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- Volume 68, Issue 3, 2020
Geophysical Prospecting - Volume 68, Issue 3, 2020
Volume 68, Issue 3, 2020
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Markov chain Monte Carlo algorithms for target‐oriented and interval‐oriented amplitude versus angle inversions with non‐parametric priors and non‐linear forward modellings
Authors Mattia Aleardi and Alessandro SalustiABSTRACTIn geophysical inverse problems, the posterior model can be analytically assessed only in case of linear forward operators, Gaussian, Gaussian mixture, or generalized Gaussian prior models, continuous model properties, and Gaussian‐distributed noise contaminating the observed data. For this reason, one of the major challenges of seismic inversion is to derive reliable uncertainty appraisals in cases of complex prior models, non‐linear forward operators and mixed discrete‐continuous model parameters. We present two amplitude versus angle inversion strategies for the joint estimation of elastic properties and litho‐fluid facies from pre‐stack seismic data in case of non‐parametric mixture prior distributions and non‐linear forward modellings. The first strategy is a two‐dimensional target‐oriented inversion that inverts the amplitude versus angle responses of the target reflections by adopting the single‐interface full Zoeppritz equations. The second is an interval‐oriented approach that inverts the pre‐stack seismic responses along a given time interval using a one‐dimensional convolutional forward modelling still based on the Zoeppritz equations. In both approaches, the model vector includes the facies sequence and the elastic properties of P‐wave velocity, S‐wave velocity and density. The distribution of the elastic properties at each common‐mid‐point location (for the target‐oriented approach) or at each time‐sample position (for the time‐interval approach) is assumed to be multimodal with as many modes as the number of litho‐fluid facies considered. In this context, an analytical expression of the posterior model is no more available. For this reason, we adopt a Markov chain Monte Carlo algorithm to numerically evaluate the posterior uncertainties. With the aim of speeding up the convergence of the probabilistic sampling, we adopt a specific recipe that includes multiple chains, a parallel tempering strategy, a delayed rejection updating scheme and hybridizes the standard Metropolis–Hasting algorithm with the more advanced differential evolution Markov chain method. For the lack of available field seismic data, we validate the two implemented algorithms by inverting synthetic seismic data derived on the basis of realistic subsurface models and actual well log data. The two approaches are also benchmarked against two analytical inversion approaches that assume Gaussian‐mixture‐distributed elastic parameters. The final predictions and the convergence analysis of the two implemented methods proved that our approaches retrieve reliable estimations and accurate uncertainties quantifications with a reasonable computational effort.
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Look‐ahead vertical seismic profiling inversion approach for density and compressional wave velocity in Bayesian framework
Authors Ahmed M. Daoud and Muhammad A. Abd El DayemABSTRACTTo reduce drilling uncertainties, zero‐offset vertical seismic profiles can be inverted to quantify acoustic properties ahead of the bit. In this work, we propose an approach to invert vertical seismic profile corridor stacks in Bayesian framework for look‐ahead prediction. The implemented approach helps to successfully predict density and compressional wave velocity using prior knowledge from drilled interval. Hence, this information can be used to monitor reservoir depth as well as quantifying high‐pressure zones, which enables taking the correct decision during drilling. The inversion algorithm uses Gauss–Newton as an optimization tool, which requires the calculation of the sensitivity matrix of trace samples with respect to model parameters. Gauss–Newton has quadratic rate of convergence, which can speed up the inversion process. Moreover, geo‐statistical analysis has been used to efficiently utilize prior information supplied to the inversion process. The algorithm has been tested on synthetic and field cases. For the field case, a zero‐offset vertical seismic profile data taken from an offshore well were used as input to the inversion algorithm. Well logs acquired after drilling the prediction section was used to validate the inversion results. The results from the synthetic case applications were encouraging to accurately predict compressional wave velocity and density from just a constant prior model. The field case application shows the strength of our proposed approach in inverting vertical seismic profile data to obtain density and compressional wave velocity ahead of a bit with reasonable accuracy. Unlike the commonly used vertical seismic profile inversion approach for acoustic impedance using simple error to represent the prior covariance matrix, this work shows the importance of inverting for both density and compressional wave velocity using geo‐statistical knowledge of density and compressional wave velocity from the drilled section to quantify the prior covariance matrix required during Bayesian inversion.
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Multiscale phase inversion for 3D ocean‐bottom cable data
Authors Lei Fu, Zongcai Feng and Gerard T. SchusterABSTRACTWe invert three‐dimensional seismic data by a multiscale phase inversion scheme, a modified version of full waveform inversion, which applies higher order integrations to the input signal to produce low‐boost signals. These low‐boost signals are used as the input data for the early iterations, and lower order integrations are computed at the later iterations. The advantages of multiscale phase inversion are that it (1) is less dependent on the initial model compared to full waveform inversion, (2) is less sensitive to incorrectly modelled magnitudes and (3) employs a simple and natural frequency shaping filtering. For a layered model with a three‐dimensional velocity anomaly, results with synthetic data show that multiscale phase inversion can sometimes provide a noticeably more accurate velocity profile than full waveform inversion. Results with the Society of Exploration Geophysicists/European Association of Geoscientists and Engineers overthrust model shows that multiscale phase inversion more clearly resolves meandering channels in the depth slices. However, the data and model misfit functions achieve about the same values after 50 iterations. The results with three‐dimensional ocean‐bottom cable data show that, compared to the full waveform inversion tomogram, the three‐dimensional multiscale phase inversion tomogram provides a better match to the well log, and better flattens angle‐domain common image gathers. The problem is that the tomograms at the well log provide an incomplete low‐wavenumber estimate of the log's velocity profile. Therefore, a good low‐wavenumber estimate of the velocity model is still needed for an accurate multiscale phase inversion tomogram.
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Assumptions and goals for least squares migration
By Daniel TradABSTRACTLeast squares migration uses the assumption that, if we have an operator that can create data from a reflectivity function, the optimal image will predict the actual recorded data with minimum square error. For this assumption to be true, it is also required that: (a) the prediction operator must be error‐free, (b) model elements not seen by the operator should be constrained by other means and (c) data weakly predicted by the operator should make limited contribution to the solution. Under these conditions, least squares migration has the advantage over simple migration of being able to remove interference between different model components. Least squares migration does that by de‐convolving or inverting the so‐called Hessian operator. The Hessian is the cascade of forward modelling and migration; for each image point, it computes the effects of interference from other image points (point‐spread function) given the actual recording geometry and the subsurface velocity model. Because the Hessian contains illumination information (along its diagonal), and information about the model cross‐correlation produced by non‐orthogonality of basis functions, its inversion produces illumination compensation and increases resolution. In addition, sampling deficiencies in the recording geometry map to the Hessian (both diagonal and non‐diagonal elements), so least squares migration has the potential to remove sampling artefacts as well. These (illumination compensation, resolution and mitigating recording deficiencies) are the three main goals of least squares migration, although the first one can be achieved by cheaper techniques. To invert the Hessian, least squares migration relies on the residual errors during iterations. Iterative algorithms, like conjugate gradient and others, use the residuals to calculate the direction and amplitudes (gradient and step size) of the necessary corrections to the reflectivity function or model. Failure of conditions (a), (b) or (c) leads the inversion to calculate incorrect model updates, which translate to noise in the final image. In this paper, we will discuss these conditions for Kirchhoff migration and reverse time migration.
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Understanding acquisition and processing error in microseismic data: An example from Pouce Coupe Field, Canada
Authors Matthew Bray and Isabel WhiteABSTRACTA challenge in microseismic monitoring is quantification of survey acquisition and processing errors, and how these errors jointly affect estimated locations. Quantifying acquisition and processing errors and uncertainty has multiple benefits, such as more accurate and precise estimation of locations, anisotropy, moment tensor inversion and, potentially, allowing for detection of 4D reservoir changes. Here, we quantify uncertainty due to acquisition, receiver orientation error, and hodogram analysis. Additionally, we illustrate the effects of signal to noise ratio variances upon event detection. We apply processing steps to a downhole microseismic dataset from Pouce Coupe, Alberta, Canada. We use a probabilistic location approach to identify the optimal bottom well location based upon known source locations. Probability density functions are utilized to quantify uncertainty and propagate it through processing, including in source location inversion to describe the three‐dimensional event location likelihood. Event locations are calculated and an amplitude stacking approach is used to reduce the error associated with first break picking and the minimization with modelled travel times. Changes in the early processing steps have allowed for understanding of location uncertainty of the mapped microseismic events.
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Detection of point scatterers using diffraction imaging and deep learning
Authors Valentin Tschannen, Norman Ettrich, Matthias Delescluse and Janis KeuperABSTRACTDiffracted waves carry high‐resolution information that can help interpreting fine structural details at a scale smaller than the seismic wavelength. However, the diffraction energy tends to be weak compared to the reflected energy and is also sensitive to inaccuracies in the migration velocity, making the identification of its signal challenging. In this work, we present an innovative workflow to automatically detect scattering points in the migration dip angle domain using deep learning. By taking advantage of the different kinematic properties of reflected and diffracted waves, we separate the two types of signals by migrating the seismic amplitudes to dip angle gathers using prestack depth imaging in the local angle domain. Convolutional neural networks are a class of deep learning algorithms able to learn to extract spatial information about the data in order to identify its characteristics. They have now become the method of choice to solve supervised pattern recognition problems. In this work, we use wave equation modelling to create a large and diversified dataset of synthetic examples to train a network into identifying the probable position of scattering objects in the subsurface. After giving an intuitive introduction to diffraction imaging and deep learning and discussing some of the pitfalls of the methods, we evaluate the trained network on field data and demonstrate the validity and good generalization performance of our algorithm. We successfully identify with a high‐accuracy and high‐resolution diffraction points, including those which have a low signal to noise and reflection ratio. We also show how our method allows us to quickly scan through high dimensional data consisting of several versions of a dataset migrated with a range of velocities to overcome the strong effect of incorrect migration velocity on the diffraction signal.
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Attenuation of marine seismic interference noise employing a customized U‐Net
Authors Jing Sun, Sigmund Slang, Thomas Elboth, Thomas Larsen Greiner, Steven McDonald and Leiv‐J. GeliusABSTRACTMarine seismic interference noise occurs when energy from nearby marine seismic source vessels is recorded during a seismic survey. Such noise tends to be well preserved over large distances and causes coherent artefacts in the recorded data. Over the years, the industry has developed various denoising techniques for seismic interference removal, but although well performing, they are still time‐consuming in use. Machine‐learning‐based processing represents an alternative approach, which may significantly improve the computational efficiency. In the case of conventional images, autoencoders are frequently employed for denoising purposes. However, due to the special characteristics of seismic data as well as the noise, autoencoders failed in the case of marine seismic interference noise. We, therefore, propose the use of a customized U‐Net design with element‐wise summation as part of the skip‐connection blocks to handle the vanishing gradient problem and to ensure information fusion between high‐ and low‐level features. To secure a realistic study, only seismic field data were employed, including 25,000 training examples. The customized U‐Net was found to perform well, leaving only minor residuals, except for the case when seismic interference noise comes from the side. We further demonstrate that such noise can be treated by slightly increasing the depth of our network. Although our customized U‐Net does not outperform a standard commercial algorithm in quality, it can (after proper training) read and process one single shot gather in approximately 0.02 s. This is significantly faster than any existing industry denoising algorithm. In addition, the proposed network processes shot gathers in a sequential order, which is an advantage compared with industry algorithms that typically require a multi‐shot input to break the coherency of the noise.
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Random noise attenuation via the randomized canonical polyadic decomposition
Authors Wenlei Gao and Mauricio D. SacchiABSTRACTTensor algebra provides a robust framework for multi‐dimensional seismic data processing. A low‐rank tensor can represent a noise‐free seismic data volume. Additive random noise will increase the rank of the tensor. Hence, tensor rank‐reduction techniques can be used to filter random noise. Our filtering method adopts the Candecomp/Parafac decomposition to approximates a N‐dimensional seismic data volume via the superposition of rank‐one tensors. Similar to the singular value decomposition for matrices, a low‐rank Candecomp/Parafac decomposition can capture the signal and exclude random noise in situations where a low‐rank tensor can represent the ideal noise‐free seismic volume. The alternating least squares method is adopted to compute the Candecomp/Parafac decomposition with a provided target rank. This method involves solving a series of highly over‐determined linear least‐squares subproblems. To improve the efficiency of the alternating least squares algorithm, we uniformly randomly sample equations of the linear least‐squares subproblems to reduce the size of the problem significantly. The computational overhead is further reduced by avoiding unfolding and folding large dense tensors. We investigate the applicability of the randomized Candecomp/Parafac decomposition for incoherent noise attenuation via experiments conducted on a synthetic dataset and field data seismic volumes. We also compare the proposed algorithm (randomized Candecomp/Parafac decomposition) against multi‐dimensional singular spectrum analysis and classical prediction filtering. We conclude the proposed approach can achieve slightly better denoising performance in terms of signal‐to‐noise ratio enhancement than traditional methods, but with a less computational cost.
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Low‐rank seismic denoising with optimal rank selection for hankel matrices
Authors Chong Wang, Zhihui Zhu and Hanming GuABSTRACTBased on the fact that the Hankel matrix representing clean seismic data is low rank, low‐rank approximation methods have been widely utilized for removing noise from seismic data. A common strategy for real seismic data is to perform the low‐rank approximations for small local windows where the events can be approximately viewed as linear. This raises a fundamental question of selecting an optimal rank that best captures the number of events for each local window. Gavish and Donoho proposed a method to select the rank when the noise is independent and identically distributed. Gaussian matrix by analysing the statistical performance of the singular values of the Gaussian matrices. However, such statistical performance is not available for noisy Hankel matrices. In this paper, we adopt the same strategy and propose a rule that computes the number of singular values exceed the median singular value by a multiplicative factor. We suggest a multiplicative factor of 3 based on simulations which mimic the theories underlying Gavish and Donoho in the independent and identically distributed Gaussian setting. The proposed optimal rank selection rule can be incorporated into the classical low‐rank approximation method and many other recently developed methods such as those by shrinking the singular values. The low‐rank approximation methods with optimally selected rank rule can automatically suppress most of the noise while preserving the main features of the seismic data in each window. Experiments on both synthetic and field seismic data demonstrate the superior performance of the proposed rank selection rule for seismic data denoising.
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S wave attenuation based on Stokes boundary layer
By Guangquan LiABSTRACTBiot theory was based on two ideas: the coupling factor to quantify the kinetic energy of fluid and Darcy permeability to quantify the dissipation function. As Biot theory did not well predict attenuation of ultrasonic S wave, we modify the theory to better characterize the S wave attenuation. The range of the coupling factor is at first estimated in view of fluid mechanics. Application of the original theory to water‐saturated Boise sandstone and brine‐saturated Berea sandstone shows that the model prediction significantly underestimates the S wave attenuation ultrasonically measured. For this reason, we replace Darcy permeability with variable permeability to improve the fluid momentum equation. The new model yields predictions of phase velocity and the quality factor both close to the ultrasonic measurements. The reason why the improved model is superior to Biot theory is that variable permeability is based on the Stokes boundary layer at the fluid–solid interface, thus accurately quantifying the viscous stress between the two phases. Finally, the length scale of the viscous stress is calculated in the mesoscopic sense.
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Approximate traveltime inversion in downhole microseismic monitoring
Authors Sergey Yaskevich, Anton A. Duchkov and Yuriy IvanovABSTRACTIn downhole microseismic monitoring, accurate event location relies on the accuracy of the velocity model. The model can be estimated along with event locations. Anisotropic models are important to get accurate event locations. Taking anisotropy into account makes it possible to use additional data – two S‐wave arrivals generated due to shear‐wave splitting. However, anisotropic ray tracing requires iterative procedures for computing group velocities, which may become unstable around caustics. As a result, anisotropic kinematic inversion may become time consuming. In this paper, we explore the idea of using simplified ray tracing to locate events and estimate medium parameters. In the simplified ray‐tracing algorithm, the group velocity is assumed to be equal to phase velocity in both magnitude and direction. This assumption makes the ray‐tracing algorithm five times faster compared to ray tracing based on exact equations. We present a set of tests showing that given perforation‐shot data, one can use inversion based on simplified ray‐tracing even for moderate‐to‐strong anisotropic models. When there are no perforation shots, event‐location errors may become too large for moderately anisotropic media.
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Frequency‐dependent PP and PS reflection coefficients in fractured media
Authors Shuo Pang and Alexey StovasABSTRACTWhen a porous layer is permeated by mesoscale fractures, wave‐induced fluid flow between pores and fractures can cause significant attenuation and dispersion of velocities and anisotropy parameters in the seismic frequency band. This intrinsic dispersion due to fracturing can create frequency‐dependent reflection coefficients in the layered medium. In this study, we derive the frequency‐dependent PP and PS reflection coefficients versus incidence angle in the fractured medium. We consider a two‐layer vertical transverse isotropy model constituted by an elastic shale layer and an anelastic sand layer. Using Chapman's theory, we introduce the intrinsic dispersion due to fracturing in the sand layer. Based on the series coefficients that control the behaviour of velocity and anisotropy parameters in the fractured medium at low frequencies, we extend the conventional amplitude‐versus‐offset equations into frequency domain and derive frequency‐dependent amplitude‐versus‐offset equations at the elastic–anelastic surface. Increase in fracture length or fracture density can enlarge the frequency dependence of amplitude‐versus‐offset attributes of PP and PS waves. Also, the frequency dependence of magnitude and phase angle of PP and PS reflection coefficients increases as fracture length or fracture density increases. Amplitude‐versus‐offset type of PP and PS reflection varies with fracture parameters and frequency. What is more, fracture length shows little impact on the frequency‐dependent critical phase angle, while the frequency dependence of the critical phase angle increases with fracture density.
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A high‐precision time–frequency analysis for thin hydrocarbon reservoir identification based on synchroextracting generalized S‐transform
Authors Ying Hu, Hui Chen, Hongyan Qian, Xinyue Zhou, Yuanjun Wang and Bin LyuABSTRACTImproving the seismic time–frequency resolution is a crucial step for identifying thin reservoirs. In this paper, we propose a new high‐precision time–frequency analysis algorithm, synchroextracting generalized S‐transform, which exhibits superior performance at characterizing reservoirs and detecting hydrocarbons. This method first calculates time–frequency spectra using generalized S‐transform; then, it squeezes all but the most smeared time–frequency coefficients into the instantaneous frequency trajectory and finally obtains highly accurate and energy‐concentrated time–frequency spectra. We precisely deduce the mathematical formula of the synchroextracting generalized S‐transform. Synthetic signal examples testify that this method can correctly decompose a signal and provide a better time–frequency representation. The results of a synthetic seismic signal and real seismic data demonstrate that this method can identify some reservoirs with thincknesses smaller than a quarter wavelength and can be successfully applied for hydrocarbon detection. In addition, examples of synthetic signals with different levels of Gaussian white noise show that this method can achieve better results under noisy conditions. Hence, the synchroextracting generalized S‐transform has great application prospects and merits in seismic signal processing and interpretation.
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Rock physics modelling of porous rocks with multiple pore types: a multiple‐porosity variable critical porosity model
Authors Jiajia Zhang, Yingyao Yin and Guangzhi ZhangABSTRACTA critical porosity model establishes the empirical relationship between a grain matrix and a dry rock by the concept of critical porosity. The model is simple and practical and widely used. But the critical porosity in the model is a fixed value that cannot relate to pore structure. The aim of this paper is to establish the theoretical relationship between critical porosity and pore structure by combining Kuster–Toksöz theory with the critical porosity model. Different from the traditional critical porosity model, critical porosity is not an empirical value but varied with pore shape and the ratio of bulk modulus versus shear modulus of the grain matrix. The substitution of the theoretical relationship into Kuster–Toksöz theory will generate the formulae for the bulk and shear moduli of multiple‐porosity dry rocks, which is named the multiple‐porosity variable critical porosity model. The new model has been used to predict elastic moduli for sandstone and carbonate rock. We compare the modelling results for P‐ and S‐wave velocities and elastic moduli with the experimental data. The comparison shows that the new model can be used to describe the elastic properties for the rocks with multiple pore types.
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Effects of stress reduction on geomechanical and acoustic relationship of overconsolidated sands
Authors Sirikarn Narongsirikul, Nazmul Haque Mondol and Jens JahrenABSTRACTRelationship between different geomechanical and acoustic properties measured from seven laboratory‐tested unconsolidated natural sands with different mineralogical compositions and textures were presented. The samples were compacted in the uniaxial strain configuration from 0.5 to 30 MPa effective stress. Each sand sample was subjected to three loading–unloading cycles to study the influence of stress reduction. Geomechanical, elastic and acoustic parameters are different between normal compaction and overconsolidation (unloaded and reloaded). Stress path (K0) data differ between normal consolidated and overconsolidated sediments. The K0 value of approximately 0.5 is founded for most of the normal consolidated sands, but varies during unloading depending on mineral compositions and textural differences. The K0 and overconsolidation ratio relation can be further simplified and can be influenced by the material compositions. K0 can be used to estimate horizontal stress for drilling applications. The relationship between acoustic velocity and geomechanical is also found to be different between loading and unloading conditions. The static moduli of the overconsolidated sands are much higher than normal consolidated sands as the deformation is small (small strain). The correlation between dynamic and static elastic moduli is stronger for an overconsolidation stage than for a normal consolidation stage. The results of this study can contribute to geomechanical and acoustic dataset which can be applied for many seismic‐geomechanics applications in shallow sands where mechanical compaction is the dominant mechanism.
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Coal top detection by conductively guided borehole radar wave imaging for open cut blast‐hole drilling
Authors Binzhong Zhou, Matthew van de Werken and Jianjian HuoABSTRACTDamage to the top of coal seams, caused by incorrect blast stand‐off distances, results in coal losses of up to 10–15% to the Australian open cut coal mining operations. This is a serious issue to be addressed. Here we propose to use a new forward‐looking imaging technique based on the borehole radar technology to predict the coal seam top in real time while drilling blast holes. This is achieved by coupling the conventional borehole radar waves on to a steel drill rod to induce a guided wave along the axial drill rod. The drill rod ahead of the borehole radar behaves as a forward‐looking antenna for the guided waves. Both numerical modelling and field trials simulating a drill rod as an antenna are used to investigate the feasibility of the proposed technique for prediction of the coal top under typical open cut environments. Numerical modelling demonstrated that conductivity of the overburden is the most important factor affecting our ability to see coal seams ahead of the drill bit, the guided borehole radar waves could be used for top coal prediction and a theoretical prediction error less than 10 cm and a forward‐looking capability of 4–6 m can be achieved. Field trials at Australian open cut coal mines also demonstrated that guided borehole radar waves can be observed and used for prediction of coal top ahead of drill bit during blast‐hole drilling in resistive, open cut environments (the average resistivity should be higher than 75 Ωm).
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On the exploration of a marine aquifer offshore Israel by long‐offset transient electromagnetics
Authors Klaus Lippert and Bülent TezkanABSTRACTThe existence of aquifers extending from land beneath the sea floor up to a distance of several kilometres has been observed and examined all over the world. The coastal aquifer of Israel is a heavily used groundwater reservoir which has to be constantly monitored to ensure the drinking water supply. Former land‐based electromagnetic measurements show that it is, in several places, blocked to seawater intrusion and is consequently a candidate for submarine extension. Multicomponent long‐offset transient electromagnetic measurements were carried out offshore on the coast of Israel. We deployed a 400‐m‐long grounded dipole as transmitter and several electric and magnetic receivers on the sea floor up to a distance of 4.8 km from the coast. Altogether, we deployed 8 transmitter positions and received data sets at 14 receiver stations onshore and offshore, with offsets of mostly 400 and 800 m. In this paper, we present the survey and 1D Occam and Marquardt inversions of the offshore horizontal electric components in the broadside and inline configuration. In addition, the vertical magnetic component in the broadside position is also considered. Only single inversions, both single offset and single component, were used to detect the aquifer under sea sediments. We prove the submarine existence of the Israeli coastal aquifer up to a distance to the coast of approximately 3.2 to 3.6 km using all measured long‐offset transient electromagnetic components. In addition, we present modelling studies with synthetic data derived from a subsurface model adjusted to our survey area with very shallow water from 10 to 50 m. As part of the planning before the survey, a parameter study of the expected subsurface, the examination of the airwave phenomenon and the justification for our 1D inversion strategy are shown. More detailed eigen parameter analyses are added to explain the measured data.
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Magnetotelluric responses of three‐dimensional conductive and magnetic anisotropic anomalies
Authors Tiaojie Xiao, Yun Wang, Xiangyu Huang, Guoli He and Jie LiuABSTRACTA magnetotelluric finite‐element modelling algorithm is developed, which is capable of handling three‐dimensional conductive and magnetic anisotropic anomalies. Different from earlier three‐dimensional magnetotelluric anisotropic modelling methods, the algorithm we presented has taken the magnetic anisotropy into consideration. The variational equations are produced by the Galerkin method and the governing equations are solved using a hexahedral vector edge finite‐element method. The accuracy of this algorithm is firstly validated by comparing its solutions with the results of finite‐difference method for a three‐dimensional conductive arbitrary anisotropic model, and then validated by comparing with analytical solutions for a one‐dimensional magnetic model. The responses of four kinds of models under different conditions are studied, and some conclusions are obtained. It shows that for materials with a high magnetic permeability, its influence on magnetotelluric responses cannot be ignored in some circumstances. Especially, if the magnetic susceptibility is exceptionally high, it may really distort the apparent resistivities of lower resistive anomalies. These conclusions are also beneficial for magnetotelluric survey.
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An optimal Savitzky–Golay derivative filter with geophysical applications: an example of self‐potential data
More LessABSTRACTWe propose a strategy in designing an optimal set of filter parameters, such as the order of interpolating polynomial and the filter length for a Savitzky–Golay derivative filter. The proposed strategy is based on the ‘principle of parsimony’ while satisfying the optimality conditions. The optimality conditions are based on the Durbin–Watson lag‐1 test statistic and the Derringer–Suich desirability function. While the former checks for an appropriate data fitting, the latter, on the other hand, ensures minimal shape distortion of the reconstructed response. The proposed strategy of designing filter parameters is developed and validated through numerical experiments using Gaussian pulse as a test function which is contaminated with additive white Gaussian noise. In the numerical tests, the polynomial orders used were 3, 5 and 7, but the filter length for each polynomial was varying until the optimality conditions were satisfied. The Savitzky–Golay derivative filtering is used in obtaining the robust reconstruction of noisy geophysical anomaly and the robust estimation of its first‐ and second‐order derivatives. We validated the proposed technique on the published self‐potential anomaly data using a data‐based interpretation technique where the reconstructed anomaly and its first‐ and second‐order derivatives were used in estimating model parameters. The data‐based interpretation using the proposed technique of Savitzky–Golay derivative filtering provides a close agreement with the published results.
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