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- A priori and effective estimation of variance factors based on the code
chipping rate in BeiDou navigation satellite system positioning-
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Abstract: In multi-frequency and multi-constellation BeiDou Navigation Satellite System (BDS), the observation type is increased, and the observation precision is inevitably different. Consequently, it is difficult to determine the variance factors of various observations. Variance component estimation can reasonably determine the weights of different types of observations and greatly improve positioning accuracy, but the prerequisite is that there are enough redundant observations, which may not be met in the case of BDS. In addition, it has relatively high time and space complexity. In this study, a priori and effective estimation of variance factors based on the code chipping rate is proposed to properly adjust and determine the observation weights in BDS, thus better characterizing the observation precision while simplifying the calculation. Both static and kinematic experiments were conducted to verify the effectiveness of the new method. The results show that the proposed method is suitable for both open and obstructed environments, and the accuracy and reliability of single point positioning are improved while high efficiency is met.
PubDate: 2023-05-18
- On the topographic bias by analytical continuation in geoid determination
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Abstract: We consider the topographic bias in gravimetric geoid determination when analytically downward continuing the disturbing potential from the Earth’s surface to sea level. The total bias is subdivided into those of the Bouguer shell or plate and the terrain. In this process, the potential of the Bouguer shell always has a downward continuation bias in the process, which increases with the square of the topographic height and typically exceeds 1–2 cm for elevations higher than 1 km. The main conclusion is that the terrain does not provide a potential bias except possibly for masses located inside a dome of height of about 0.4 times the height of the computation point, and base radius equal to the height of the computation point. This result implies that the potential of all terrain masses of arbitrary density located exterior to the Bouguer shell as well as those outside the dome are unbiasedly downward continued to sea level.
PubDate: 2023-04-25
- The gravitational potential and its first- and second-order partial
derivatives of an ellipsoidal tesseroid based on the Cartesian integral
kernel-
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Abstract: Gravity forward modelling is a fundamental problem in the fields of geophysics and geodesy at regional and global scales. Considering the curvature of the Earth, tesseroids are suitable to accurately simulate the theoretical gravity field. In general, the spherical tesseroid is regarded as an ideal model, but it cannot consider the oblateness of the Earth. Therefore, we define an ellipsoidal tesseroid at the local Cartesian coordinate system. Then we propose the formulas of the gravitational potential and its first- and second-order partial derivatives of the ellipsoidal tesseroid based on the Cartesian integral kernel. To enhance the practicality, we approximate the ellipsoidal tesseroid to the spherical tesseroid and derive the formulas of the gravitational potential and its partial derivatives. Moreover, we discuss the formulas of the gravity field for the model with linear variable density. The ellipsoidal tesseroid, which is selected as the fundamental mass element, can more accurately simulate the gravity and gravity gradient anomalies of the Earth. Compared with methodologies that make use of integral kernels expressed in spherical coordinate system, the formulas based on the Cartesian integral kernel are given in compact and computationally attractive form. Besides, these formulas can avoid the polar singularity of the spherical coordinate system. The numerical simulation and comparison with previous methods validate the new ellipsoidal tesseriod formulas.
PubDate: 2023-03-21
DOI: 10.1007/s11200-022-0344-5
- Numerical simulation and time-frequency analysis of borehole acoustic
logging in fractured formations-
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Abstract: Currently, using the finite difference method to simulate millimeter-sized fractures in formations requires intensive calculations. However, only the time domain characteristics of the calculated borehole acoustic signal are often analysed, while the frequency domain characteristics are ignored. This study aims to obtain the time-frequency characteristics of full acoustic waveforms in different types of fractured formations while reducing operational time and to analyze more comprehensively the influence of fractures on time-frequency characteristics. Therefore, the variable grid finite difference method is used to simulate full acoustic waveforms in boreholes in formations with millimeter-sized horizontal fractures to reduce the computational time of the finite difference method. Afterwards, the wavelet transform is used to analyze the influence of fracture width, fracture number, and radial extension length on the waveform time-frequency characteristics. The results show that with increasing fracture width or number, the P- and S-wave arrival times are delayed, amplitude attenuation is enhanced, and the dominant frequency increases gradually. The frequency and amplitude attenuation of each Stoneley wave component also increases, and the arrival time of the 20–28 kHz high-frequency Stoneley wave is delayed. When the fracture radial length is limited, an increase in radial length delays the P- and S-wave arrival times, and the amplitude attenuation increases. The main S-, Stoneley, and pseudo-Rayleigh wave frequencies increase, and the Stoneley wave and pseudo-Rayleigh wave amplitude attenuation increases. When the fracture radial length is infinite, the P-wave and pseudo-Rayleigh wave amplitude attenuation increases, whereas that of the S-wave and Stoneley wave decreases. This study reveals the influence of fractures on the time-frequency characteristics of full acoustic waveforms in boreholes, provides a theoretical basis for the time-frequency analysis of full acoustic waveforms, and is significant for further clarification of the propagation characteristics of borehole acoustic waves in fractured formations.
PubDate: 2023-01-11
DOI: 10.1007/s11200-021-1145-y
- Magnetic susceptibility in soil pedons developed on different parent rocks
in Kerman province (Iran)-
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Abstract: Lithology and soil formation affect magnetic susceptibility and its distribution along soil pedons. Kerman province in Iran is typical for variable lithology. However, only limited data on soil magnetic susceptibility in this province and its relation to the lithology are available. We investigate the effect of soil properties and processes on magnetic susceptibility values of soils with different geology in central Iran. Seven soil pedons with different lithology including sedimentary and igneous bedrocks were selected in arid and semi-arid parts of northern Kerman. Routine physical and chemical properties, different forms of iron, and mass-specific magnetic susceptibility values were measured in all the collected samples. Four selected samples underwent magnetic separation. Two of them, which yielded the highest amount of magnetically extracted material, were subjected to X-ray diffraction analysis. In addition, polished sections from the sample with the highest mass-specific magnetic susceptibility were prepared. The results show that lithology strongly affects the magnetic susceptibility in the studied soils, ranging from the minimum value of 4.3 × 10−8 m3 kg−1 (in the soils developed on Cretaceous marls and limestone) to 1264 × 10−8 m3 kg−1 (on andesite rocks). Frequency-dependent susceptibility values of soils (from 0 to 5.3%) showed that coarse multi domain grains inherited from parent material were the main source of magnetism in the area under study. The average amount of free, non-crystalline, crystalline, and active iron oxides in the studied samples were 0.5, 9.96, 8.45, and 0.05 g kg−1, respectively. The weighted mean for different factors was calculated in three depth ranges. Slope of linear regression was used to investigate the relation between mass-specific susceptibility and physicochemical parameters for different soil depths. The argilluviation process caused a decrease in the magnetic susceptibility in moderately developed soils of the region. The highest magnetic susceptibility values were found for Cambic Calcisols, followed by the Abruptic Solonetz, both developed on the andesite and gypsiferous marl. A positive relationship between magnetic susceptibility and Feo, Fed and Fed — Feo, and a negative correlation between magnetic susceptibility and Feo/Fed were found. According to X-ray diffraction analyses, diamagnetic minerals are dominant, while antiferromagnetic minerals are rare. The results suggest that changes in the magnetic susceptibility values are highly affected by the processes of soil formation, lithology, and soil classification.
PubDate: 2023-01-11
DOI: 10.1007/s11200-021-0771-8
- Regionally adjusted ground motion model: Case study of the ML6.2 (Mw6.4)
Petrinja (Croatia) 2020 earthquake-
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Abstract: In the wake of recent 2020 ML ≥ 5.5 earthquakes in Croatia, Zagreb ML5.5 and Petrinja ML6.2, the insufficient instrumental network as well as the lack of regional ground motion prediction equation (GMPE) were identified as the drawbacks of our engineering community. The former is related to the quality definition of active seismicity (most of the instruments are installed in the southern part of Croatia with fewer installed around Zagreb in the northwestern part of Croatia), and the latter is related to the proper number of strong motion recordings. In Croatia, there is a sparse database of ground motion recordings for moderate earthquakes which makes a well-designed ground motion selecting procedure hardly achievable. Following this, strong motion BSHAP database for empirical estimation of the response spectrum based on Fourier amplitude spectrum and the ground motion duration using Random Vibration Theory approach adjusted to source, propagation, and local site conditions was used. Regionally adjusted ground motion model estimations for the ML6.2 Petrinja 2020 earthquake scenario are comparable with the previously published GMPEs models for this part of Europe and for the Western part of North America. However, model-to-model variability and uncertainties in local GMPE exceeded those of global GMPEs and are influenced by statistically less stable and more limited datasets. Model is applicable for magnitudes up to Mw6.5 and Joyner-Boore distances up to 200 km with usable frequency range between 0.4 and 33 Hz. The presented model is a step forward toward performing hybrid-empirical seismic hazard studies in areas with sparse ground motions such as the region of Croatia.
PubDate: 2022-10-01
DOI: 10.1007/s11200-022-0914-6
- Hybrid geoid model over peninsular Malaysia (PMHG2020) using two
approaches-
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Abstract: We describe the development of a hybrid geoid model for Peninsular Malaysia, based on two approaches. The first approach is utilising an ordinary method fitting the gravimetric geoid to the geometric undulation derived from GNSS-levelling data; the second approach directly fits the gravimetric geoid to the reference mean sea level derived from the tide measurements of Port Klang tide gauge station. The hybrid geoid model fitted to Port Klang (PMHGG2020_PK) is produced by adding an offset of 0.446 m to the gravimetric geoid, based on the comparison at the tide gauge benchmark. To calculate the gravimetric geoid, a new model for Peninsular Malaysia (PMGG2020) has been developed based on Least-Squares Modification of Stokes’ Formula with Additive correction (LSMSA). Three different sources of gravity data which are terrestrial, airborne, and satellite altimetry-derived gravity anomaly (DTU17) have been combined to construct the geoid model. The height information has been extracted from the newly released global digital elevation model, TanDEM-X DEM. GO_CONS_GCF_2_SPW_R4 model derived from GOCE data provides long-wavelengths gravity field up to maximum degree and order 130. The gravity datasets are gridded by 3D Least-Squares Collocation method. The PMGG2020 model is consistent with the geometric geoid heights from 173 GNSS-levelling measurements, with a standard deviation of ±5.8 cm. Evaluation of the hybrid geoid model constructed from the first approach shows a significant improvement over the two existing hybrid geoid models. The accuracy of ±4.6 cm has been achieved after evaluating by 20 GNSS-levelling points, externally. Hybrid geoid model fitted to Port Klang has also been evaluated via 173 GNSS-levelling points, and the result shows that 71% of the total data exhibit height differences lower than 10 cm. The overall results indicate that the hybrid geoid model developed in this study can be valuable as an alternative to the current modern height system in Peninsular Malaysia for surveying and mapping.
PubDate: 2022-10-01
DOI: 10.1007/s11200-021-0769-2
- New methods for numerical evaluation of ultra-high degree and order
associated Legendre functions-
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Abstract: We improve the precision and computation speed of the fully-normalized associated Legendre functions (fnALFs) for ultra-high degrees and orders of spherical harmonic transforms. We take advantage of their numerical behaviour of and propose two new methods for solving an underflow/overflow problem in their calculation. We specifically discuss the application of the two methods in the fixed-order increasing-degree recursion computation technique. The first method uses successive ratios of fnALFs and the second method, called the Midway method, starts iteration from tiny initial values, which are still in the range of the IEEE double-precision environment, rather than from sectorial fnALFs. The underflow/overflow problem in the successive ratio method is handled by using a logarithm-based method and the extended range arithmetic. We validate both methods using numerical tests and compare their results with the X-number method in terms of precision, stability, and speed. The results show that the relative precision of the proposed methods is better than 10−9 for the maximum degree of 100000, compared to results derived by the high precision Wolfram’s Mathematica software. Average CPU times required for evaluation of fnALFs over different latitudes demonstrate that the two proposed methods are faster by about 10–30% and 20–90% with respect to the X-number method for the maximum degree in the range of 50–65000.
PubDate: 2022-10-01
DOI: 10.1007/s11200-022-0830-9
- Tracing real-valued reference rays in anisotropic viscoelastic media
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Abstract: The eikonal equation in an attenuating medium has the form of a complex—valued Hamilton—Jacobi equation and must be solved in terms of the complex—valued travel time. A very suitable approximate method for calculating the complex—valued travel time right in real space is represented by the perturbation from the reference travel time calculated along the real—valued reference rays to the complex—valued travel time defined by the complex—valued Hamilton—Jacobi equation. The real—valued reference rays are calculated using the reference Hamiltonian function. The reference Hamiltonian function is constructed using the complex—valued Hamiltonian function corresponding to a given complex—valued Hamilton—Jacobi equation. The ray tracing equations and the corresponding equations of geodesic deviation are often formulated in terms of the eigenvectors of the Christoffel matrix. Unfortunately, a complex—valued Christoffel matrix need not have all three eigenvectors at an S—wave singularity. We thus formulate the ray tracing equations and the corresponding equations of geodesic deviation using the eigenvalues of a complex—valued Christoffel matrix, without the eigenvectors of the Christoffel matrix. The resulting equations for the real—valued reference P—wave rays and the real—valued reference common S—wave rays are applicable everywhere, including S—wave singularities.
PubDate: 2022-10-01
DOI: 10.1007/s11200-022-0906-6
- Extremely strong anisotropy of out-of-phase component of AC magnetic
susceptibility in hematite single crystals and its origin-
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Abstract: So far unknown anisotropic properties of out-of-phase magnetic susceptibility (opMS) of hematite and their bearing on understanding the origin of hematite magnetism were investigated on example of four single crystals from Minas Gerais, Brazil. The research comprised measurement of directional variation of field-dependent and frequency-dependent opMS and testing whether the anisotropy of opMS can be represented by the second rank tensor. It was found that the opMS shows strong field dependence along basal plane and only weak field dependence along c-axis. As for the frequency dependence, it is virtually non-existent along the basal plane. Along the c-axis, the opMS is similar at frequencies 976 and 3904 Hz, while at 15616 Hz it is clearly higher. Consequently, opMS is dominantly due to weak field hysteresis. The minimum opMS directions are parallel to the c-axis, while the other two principal directions lie within basal plane. The degree of anisotropy is extremely high (ratio of maximum opMS to minimum opMS ≫100) and the anisotropy ellipsoid is very oblate. In stereographic diagrams, the opMS contours very roughly resemble the theoretical contours calculated from opMS tensor. However, the differences between measured and theoretical values in directional opMS are clearly higher than the measuring error and are distributed very non-homogeneously. This indicates that the second rank tensor is only very approximate representative of the spatial variation of the directional opMS of hematite single crystals. Earlier model of hematite magnetism assuming slightly scanted antiferromagnetism along the c-axis and strongly canted antiferromagnetism or even weak ferromagnetism along the basal plane is supported.
PubDate: 2022-08-30
DOI: 10.1007/s11200-022-0829-2
- Anisotropic VSP reverse-time migration with optimal pure acoustic wave and
complex wavefield separation-
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Abstract: Ignoring anisotropy characteristic of subsurface media may lead to misplaced images and low resolution of the target for the reverse-time migration (RTM). The mature anisotropic RTM methods are mainly based on the pseudoacoustic wave approximation. Although these schemes have high computational efficiency, most of pseudo-acoustic wave equations (PWEs) inevitably encounter SV-wave artifacts or instability for anisotropic modeling and imaging. To improve the anisotropic RTM quality, we develop a combination of optimal pure acoustic wave and complex wavefield separation to conduct anisotropic RTM for both surface and vertical seismic profiling (VSP) acquisition geometries. Among the proposed scheme, we derive an optimal pure acoustic wave dispersion relation, and solve the corresponding wave equation by incorporating finite-difference and Poisson solver. The modified equation can remove SV-wave artifacts and instability of PWEs. Wavefield separation approach can choose desired wavefield components along different directions to carry out the final imaging, which can effectively suppress low-frequency imaging noise. Moreover, the hybrid absorbing boundary condition is adopted to suppress artificial boundary reflections during wavefield extrapolation. Basic theory and modeling examples demonstrate that the developed schemes can generate RTM results with high accuracy.
PubDate: 2022-08-27
DOI: 10.1007/s11200-022-0717-9
- Velocity analysis of noisy seismic data based on RSVD weighted semblance
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Abstract: The accuracy of velocity spectrum affects the subsequent processing of seismic data. Though the singular value decomposition (SVD) weighted semblance has a higher velocity resolution than conventional semblance, its performance is degraded for noisy seismic data. A rectified SVD weighted semblance method (RSVD), aiming to improve the accuracy of velocity spectrum for seismic data contaminated by noise, is proposed. In this approach, the weighting function is constructed from the first two singular values and their mean square error obtained via SVD of noisy seismic data after normal moveout (NMO) with scanning velocity. Synthetic and field examples demonstrate that the proposed method performs better than the SVD weighted semblance in enhancing the accuracy of velocity spectra for noisy near-offset common midpoint gathers in layered isotropic media.
PubDate: 2022-04-30
DOI: 10.1007/s11200-021-0327-y
- Erratum to “Improving the accuracy of the expanded anisotropic eikonal
equation at larger offsets using Levin’s T-transformation”-
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PubDate: 2022-04-04
DOI: 10.1007/s11200-022-0901-y
- Expressions for the calculation of isotropic Gaussian filter kernels in
the spherical harmonic domain-
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Abstract: The isotropic Gaussian filter has been used extensively in Gravity Recovery and Climate Experiment (GRACE) temporal gravity field solutions, and is still being applied to GRACE Follow-On products to remove high-frequency errors and improve the estimation of mass transport events on the Earth’s surface. For such applications, the only known rigorous method to calculate the spherical harmonic coefficients of an isotropic Gaussian filter is by the use of a second-order recurrence relation. As an alternative, an approximate expression is also used frequently. In this paper, we provide some additional expressions for the calculation of isotropic Gaussian filter kernels in the spherical harmonic domain. Specifically, we derive a new recurrence relation, a closed-form expression, expressions involving modified Bessel functions of the first kind, and a new approximate expression. We also examine and compare them from a computational viewpoint. The results of our numerical investigations indicate that the new recurrence relation and the closed-form expression are unstable in a way similar to the second-order recurrence relation that has been used so far. The expressions involving modified Bessel functions, and particularly the ones using exponentially scaled modified Bessel functions, provide a simple, elegant and stable way of calculating isotropic Gaussian filter coefficients, since routines for their stable evaluation are readily available in many programming languages. Alternatively, the new approximate expression can be used, which is also stable and offers better accuracy than previous approximations.
PubDate: 2022-04-04
DOI: 10.1007/s11200-021-0272-9
- Evaluation of latest marine gravity field models derived from satellite
altimetry over the Gulf of Guinea (Central Africa) with shipborne gravity
data-
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Abstract: The marine gravity field is vital for mapping various submarine geological and tectonic structures, also for computation of high-resolution gravimetric geoid. This study aims to evaluate the accuracy of two latest high-resolution marine gravity models derived from satellite altimetry (DTU17 and SSv27.1) using shipborne gravity data and to pruduce high-precision gravity field over the Gulf of Guinea. The gross-errors affecting the shipborne gravity data have been removed by cross-validation technique to ensure better evaluation of gravity field models. The standard deviation σ of the differences between the measured and model gravity data drops from 9.96 mGal before the cross-validation to 6.28 mGal after this process. The comparison between the DTU17 and SSv27.1 gravity field models has been done in order to detect significant differences between them. The differences between the two models are quite small with a mean of 1.73 mGal and σ of 6.55 mGal. The discrepancies between them are found around coastal areas and along islands. This shows the poor accuracy of satellite altimetry near coastal areas. Afterwards, the accuracy of each marine gravity field models was evaluated using shipborne gravity data free of gross-errors. The SSv27.1 model fits better to the shipborne gravity data with a mean of −4.88 mGal and σ of 7.18 mGal. Hence, the SSv27.1 model has a better performance than the DTU17 model on the Gulf of Guinea. Finally, we used the least-squares collocation technique associated to the Markov model of second-order covariance to combine the SSv27.1 model with the shipborne gravity data. We produced here a marine gravity field of good accuracy around the Gulf of Guinea with no data gaps. The precision of this combined gravity field is estimated to be 5.54 mGal with a spatial resolution of 1 arc-minute.
PubDate: 2022-04-04
DOI: 10.1007/s11200-021-0157-y
- Rotationally invariant viscoelastic medium with a non-symmetric stiffness
matrix-
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Abstract: The stiffness matrix of a viscoelastic medium is symmetric in the low—frequency and high—frequency limits, but not for finite frequencies. We thus consider a non—symmetric stiffness matrix in this paper. We determine the general form of a rotationally invariant non—symmetric stiffness matrix of a viscoelastic medium. It is described by three additional complex—valued parameters in comparison with a rotationally invariant symmetric stiffness matrix of a transversely isotropic (uniaxial) viscoelastic medium with a symmetric stiffness matrix. As a consequence, we find that the stiffness matrix of an isotropic viscoelastic medium is always symmetric.
PubDate: 2022-02-08
DOI: 10.1007/s11200-021-1106-5
- Geoelectrical imaging and spatial distribution of alluvial materials of
the Sanaga River (Cameroon, Central Africa)-
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Abstract: Recent research has shown that the floodplains of the Sanaga river catchment contain significant resources of minerals suitable for industry. However, these studies were carried out in large grids, and do not allow for detailed specification of the geometry of the identified resources. The aim of this study is to minimize the uncertainties observed in the characterization of these deposits using two techniques: geoelectric imaging and manual drilling. The geoelectrical results associated with the lithological sections show, from surface to depth, three main units: semi-resistive (66–500 Ωm) corresponding to clayey sands; conductive (42–100 Ωm) associated with sandy clays/silt clays, and resistive (<1000 Ωm) corresponding to sands and gravels. The data show good correlation between resistivity and clay content, and resistivity and sand content. Similarly, good correlation exists between the thickness determined by geoelectrical method and the actual thickness. The average thickness of these layers on the banks of the Sanaga river, suitable for exploitation, is 4 m over an area of about 100 m2. Our results show that the combination of geoelectric technique and manual drilling allows a good assessment of the volume of deposits of useful resources and can better constrain their exploitation in the field of civil engineering.
PubDate: 2022-02-08
DOI: 10.1007/s11200-020-1083-0
- Two S-wave eigenvectors of the Christoffel matrix need not exist in
anisotropic viscoelastic media-
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Abstract: The 3×3×3×3 frequency-domain stiffness tensor is complex-valued in viscoelastic media. The 3 × 3 Christoffel matrix is then also complex-valued. Using a simple example, we demonstrate that a complex-valued Christoffel matrix need not have all three eigenvectors at an S-wave singularity, and we thus cannot apply the eigenvectors to calculating the phase-space derivatives of the Hamiltonian function.
PubDate: 2021-11-15
DOI: 10.1007/s11200-021-0824-z
- Sensitivity Gaussian packets
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Abstract: Perturbations of elastic moduli and density can be decomposed into Gabor functions. The wave field scattered by the perturbations is then composed of waves scattered by the individual Gabor functions. The scattered waves can be estimated using the first-order Born approximation with the paraxial ray approximation. For a particular source generating a short-duration broad-band incident wave field with a smooth frequency spectrum, each Gabor function generates at most a few scattered sensitivity Gaussian packets propagating in determined directions. Each of these scattered Gaussian packets is sensitive to just a single linear combination of the perturbations of elastic moduli and density corresponding to the Gabor function. This information about the Gabor function is lost if the scattered sensitivity Gaussian packet does not fall into the aperture covered by the receivers and into the recording frequency band. We illustrate this loss of information using the difference between the 2-D Marmousi model and the corresponding smooth velocity model. We decompose the difference into Gabor functions. For each of the 240 point shots, we consider 96 receivers. For each shot and each Gabor function, we trace the central ray of each sensitivity Gaussian packet. If a sensitivity Gaussian packet arrives to the receiver array within the recording time interval and frequency band, the recorded wave field contains information on the corresponding Gabor function. We then decompose the difference into the part influencing some recorded seismograms, and the part on which we recorded no information and which thus cannot be recovered from the reflection experiment.
PubDate: 2021-11-15
DOI: 10.1007/s11200-021-0931-x
- Erratum to “Bounds and averages of seismic quality factor Q”
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PubDate: 2021-07-28
DOI: 10.1007/s11200-021-0428-7
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