated cycles of deformation are associ-ated with progressively weaker current emission, indicating the strong dependence of electrification on the residual damage (hence Young modulus). Explanation of these observations was attempted with a theoretical model involving microfracturing and the Motion of Charged edge Dislocations (MCD) as the primary electrification mech-anism (see Tzanis and Vallianatos, 2002). The PSC waveforms and mode of appear-ance accord with the predictions of the MCD model and it appears that if this process could scale up to the size of seismogenic zones, it would yield observable EEP. The MCD model also makes specific predictions of the waveforms of the expected EEP signals. Furthermore some resent results based on the statistical description of PSC from rock fracture experiments obtained from samples of calcite are presented. In all considered cases, the waiting time distribution can be described by a unique scaling function. A frequency-energy distribution similar to the well known from Seismology Gutenberg-Richter law was obtained, as well as an accelerating power-law time-to-failure be-haviour of energy release rates, also analogous to the well known acceleration of cumulative Benniof strain, observed prior to large earthquakes. The resulting scal-ing functions are similar with those for earthquakes and acoustic emissions and are strongly suggestive of the universality in the manifestation of fracture processes and derivative effects.
The present work introduces a curvelet-like directional filter and discusses its application to e... more The present work introduces a curvelet-like directional filter and discusses its application to edge detection in general images and fracture detection in GPR data. The filter is essentially a curvelet of adjustable anisotropy and orientation that can be tuned on any given (target) wavenumber; while retaining the properties of curvelets, it is not bound to the scaling rules of the Curvelet Frame but is individually steerable to any local trait of the data, hence it is dubbed " Curveletiform ". Curveletiforms can be used in single-or multi-directional modes in a manner simple, computationally inexpensive and demonstrably efficient. GPR data generally contains straight or curved edge-like objects comprising reflections from planar interfaces and is notoriously susceptible to broadband noise. Fractures are an important class of interfaces as they determine the health state of rocks or man-made structures and are primary targets of GPR surveys in geotechnical, engineering and environmental applications. As demonstrated with examples, Curveletiforms can efficiently recover information of specific scale and geometry from straight or curved edges in general images. In GPR data they may distinguish reflections from small and large fractures, discriminate between groups of fractures, resolve fracture density and aid the assessment of damage in rocks and structures.
This work presents the results of gravity, TEM and geological surveys conducted in the area of Ma... more This work presents the results of gravity, TEM and geological surveys conducted in the area of Marathon - Kato Souli Plain, as part of an effort to study its hydrogeological characteristics. The gravity survey offered a rather detailed image of the alpine basement and together with surface geological observations, insight into the post-alpine tectonic processes that have controlled the development of the area. The TEM survey produced detailed three-dimensional images of the aquifer systems and salination conditions. The results have shown that (a) the alpine basement is located much deeper than previously thought and, (b) that the sea water intrusion takes place both near sea level and at depth. The depth and morphology of the alpine basement are believed to have been fashioned by faults that either have not been active during the Quaternary, or are buried under thick terrestrial and alluvial deposits. Sea water intrusion forms at least two distinct salination horizons, presumably a...
We examine the nature of the seismogenetic system along the San Andreas Fault (SAF), California, ... more We examine the nature of the seismogenetic system along the San Andreas Fault (SAF), California, USA, by searching for evidence of complexity and non-extensivity in the earthquake record. We use accurate, complete and homogeneous earthquake catalogues in which aftershocks are included (raw catalogues), or have been removed by a stochastic declustering procedure (declustered catalogues). On the basis of Non-Extensive Statistical Physics (NESP), which generalizes the Boltzmann-Gibbs formalism to non-equilibrating (complex) systems, we investigate whether earthquakes are generated by an extensive self-excited Poisson process or by a non-extensive complex process. We examine bivariate cumulative frequency distributions of earthquake magnitudes and interevent times and determine the size and time dependence of the respective magnitude and temporal entropic indices, which indicate the level on non-equilibrium (correlation). It is shown that the magnitude entropic index is very stable and corresponds to proxy b-values that are remarkably consistent with the b-values computed by conventional means. The temporal entropic index computed from the raw catalogues indicate moderately to highly correlated states during the aftershock sequences of large earthquakes, progressing to quasi-uncorrelated states as these die out and before the next large event. Conversely, the analysis of the declustered catalogues shows that background seismicity exhibits moderate to high correlation that varies significantly albeit smoothly with time. This indicates a persistent sub-extensive seismogenetic system. The degree of correlation is generally higher in the southern SAF segment, which is consistent with the observation of shorter return periods for large earthquakes. A plausible explanation is that because aftershock sequences are localized in space and time, their efficient removal unveils long-range background interactions which are obscured by their presence! Our results indicate complexity in the expression of background seismicity along the San Andreas Fault, with criticality being a very likely mechanism as a consequence of the persistent non-equilibrium inferred from the temporal entropic index. However, definite conclusions cannot be drawn until the earthquake record is exhaustively studied in all its forms.
The Ground Probing Radar (GPR) is a valuable tool for near surface geological, geotechnical, engi... more The Ground Probing Radar (GPR) is a valuable tool for near surface geological, geotechnical, engineering, environmental, archaeological and other work. GPR images of the subsurface frequently contain geometric information (constant or variable-dip reflections) from various structures such as bedding, cracks, fractures etc. Such features are frequently the target of the survey; however, they are usually not good reflectors and they are highly localized in time and in space. Their scale is therefore a factor significantly affecting their detectability. At the same time, the GPR method is very sensitive to broadband noise from buried small objects, electromagnetic anthropogenic activity and systemic factors, which frequently blurs the reflections from such targets. The purpose of this paper is to investigate the Curvelet Transform (CT) as a means of S/N enhancement and information retrieval from 2-D GPR sections, with particular emphasis on the recovery of features associated with specific temporal or spatial scales and geometry (orientation/dip).
The CT is a multiscale and multidirectional expansion that formulates an optimally sparse representation of bivariate functions with singularities on twice-differentiable (C^2-continuous) curves (e.g. edges) and allows for the optimal, whole or partial reconstruction of such objects. The CT can be viewed as a higher dimensional extension of the wavelet transform: whereas discrete wavelets are isotropic and provide sparse representations of functions with point singularities, curvelets are highly anisotropic and provide sparse representations of functions with singularities on curves. A GPR section essentially comprises a spatio-temporal sampling of the transient wavefield which contains different arrivals that correspond to different interactions with wave scatterers in the subsurface (wavefronts). These are generally longitudinally piecewise smooth and transversely oscillatory, i.e. they comprise edges. Curvelets can detect wavefronts at different angles and scales because curvelets of a given angle and scale locally correlate with aligned wavefronts of the same scale.
The utility of the CT in processing noisy GPR data is investigated with software based on the Fast Discrete CT and adapted for use with a set of interactive driver functions that compute and display the curvelet decomposition and then allow the manipulation of data (wavefront) components at different scales and angles via the corresponding manipulation (cancelation or restoration) of their associated curvelets. The method is demonstrated with data from archaeometric, geotechnical and hydrogeological surveys, contaminated by high levels of noise, or featuring straight and curved reflections in complex propagation media, or both. It is shown that the CT is very effective in enhancing the S/N ratio by isolating and cancelling directional noise wavefronts of any scale and angle of emergence, sometimes with surgical precision and with particular reference to clutter. It can as successfully be used to retrieve waveforms of specific scale and geometry for further scrutiny, also with surgical precision, as for instance distinguish signals from small and large aperture fractures and faults, different phases of fracturing and faulting, bedding etc. Moreover, it can be useful in investigating the characteristics of signal propagation (hence material properties), albeit indirectly. This is possible because signal attenuation and temporal localization are closely associated, so that scale and spatio-temporal localization are also closely related. Thus, interfaces embedded in low attenuation domains will tend to produce sharp reflections and fine-scale localization. Conversely, interfaces in high attenuation domains will tend to produce dull reflections with broad localization.
The Ground Probing Radar (GPR) is a valuable tool for near surface geological, geotechnical, engi... more The Ground Probing Radar (GPR) is a valuable tool for near surface geological, geotechnical, engineering, environmental, archaeological and other work. GPR data frequently contain anisotropic geometric information from various structures such as bedding, cracks, fractures etc. Such features are generally recorded as wavefronts and are frequently the target of a survey. However, they are usually not good reflectors and they are highly localized in time and in space. Their scale is therefore a factor significantly affecting detectability. At the same time, GPR is sensitive to broadband noise from buried small objects, electromagnetic anthropogenic activity and systemic factors, which frequently obscures reflections from such targets. This presentation discusses the case of de-noising GPR data and retrieving geometric information with advanced scale-and-orientation-sensitive filtering methods. These include orthogonal steerable wavelet arrays and the Gabor wavelet in particular, as well as multiscale/multidirectional optimally sparse representations of bivariate functions with singularities on curves, the archetypal example of the latter being the Curvelet Transform. This approach to processing is demonstrated with examples to be exceptionally adaptive and highly effective. It is also not limited to GPR data.
It is shown that the Magnetotelluric (MT) impedance tensor admits an anti-symmetric generalized e... more It is shown that the Magnetotelluric (MT) impedance tensor admits an anti-symmetric generalized eigenvalue - eigenstate decomposition consistent with the anti-symmetry of electric and magnetic fields referred to the same coordinate frame: this is achieved by anti-diagonalization through rotation by 2x2 complex operators of the SU(2) rotation group. The eigenstates comprise simple proportional relationships between linearly polarized eigenvalues of the input magnetic and output electric field along the locally resistive and conductive propagation path into the Earth, respectively mediated by the maximum and minimum characteristic values of the tensor (eigen-impedances). It is shown from first principles that the eigen-impedances are expected to be positive real (passive) functions, analytic in the entire lower-half complex frequency plane and with singularities confined on the positive imaginary frequency axis. Insofar as the impedance tensor is generated by isometric transformation of the eigen-impedances, it is also passive. The expected passivity is an effective means of appraising measured tensors for compliance with the basic tenets of the MT method: it can be violated only in the presence of sources in the Earth. In addition to extrinsic effects (e.g. noise), it is demonstrated with examples, that such sources may be secondary large or small scale inductive phenomena generated by realistic conductivity configurations. However, they may not be time-independent effects taking place in a passive induction context, such as steady-state current channelling, galvanic distortion and electric field reversals. In general, to assert whether violation of passivity has occurred, it is necessary to decompose the impedance tensor, refer it to its intrinsic coordinate frame and evaluate the compliance of the eigen-impedances with their expected analytic properties
NATO Science for Peace and Security Series C: Environmental Security, 2009
This work presents the results of gravity, TEM and geological surveys conducted in the area of Ka... more This work presents the results of gravity, TEM and geological surveys conducted in the area of Kato Souli (Greece), as study of its hydro-geological characteristics and mapping through GIS. The gravity survey offered a rather detailed image of the alpine basement and together with surface geological observations, insight into the post-alpine tectonic processes that have controlled the development of the area. The TEM survey produced detailed three-dimensional images of the aquifer systems and salination conditions. The results have shown that (a) the alpine basement is located much deeper than previously thought and, (b) that the sea water intrusion takes place both near sea level and at depth. The depth and morphology of the alpine basement are believed to have been fashioned by faults that either have not been active during the Quaternary, or are buried under thick terrestrial and alluvial deposits. Sea water intrusion forms at least two distinct salination horizons, presumably as...
The accumulation of stress and strain is known to induce changes in the electrical properties of ... more The accumulation of stress and strain is known to induce changes in the electrical properties of rocks, which can be monitored for signs of earthquake prepara-tion processes. To this effect, the Magnetotelluric sounding method presents some unique advantages. However, single-site MT data are notoriously susceptible to natural or anthropogenic time-varying coherent noise, which may severely bias the response function estimators and degrade their repeatability, unless treated with dedicated processing techniques. Such a technique is presented herein, involving the W-estimator with random error weighting, followed by an iterative robustification scheme based on an influence function approach. The algorithm is demonstrated on a set of severely distorted data exhibiting a marginal distribution of outliers and is shown to effectively reduce the bias errors and the variance. It is also applied to the long-term monitoring of crustal resistivity with MT response functions at a noisy site loc...
We present a model of spontaneous electric current generation (electrification) involving the mot... more We present a model of spontaneous electric current generation (electrification) involving the motion of charged edge dislocation arrays, during crack formation and propagation (microfracturing) in rocks under stress. Inasmuch as the seismogenic zone comprises a rock volume filled with cracks, massive pre-seismic crack propagation may produce macroscopic electrical earthquake precursors by the superposition of the electric fields of propagating cracks, varying proportionally to the changes in dislocation /crack density and rock resistivity. The motion of dislocations occurs parallel to the applied shear stress, generating a dipole electric field parallel to their slip vector, hence quasi-parallel to the slip vector of shear cracks and by the self-similarity of fragmentation processes, to the slip vector of the upcoming earthquake. We simulate the evolution of crack populations using an ad hoc kinetic theory based on Maxwell-Boltzman statistics. Our results indicate that crack propaga...
We outline a theoretical framework to derive the time-to-failure power-law from basic principles.... more We outline a theoretical framework to derive the time-to-failure power-law from basic principles. We use energy conservation in a faulted crustal volume undergoing stress changes and we assume that the fault system obeys a fractal / hierarchical distribution law. Furthermore, we assume that the precursory seismic activation extends over a broad area around the impending failure and rapidly converges to the rupture zone as a function of the time-to-failure. By considering the analytic conditions near the time of failure, we derive the time-to-failure power-law and show that the critical exponent m is a function of the fractal dimension and that the cumulative precursory crustal deformation (A) is a function of the energy supplied to the system and the size of the rupture. The fractallity of the fault system is a necessary condition for the appearance of power-law acceleration in the seismic release rates. From a theoretical point of view things appear to be quite illuminated but real...
Surface and groundwater resources are important assets for the economic and so-cial welfare of th... more Surface and groundwater resources are important assets for the economic and so-cial welfare of the population of a given area; they are renewable and with proper management, they contribute substantially to sustained development. Unfortunately, the expansion of industrial and agricultural activities lead to increasing withdrawal of wa-ter from the aquifers, excessive use of pesticides and fertilizers and (frequently illicit) disposal of industrial waste. This results in rapid reduction in the quantity and quality of water reserves. There's a compelling call for the development of advanced and inex-pensive means to monitor the quantitative and qualitative characteristics of water re-sources. This paper introduces a multi-channel system, developed in response to the re-quirements specified by Directive 2000/60 of the European Commission, to measure up to 15 physical and chemical parameters (pollutant concentration) facilitating the evalu-ation of water quality in real time. The sy...
On the assumption of causality, it is shown that for general (three-dimensional) conduc- tivity d... more On the assumption of causality, it is shown that for general (three-dimensional) conduc- tivity distributions integrable over any cuboid region of the Earth, the magnetotelluric field possesses very restrictive analytic properties: the singularities of the electric and magnetic field components are all simple zeros confined on the positive imaginary axis of the complex frequency plane. This means that transfer functions comprising simple ratios of orthogonal electric and magnetic field components should also have simple poles and zeros located on the positive imaginary axis. Three-dimensional impedance tensors can be reduced to diagonal or anti-diagonal forms with elements comprising simple ratios of orthogonal field components, using such methods, as the Canonical Decomposition or the SVD, which can be shown to constitute 3-D rotations. Then, it can be shown that the Schmucker Response function derived from the characteristic (singular) values of the impedance tensor can be cast in...
In a prequel to this paper, it was shown that Schmucker response function derived from the charac... more In a prequel to this paper, it was shown that Schmucker response function derived from the characteristic values of the impedance tensor can be cast into a simple Cauer form (expansion). This representation epitomizes the properties of the magnetotelluric responses, which are a direct consequence of its sensu stricto causality. It may also form the basis for a practical means to test a measured magnetotelluric response function for realizability, i.e. physical validity and origin in a real and recoverable Earth structure. Moreover, owing to its analyticity, the Cauer representation may also be used to interpolate (reconstruct) distorted portions of the observed response functions. A procedure and algorithm to realize these objectives is presented herein and its effectiveness is demonstrated with a number of applications to synthetic and measured MT data.
The study of vertical and horizontal crustal movements at the Santorini Volcanic Complex (SVC), a... more The study of vertical and horizontal crustal movements at the Santorini Volcanic Complex (SVC), as deduced by Differential GPS measurements revealed that an intricate pattern of five distinct domains with different horizontal kinematics: The West SVC (Akrotiri peninsula and Therassia) with very significant NNW-ward motion, North Thera with rather significant NW-ward motion, East Thera (Monolithos), with significant SE-ward motion, South Thera with significant NW-ward motion and, finally, central Thera with small westward motion. An apparently dextral, NNW-SSE oblique-to-strike-slip fault emerges as a prominent tectonic structure, separating the West SVC from the rest of the complex; this is the "Santorini Fault Zone". Additional insight is afforded by the results of MT and GDS surveys: a significant NNW-SSE conductive zone was detected, which is collocated with the purported NNW-SSE fault zone indicated by DGPS analysis and may be explained as an epiphenomenal conductivity...
ABSTRACT The possibility of electrical earthquake precursors (EEP) has long been appreciated, but... more ABSTRACT The possibility of electrical earthquake precursors (EEP) has long been appreciated, but it has proven difficult to construct a solid theory to describe their generation and expected characteristics, or proven techniques to identify and discriminate true pre-cursors from noise. To this end, a large number of laboratory experiments have been conducted, which have demonstrated the generation of transient electric potential prior to rupture in both dry and wet rock specimens. The principal mechanisms proposed to explain these observations are the piezo-electric and electrokinetic effects. The piezo-electric effect, often quoted in the literature as the principal generator of EEP, cannot explain why non-piezoelectric rock specimens also generate precursory electric phe-nomena. Streaming potential can be generated in many kinds of saturated rocks, but the fracturing of dry rocks also produces transient electric effects. Therefore, these two mechanisms may not be the basic contributors to the precursory phenomena ob-served in the laboratory (and to possible EEP signals). Herein we present a series of laboratory experiments on the microfracturing electrification of dry marble samples under stress, and discuss their possible relationship to field observations of purported EEP. The marble samples were subjected to uniaxial compression,at constant and variable stress-rates, in both the elastic and the plastic domains. During the experiments, pres-sure stimulated currents (PSC) were observed. The PSC was linearly related to the stress rate, so long as the stressed material deformed elastically. Deviation from lin-earity arose when the specimen was driven into the plastic deformation range; this effect has been counterparted with the dependence of the PSC on the stress rate and, ultimately, with the inverse of the changing (decreasing) Young's modulus. The emit-ted current appears very intense and non-linear just prior to failure. The dependence of the emitted pressure stimulated currents on the stress rate and the Young modulus of the material was thus demonstrated. Repeated cycles of deformation are associ-ated with progressively weaker current emission, indicating the strong dependence of electrification on the residual damage (hence Young modulus). Explanation of these observations was attempted with a theoretical model involving microfracturing and the Motion of Charged edge Dislocations (MCD) as the primary electrification mech-anism (see Tzanis and Vallianatos, 2002). The PSC waveforms and mode of appear-ance accord with the predictions of the MCD model and it appears that if this process could scale up to the size of seismogenic zones, it would yield observable EEP. The MCD model also makes specific predictions of the waveforms of the expected EEP signals. Furthermore some resent results based on the statistical description of PSC from rock fracture experiments obtained from samples of calcite are presented. In all considered cases, the waiting time distribution can be described by a unique scaling function. A frequency-energy distribution similar to the well known from Seismology Gutenberg-Richter law was obtained, as well as an accelerating power-law time-to-failure be-haviour of energy release rates, also analogous to the well known acceleration of cumulative Benniof strain, observed prior to large earthquakes. The resulting scal-ing functions are similar with those for earthquakes and acoustic emissions and are strongly suggestive of the universality in the manifestation of fracture processes and derivative effects.
The present work introduces a curvelet-like directional filter and discusses its application to e... more The present work introduces a curvelet-like directional filter and discusses its application to edge detection in general images and fracture detection in GPR data. The filter is essentially a curvelet of adjustable anisotropy and orientation that can be tuned on any given (target) wavenumber; while retaining the properties of curvelets, it is not bound to the scaling rules of the Curvelet Frame but is individually steerable to any local trait of the data, hence it is dubbed " Curveletiform ". Curveletiforms can be used in single-or multi-directional modes in a manner simple, computationally inexpensive and demonstrably efficient. GPR data generally contains straight or curved edge-like objects comprising reflections from planar interfaces and is notoriously susceptible to broadband noise. Fractures are an important class of interfaces as they determine the health state of rocks or man-made structures and are primary targets of GPR surveys in geotechnical, engineering and environmental applications. As demonstrated with examples, Curveletiforms can efficiently recover information of specific scale and geometry from straight or curved edges in general images. In GPR data they may distinguish reflections from small and large fractures, discriminate between groups of fractures, resolve fracture density and aid the assessment of damage in rocks and structures.
This work presents the results of gravity, TEM and geological surveys conducted in the area of Ma... more This work presents the results of gravity, TEM and geological surveys conducted in the area of Marathon - Kato Souli Plain, as part of an effort to study its hydrogeological characteristics. The gravity survey offered a rather detailed image of the alpine basement and together with surface geological observations, insight into the post-alpine tectonic processes that have controlled the development of the area. The TEM survey produced detailed three-dimensional images of the aquifer systems and salination conditions. The results have shown that (a) the alpine basement is located much deeper than previously thought and, (b) that the sea water intrusion takes place both near sea level and at depth. The depth and morphology of the alpine basement are believed to have been fashioned by faults that either have not been active during the Quaternary, or are buried under thick terrestrial and alluvial deposits. Sea water intrusion forms at least two distinct salination horizons, presumably a...
We examine the nature of the seismogenetic system along the San Andreas Fault (SAF), California, ... more We examine the nature of the seismogenetic system along the San Andreas Fault (SAF), California, USA, by searching for evidence of complexity and non-extensivity in the earthquake record. We use accurate, complete and homogeneous earthquake catalogues in which aftershocks are included (raw catalogues), or have been removed by a stochastic declustering procedure (declustered catalogues). On the basis of Non-Extensive Statistical Physics (NESP), which generalizes the Boltzmann-Gibbs formalism to non-equilibrating (complex) systems, we investigate whether earthquakes are generated by an extensive self-excited Poisson process or by a non-extensive complex process. We examine bivariate cumulative frequency distributions of earthquake magnitudes and interevent times and determine the size and time dependence of the respective magnitude and temporal entropic indices, which indicate the level on non-equilibrium (correlation). It is shown that the magnitude entropic index is very stable and corresponds to proxy b-values that are remarkably consistent with the b-values computed by conventional means. The temporal entropic index computed from the raw catalogues indicate moderately to highly correlated states during the aftershock sequences of large earthquakes, progressing to quasi-uncorrelated states as these die out and before the next large event. Conversely, the analysis of the declustered catalogues shows that background seismicity exhibits moderate to high correlation that varies significantly albeit smoothly with time. This indicates a persistent sub-extensive seismogenetic system. The degree of correlation is generally higher in the southern SAF segment, which is consistent with the observation of shorter return periods for large earthquakes. A plausible explanation is that because aftershock sequences are localized in space and time, their efficient removal unveils long-range background interactions which are obscured by their presence! Our results indicate complexity in the expression of background seismicity along the San Andreas Fault, with criticality being a very likely mechanism as a consequence of the persistent non-equilibrium inferred from the temporal entropic index. However, definite conclusions cannot be drawn until the earthquake record is exhaustively studied in all its forms.
The Ground Probing Radar (GPR) is a valuable tool for near surface geological, geotechnical, engi... more The Ground Probing Radar (GPR) is a valuable tool for near surface geological, geotechnical, engineering, environmental, archaeological and other work. GPR images of the subsurface frequently contain geometric information (constant or variable-dip reflections) from various structures such as bedding, cracks, fractures etc. Such features are frequently the target of the survey; however, they are usually not good reflectors and they are highly localized in time and in space. Their scale is therefore a factor significantly affecting their detectability. At the same time, the GPR method is very sensitive to broadband noise from buried small objects, electromagnetic anthropogenic activity and systemic factors, which frequently blurs the reflections from such targets. The purpose of this paper is to investigate the Curvelet Transform (CT) as a means of S/N enhancement and information retrieval from 2-D GPR sections, with particular emphasis on the recovery of features associated with specific temporal or spatial scales and geometry (orientation/dip).
The CT is a multiscale and multidirectional expansion that formulates an optimally sparse representation of bivariate functions with singularities on twice-differentiable (C^2-continuous) curves (e.g. edges) and allows for the optimal, whole or partial reconstruction of such objects. The CT can be viewed as a higher dimensional extension of the wavelet transform: whereas discrete wavelets are isotropic and provide sparse representations of functions with point singularities, curvelets are highly anisotropic and provide sparse representations of functions with singularities on curves. A GPR section essentially comprises a spatio-temporal sampling of the transient wavefield which contains different arrivals that correspond to different interactions with wave scatterers in the subsurface (wavefronts). These are generally longitudinally piecewise smooth and transversely oscillatory, i.e. they comprise edges. Curvelets can detect wavefronts at different angles and scales because curvelets of a given angle and scale locally correlate with aligned wavefronts of the same scale.
The utility of the CT in processing noisy GPR data is investigated with software based on the Fast Discrete CT and adapted for use with a set of interactive driver functions that compute and display the curvelet decomposition and then allow the manipulation of data (wavefront) components at different scales and angles via the corresponding manipulation (cancelation or restoration) of their associated curvelets. The method is demonstrated with data from archaeometric, geotechnical and hydrogeological surveys, contaminated by high levels of noise, or featuring straight and curved reflections in complex propagation media, or both. It is shown that the CT is very effective in enhancing the S/N ratio by isolating and cancelling directional noise wavefronts of any scale and angle of emergence, sometimes with surgical precision and with particular reference to clutter. It can as successfully be used to retrieve waveforms of specific scale and geometry for further scrutiny, also with surgical precision, as for instance distinguish signals from small and large aperture fractures and faults, different phases of fracturing and faulting, bedding etc. Moreover, it can be useful in investigating the characteristics of signal propagation (hence material properties), albeit indirectly. This is possible because signal attenuation and temporal localization are closely associated, so that scale and spatio-temporal localization are also closely related. Thus, interfaces embedded in low attenuation domains will tend to produce sharp reflections and fine-scale localization. Conversely, interfaces in high attenuation domains will tend to produce dull reflections with broad localization.
The Ground Probing Radar (GPR) is a valuable tool for near surface geological, geotechnical, engi... more The Ground Probing Radar (GPR) is a valuable tool for near surface geological, geotechnical, engineering, environmental, archaeological and other work. GPR data frequently contain anisotropic geometric information from various structures such as bedding, cracks, fractures etc. Such features are generally recorded as wavefronts and are frequently the target of a survey. However, they are usually not good reflectors and they are highly localized in time and in space. Their scale is therefore a factor significantly affecting detectability. At the same time, GPR is sensitive to broadband noise from buried small objects, electromagnetic anthropogenic activity and systemic factors, which frequently obscures reflections from such targets. This presentation discusses the case of de-noising GPR data and retrieving geometric information with advanced scale-and-orientation-sensitive filtering methods. These include orthogonal steerable wavelet arrays and the Gabor wavelet in particular, as well as multiscale/multidirectional optimally sparse representations of bivariate functions with singularities on curves, the archetypal example of the latter being the Curvelet Transform. This approach to processing is demonstrated with examples to be exceptionally adaptive and highly effective. It is also not limited to GPR data.
It is shown that the Magnetotelluric (MT) impedance tensor admits an anti-symmetric generalized e... more It is shown that the Magnetotelluric (MT) impedance tensor admits an anti-symmetric generalized eigenvalue - eigenstate decomposition consistent with the anti-symmetry of electric and magnetic fields referred to the same coordinate frame: this is achieved by anti-diagonalization through rotation by 2x2 complex operators of the SU(2) rotation group. The eigenstates comprise simple proportional relationships between linearly polarized eigenvalues of the input magnetic and output electric field along the locally resistive and conductive propagation path into the Earth, respectively mediated by the maximum and minimum characteristic values of the tensor (eigen-impedances). It is shown from first principles that the eigen-impedances are expected to be positive real (passive) functions, analytic in the entire lower-half complex frequency plane and with singularities confined on the positive imaginary frequency axis. Insofar as the impedance tensor is generated by isometric transformation of the eigen-impedances, it is also passive. The expected passivity is an effective means of appraising measured tensors for compliance with the basic tenets of the MT method: it can be violated only in the presence of sources in the Earth. In addition to extrinsic effects (e.g. noise), it is demonstrated with examples, that such sources may be secondary large or small scale inductive phenomena generated by realistic conductivity configurations. However, they may not be time-independent effects taking place in a passive induction context, such as steady-state current channelling, galvanic distortion and electric field reversals. In general, to assert whether violation of passivity has occurred, it is necessary to decompose the impedance tensor, refer it to its intrinsic coordinate frame and evaluate the compliance of the eigen-impedances with their expected analytic properties
The purpose of this paper is to investigate the Curvelet Transform (CT) as a means of S/N enhance... more The purpose of this paper is to investigate the Curvelet Transform (CT) as a means of S/N enhancement and information retrieval from 2-D GPR sections (B-scans), with particular emphasis placed on the problem of recovering features associated with specific temporal or spatial scales and geometry (orientation/dip).
This paper reports of an attempt to jointly interpret seismotectonic and magnetotelluric data in ... more This paper reports of an attempt to jointly interpret seismotectonic and magnetotelluric data in the Adapazari U Bolu stretch of the North Anatolia fault (NAF) in Turkey, which has recently been the focus of very strong earthquake activity (17 August 1999, M=7.6 and 12 November 1999, M=7.3). This area is characterized by a marked splitting of the NAF into a North and a South branch enclosing a rhomboidal block with approximate dimensions of 80 x 30 km and roughly E-W orientation. Notably, the transition from the predominantly transformational tectonics of North Anatolia to the predominantly extensional tectonics of the Marmara Sea and the Aegean is thought to begin at this same area. The study area has experienced a series of very large (M ≫ 7) earthquakes during the past few decades, (Bolu, 1944; Abant, 1957; Mudurnu, 1967; Izmit, August 1999; Düzce, November 1999), all generated by typical E-W right-lateral strike-slip faulting. A number of extraordinary events distort the perfect consistency of this picture and indicate that regional tectonics may be more complex than what is expected of purely transformational (strike-slip) deformation. For instance, the largest aftershock of the 1967 Mudurnu event (MS =5.6) was clearly normal, on NW-SE faults. Moreover, the right-lateral motion of all other major earthquakes is consistent with the footwall (south block - Anatolian plate) moving west, whereas the same lateral motion of the November 1999 Düzce event was such, that footwall was the northern block (Black Sea plate) which moved eastwards. The magnetotelluric and magnetic transfer function data from 30 sites measured in 1992 and 1994 have also provided significant evidence and constraints on the regional structure. The geoelectric signature of the area is quite different from what is expected of a vertical fault plane of great depth extent, (and very different from the results of analogous MT studies at the St. Andreas fault in California). Rather, the EM data (induction arrows, principal components of the impedance tensor and 2-D inversions), all indicate the existence of NW-SE oriented conductors. In fact, the 2-D inversion of the magnetotelluric data shows a pattern of NE and SW dipping conductors, very reminiscent of normal antithetic faulting, opening up at a NE-SW direction. The configuration of these Snormal faultsS is consistent with the orientation of the minimum principal stress (Tension axis), as computed from the focal mechanism solutions of the recent large events. It is also perfectly consistent with the mechanism of the largest aftershock of the 1967 Mudurnu earthquake. Thus, assuming that these conductors are indeed the signatures of normal faults, the geotectonic setting of the area can be interpreted in terms of a rhomboidal block caught between the south (Mudurnu) and north (Düzce) branches of the NAF, which is being stretched by the opposite motion of the Anatolian and Black Sea plates. This interpretation is also consistent with the eastward motion of the Black Sea plate during the November 1999, Düzce event. It is quite possible that a pull-apart basin has been seeded in the area of Adapazari U Bolu, whose embryonic structures reflect on the MT data. Due to its geometry and size, this hypothesized basin appears like a scaled down version of the much larger pull-apart basin of the Marmara Sea, approximately 100km to the west of the study area. If this interpretation is correct, a case has been documented, where MT and seismological data complement each other, to yield well constrained models of regional tectonic processes. This important point will also be discussed.
Observational studies indicate that many large earthquakes are preceded by accelerating seismic r... more Observational studies indicate that many large earthquakes are preceded by accelerating seismic release rates (Accelerated Seismic Deformation - ASD), characterized by a cumulative Benioff strain following a power law time to failure relation of the form S(t) = K + A(tf t)^m, (1) where tf is the failure time of the large event and m of the order of 0.2 - 0.4. More recent theoretical studies relate the behaviour of seismicity prior to a large earthquake to the excitation in proximity of a spinodal instability and show that the power-law activation associated with the spinodal instability is essentially identical to the power-law acceleration of Benioff strain observed prior to earthquakes, with m = 0.25. In the present work, we discuss a theoretical framework in which we derive the time-to-failure power-law from basic principles. We use energy conservation in a faulted crustal volume undergoing stress loading and we assume that the fault system obeys a fractal / hierarchical distribution law. Furthermore, we assume that the precursory seismic activation extends over a broad area around the impending failure and rapidly converges to the rupture zone as a function of the time-to-failure. By considering the analytic conditions near the time of failure, we derive the time-to-failure power-law and show that the critical exponent m is a function of the fractal dimension and that the cumulative precursory crustal deformation (A) is a function of the energy supplied to the system and the size of the rupture. The fractallity of the fault system is a necessary condition for the appearance of power-law acceleration in the seismic release rates. We note that this approach is based on first principles and gives a clear interpretation of the empirical parameters involved in equation (1). On the basis of these results, it is possible to explain a set of empirical laws derived by other researchers (e.g. Papazachos et al., Bull. Seism. Soc. Am., 92, 570-580, 2002), in terms of a plausible physical framework. Furthermore, by considering the relationship of the instantaneous Benioff strain rate with respect to the mean Benioff strain rate, it is possible to construct approximate analytical expressions to estimate the magnitude and time of failure of the impending earthquake. Examples and applications of this technique to observations of accelerating seismicity in Greece and abroad, will also be presented and discussed. The work of FV was supported by the grant INTAS 99-1102
It has been credibly argued that the earthquake generation process is a critical phenomenon culmi... more It has been credibly argued that the earthquake generation process is a critical phenomenon culminating with a large event that corresponds to some critical point. In this view, a great earthquake represents the end of a cycle on its associated fault network and the beginning of a new one. The dynamic organization of the fault network evolves as the cycle progresses and a great earthquake becomes more probable, thereby rendering possible the prediction of the cycle’s end by monitoring the approach of the fault network toward a critical state. This process may be described by a power-law time-to-failure scaling of the cumulative seismic release rate. Observational evidence has confirmed the power-law scaling in many cases and has empirically determined that the critical exponent in the power law is typically of the order n=0.3. There are also two theoretical predictions for the value of the critical exponent. Ben-Zion and Lyakhovsky (Pure appl. geophys., 159, 2385-2412, 2002) give n=1/3. Rundle et al. (Pure appl. geophys., 157, 2165-2182, 2000) show that the power-law activation associated with a spinodal instability is essentially identical to the power-law acceleration of Benioff strain observed prior to earthquakes; in this case n=0.25. More recently, the CP model has gained support from the development of more dependable models of regional seismicity with realistic fault geometry that show accelerating seismicity before large events. Essentially, these models involve stress transfer to the fault network during the cycle such, that the region of accelerating seismicity will scale with the size of the culminating event, as for instance in Bowman and King (Geophys. Res. Let., 38, 4039-4042, 2001). It is thus possible to understand the observed characteristics of distributed accelerating seismicity in terms of a simple process of increasing tectonic stress in a region already subjected to stress inhomogeneities at all scale lengths. Then, the region of accelerating seismic release is associated with the region defined by the stress field required to rupture a fault with a specified orientation and rake; it is thus possible to incorporate tectonic information into the analysis. Recent analysis of Greek seismicity shows definite power-law acceleration in two areas along the Hellenic Arc, with critical exponents in the expected range of 0.2-0.3. The first area is in the west Hellenic Arc, (Ionian Sea). The projected time of failure is in the interval 2003.05-2003.19 and the projected magnitude is of the order M=7. Tectonic modeling of the accelerating sequence shows that this may be interpreted in terms of stress transfer from two fault geometries generating very similar patterns of stress increase and stress shadows. The first scenario calls for a right-lateral oblique-slip fault of NE-SW orientation at the west boundary of the Aegean microplate, just east of the island of Kefallinia (Kefallinia Fault Zone). The second scenario predicts rupture in a slightly left-lateral inverse fault of NW-SE orientation, underlying Kefallinia, with mechanisms consisted with that of McKenzie (Geophys. J. R. astr. Soc., 30, 109-185, 1972) for the destructive M=7.3 earthquake of 12 August 1953. Both scenaria are consistent with the regional tectonics and kinematics and both are consistent with fault zones known to have generated large earthquakes in the past. The second area is in the SW Hellenic Arc (Mediterranean Sea). The projected time of failure is 2003.6 +/- 0.6 and the projected magnitude is M=7.1 +/- 0.4. Tectonic modelling of this sequence leads to a unique rupture scenario, on a left-lateral oblique-slip fault, probably lying at intermediate depths between Crete and the Peloponnesus, to the SW of the island of Antikythira. In both cases, the tectonic modeling has revealed the existence of a region of accelerating seismicity at the areas of positive stress transfer and, importantly, a region of power-law decelerating seismicity at the areas of negative stress transfer (stress shadows), i.e. the reverse effect which should be observed if energy was extracted from a fault system. In both cases the critical exponent of the accelerating sequence at the positive-stress-transfer regions is very close 0.25, consistent with the view of the fault network as a Self-Organizing Spinodal moving toward a first order phase transition. The reported observations are consistent with almost all of the theoretical predictions and expectations made in terms of the critical point / stress transfer model of seismogenesis. However, there are reservations as to whether they comprise bona-fide predictions. Time-to-failure modelling of accelerated seismicity is a relatively new field of study with few cases-histories whence to draw experience, most of which in fact comprise retrospective analyses of past earthquakes. Still, very little is known as to the development of real-time situations and their probability of success or failure. Also, the power-law scaling is essentially the result of a renormalisation, in which the process of failure at a small spatial scale and temporarily far from a global event can be remapped to the process of failure at a larger scale and closer to the global event. In consequence, when new elements are added, (i.e. large foreschocks), the sequence is renormalized and the predicted parameters may change, sometimes significantly. Yet another difficulty arises from the fact that even if a full-scale self-organising process is active in the critical area, it is not at all necessary that a large earthquake will occur as soon as the system enters the critical state. The critical point model merely predicts that past this time an earthquake is possible but not certain. The time of the large event may depend on several uncertain factors pertaining to the nucleation process, which may have significant time dependence of their own. Moreover, the stored energy may be dissipated with aseismic (low moment release rate) event(s). Again, the absence of a concrete case history complicates anyone’s ability to make solid inferences. In conclusion, our observations can be considered to be critical tests of the critical point / stress transfer earthquake model. If the expected earthquakes occur, then it is possible that we have a powerful tool. If not, we should contemplate the possibility that this approach has limited predictive capacity and is unsafe in evaluating seismic hazard. The answer is pending and the question is open for discussion.
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The CT is a multiscale and multidirectional expansion that formulates an optimally sparse representation of bivariate functions with singularities on twice-differentiable (C^2-continuous) curves (e.g. edges) and allows for the optimal, whole or partial reconstruction of such objects. The CT can be viewed as a higher dimensional extension of the wavelet transform: whereas discrete wavelets are isotropic and provide sparse representations of functions with point singularities, curvelets are highly anisotropic and provide sparse representations of functions with singularities on curves. A GPR section essentially comprises a spatio-temporal sampling of the transient wavefield which contains different arrivals that correspond to different interactions with wave scatterers in the subsurface (wavefronts). These are generally longitudinally piecewise smooth and transversely oscillatory, i.e. they comprise edges. Curvelets can detect wavefronts at different angles and scales because curvelets of a given angle and scale locally correlate with aligned wavefronts of the same scale.
The utility of the CT in processing noisy GPR data is investigated with software based on the Fast Discrete CT and adapted for use with a set of interactive driver functions that compute and display the curvelet decomposition and then allow the manipulation of data (wavefront) components at different scales and angles via the corresponding manipulation (cancelation or restoration) of their associated curvelets. The method is demonstrated with data from archaeometric, geotechnical and hydrogeological surveys, contaminated by high levels of noise, or featuring straight and curved reflections in complex propagation media, or both. It is shown that the CT is very effective in enhancing the S/N ratio by isolating and cancelling directional noise wavefronts of any scale and angle of emergence, sometimes with surgical precision and with particular reference to clutter. It can as successfully be used to retrieve waveforms of specific scale and geometry for further scrutiny, also with surgical precision, as for instance distinguish signals from small and large aperture fractures and faults, different phases of fracturing and faulting, bedding etc. Moreover, it can be useful in investigating the characteristics of signal propagation (hence material properties), albeit indirectly. This is possible because signal attenuation and temporal localization are closely associated, so that scale and spatio-temporal localization are also closely related. Thus, interfaces embedded in low attenuation domains will tend to produce sharp reflections and fine-scale localization. Conversely, interfaces in high attenuation domains will tend to produce dull reflections with broad localization.
The CT is a multiscale and multidirectional expansion that formulates an optimally sparse representation of bivariate functions with singularities on twice-differentiable (C^2-continuous) curves (e.g. edges) and allows for the optimal, whole or partial reconstruction of such objects. The CT can be viewed as a higher dimensional extension of the wavelet transform: whereas discrete wavelets are isotropic and provide sparse representations of functions with point singularities, curvelets are highly anisotropic and provide sparse representations of functions with singularities on curves. A GPR section essentially comprises a spatio-temporal sampling of the transient wavefield which contains different arrivals that correspond to different interactions with wave scatterers in the subsurface (wavefronts). These are generally longitudinally piecewise smooth and transversely oscillatory, i.e. they comprise edges. Curvelets can detect wavefronts at different angles and scales because curvelets of a given angle and scale locally correlate with aligned wavefronts of the same scale.
The utility of the CT in processing noisy GPR data is investigated with software based on the Fast Discrete CT and adapted for use with a set of interactive driver functions that compute and display the curvelet decomposition and then allow the manipulation of data (wavefront) components at different scales and angles via the corresponding manipulation (cancelation or restoration) of their associated curvelets. The method is demonstrated with data from archaeometric, geotechnical and hydrogeological surveys, contaminated by high levels of noise, or featuring straight and curved reflections in complex propagation media, or both. It is shown that the CT is very effective in enhancing the S/N ratio by isolating and cancelling directional noise wavefronts of any scale and angle of emergence, sometimes with surgical precision and with particular reference to clutter. It can as successfully be used to retrieve waveforms of specific scale and geometry for further scrutiny, also with surgical precision, as for instance distinguish signals from small and large aperture fractures and faults, different phases of fracturing and faulting, bedding etc. Moreover, it can be useful in investigating the characteristics of signal propagation (hence material properties), albeit indirectly. This is possible because signal attenuation and temporal localization are closely associated, so that scale and spatio-temporal localization are also closely related. Thus, interfaces embedded in low attenuation domains will tend to produce sharp reflections and fine-scale localization. Conversely, interfaces in high attenuation domains will tend to produce dull reflections with broad localization.