Local Slope Estimation Research Articles

PINNslope: Seismic data interpolation and local slope estimation with physics informed neural networks

Interpolation of aliased seismic data constitutes a key step in a seismic processing workflow to obtain high-quality velocity models and seismic images. Building on the idea of describing seismic wavefields as a superposition of local plane waves, we propose to interpolate seismic data by using a physics informed neural network (PINN). In the proposed framework, two feed-forward neural networks are jointly trained using the local plane wave differential equation as well as the available data as two terms in the objective function: a primary network assisted by positional encoding is tasked with reconstructing the seismic data, whereas an auxiliary, smaller network estimates the associated local slopes. Results on synthetic and field data validate the effectiveness of the proposed method in handling aliased (coarsely sampled) data and data with large gaps. Our method compares favorably against a classic least-squares inversion approach regularized by the local plane-wave equation as well as a PINN-based approach with a single network and precomputed local slopes. We find that introducing a second network to estimate the local slopes, whereas at the same time interpolating the aliased data enhances the overall reconstruction capabilities and convergence behavior of the primary network. Moreover, an additional positional encoding layer embedded as the first layer of the wavefield network confers to the network the ability to converge faster, improving the accuracy of the data term.

Statistically representative estimators of multi-scale surface topography: example of aluminum blasted rough samples

The topography of a rough surface determines many of its physical properties, for instance, tribology, contact mechanics, optical properties etc. Nowadays, a deep understanding of such physical phenomena requires the knowledge of the topography at appropriate length scales. Apart from performing multi-scale measurements of the surface topography, it also requires the use of proper statistical estimators for the analysis of such topography maps. Moreover, when dealing with light scattering in the visible spectral range, the scale at which the estimators of local topography properties are defined is extremely important. Here we present a multi-scale and statistical study of the surface topography of blasted aluminum samples which all have rather different visual appearance. Various statistical estimators of surface topography are examined, including estimators related to the height distribution, the lateral correlation and local topology. The combination of these various estimators unveils a scale separation between a micro-scale roughness inherited from the initial cold-rolled aluminum surface and a large scale roughness fully controlled by the blasting process. A special emphasis is given to the crucial importance of length scales in the estimation of local slopes. The present analysis establishes a quantitative link between the statistical properties of the surface topography and the blasting process used to fabricate the samples.

Robust local slope estimation by deep learning

ABSTRACTIn the seismic community, the local slope is essential for various applications, including structure prediction, seislet transform, trace interpolation and denoising. The most popular method to calculate slope is the plane‐wave destruction, which assumes that the seismic data can be represented by local plane waves. However, the plane‐wave destruction method fails when the seismic data become very noisy. Taking random noise into consideration, we adopt the deep learning method to calculate the local slope. In the deep learning architecture, the input is noisy data and the target is an accurate slope estimated from the clean data using plane‐wave destruction. The deep learning architecture includes a convolutional layer, deconvolutional layer, normalization layer and activation layer which are suitable to suppress noise and learn the nonlinear relationship between the input and the target. After training, the network is applied to other test examples to calculate local slopes. In addition, we implement the seislet transform and structure prediction applications based on the estimated slope. Both the estimated slope and the related applications indicate that the proposed method can robustly obtain the local slope from noisy seismic data, compared with the plane‐wave destruction method.

A MATLAB code package for 2D/3D local slope estimation and structural filtering

Local slope is an important attribute that can help distinguish seismic signals from noise. Based on optimal slope estimation, many filtering methods can be designed to enhance the signal-to-noise ratio of noisy seismic data. We present an open-source MATLAB code package for local slope estimation and corresponding structural filtering. This package includes 2D and 3D examples with two main executable scripts and related subfunctions. All code files are in the MATLAB format. In each main script, local slope is estimated based on the well-known plane-wave destruction algorithm. Then, the seismic data are transformed to the flattened domain by utilizing this slope information. Furthermore, the smoothing operator can be effectively applied in the flattened domain. We introduce the theory and mathematics related to these programs and present the synthetic and field data examples to show the usefulness of this open-source package. The results of local slope estimation and structural filtering demonstrate that this package can be conveniently and effectively applied to the seismic signal analysis and denoising.

Prestack diffraction separation by parameterizing the reflection local slope

Diffracted waves provide an opportunity to detect small-scale subsurface structures because they give wide illumination direction of geologic discontinuities, such as faults, pinch outs, and collapsed columns. However, separating diffracted waves is challenging because diffracted waves have greater geometric amplitude losses and are generally weaker than reflections. To retain more diffracted waves, we have developed a prestack diffraction separation method based on the local slope pattern and plane-wave destruction (PWD) method. In general, it is difficult to distinguish between hyperbolic reflections and hyperbolic diffractions using data-driven local slope estimation in the shot domain. Therefore, we transfer slope estimation in the shot domain to velocity analysis in the common-midpoint domain and ray parameter calculation in the stack domain. The connection between local slope and the normal moveout velocity and the surface-ray parameter is known, which provides a novel approach for estimating the local slope of hyperbolic reflected waves in the shot domain. The estimated slope can provide an exact slope-based operator for the PWD method, thus allowing the PWD to separate diffracted waves from reflected waves in the shot domain. Synthetic and field data tests demonstrate the feasibility and effectiveness of our prestack diffraction separation method.

A Novel Method for Estimating the Intrinsic Magnetic Field Spectrum of Kinetic-Range Turbulence

Understanding plasma turbulence below the ion characteristic scales is one of the key open problems of solar wind physics. The bulk of our knowledge about the nature of the kinetic-scale fluctuations comes from the high-cadence measurements of the magnetic field. The spacecraft frame frequencies of the sub-ion scale fluctuations are frequently around the Nyquist frequencies of the magnetic field sampling rate. Thus, the resulting ‘measured’ time series may significantly differ from the ‘true’ ones. It follows that second-order moments (e.g., power spectral density, PSD) of the signal may also be highly affected in both their amplitude and their slope. In this paper, we focus on the estimation of the PSD slope for finitely sampled data and we unambiguously define a so-called local slope in the framework of Continuous Wavelet Transform. Employing Monte Carlo simulations, we derive an empirical formula that assesses the statistical error of the local slope estimation. We illustrate the theoretical results by analyzing measurements of the magnetic field instrument (MFI) on board the Wind spacecraft. Our analysis shows that the trace power spectra of magnetic field measurements of MFI can be modeled as the sum of PSD of an uncorrelated noise and an intrinsic signal. We show that the local slope strongly depends on the signal-to-noise (S/N) ratio, stressing that noise can significantly affect the slope even for S/N around 10. Furthermore, we show that the local slopes below the frequency corresponding to proton inertial length, 5≳kλpi>1, depend on the level of the magnetic field fluctuations in the inertial range (Pin), exhibiting a gradual flattening from about −11/3 for high Pin toward about −8/3 for low Pin.

Evaluating Stereo Digital Terrain Model Quality at Mars Rover Landing Sites with HRSC, CTX, and HiRISE Images

We have used high-resolution digital terrain models (DTMs) of two rover landing sites based on mosaicked images from the High-Resolution Imaging Science Experiment (HiRISE) camera as a reference to evaluate DTMs based on High-Resolution Stereo Camera (HRSC) and Context Camera (CTX) images. The Next-Generation Automatic Terrain Extraction (NGATE) matcher in the SOCET SET and GXP® commercial photogrammetric systems produces DTMs with good (small) horizontal resolution but large vertical error. Somewhat surprisingly, results for NGATE are terrain dependent, with poorer resolution and smaller errors on smoother surfaces. Multiple approaches to smoothing the NGATE DTMs give similar tradeoffs between resolution and error; a 5 × 5 lowpass filter is near optimal in terms of both combined resolution-error performance and local slope estimation. Smoothing with an area-based matcher, the standard processing for U.S. Geological Survey planetary DTMs, yields similar errors to the 5 × 5 filter at slightly worse resolution. DTMs from the HRSC team processing pipeline fall within this same trade space but are less sensitive to terrain roughness. DTMs produced with the Ames Stereo Pipeline also fall in this space at resolutions intermediate between NGATE and the team pipeline. Considered individually, resolution and error each varied by approximately a factor of 2. Matching errors were 0.2–0.5 pixels but most results fell in the 0.2–0.3 pixel range that has been stated as a rule of thumb in multiple prior studies. Horizontal resolutions of 10–20 image pixels were found, consistently greater than the 3–5 pixel spacing generally used for stereo DTM production. Resolution and precision were inversely correlated; their product varied by ≤20% (4–5 pixels squared). Refinement of the stereo DTM by photoclinometry can yield quantitative improvement in resolution (more than a factor of 2), provided that albedo variations over distances smaller than the stereo DTM resolution are not too severe. We offer specific guidance for both producers and users of planetary stereo DTMs, based on our results.

A Refined Phase Unwrapping Method for High Noisy Dense Fringe Interferogram Based on Adaptive Cubature Kalman Filter

Cubature Kalman filter phase unwrapping (CKFPU) is an effective algorithm in unwrapping the interferograms. The local phase slope estimation is a key factor that affects the unwrapped accuracy. However, the estimation accuracy of local phase slop is relatively low in high noisy and dense stripes areas, which usually leads to the unsatisfactory unwrapped results. In order to effectively solve this issue, the rewrapped map of the unwrapped phase (obtained by CKFPU algorithm), which is a filtered interferogram with clearer fringes and more detailed information, is proposed in this paper to improve the phase slope estimation. In order to solve the problem of imprecise error variance for the new phase slope estimation, an adaptive factor is introduced into the CKFPU algorithm to increase the stability and reliability of the phase unwrapping algorithm. The proposed method is compared with the standard CKFPU algorithm using both simulated and real data. The experimental results validate the feasibility and superiority of the proposed method for processing those high noise dense fringe interferograms.

FURTHER ADVENTURES IN MARS DTM QUALITY: SMOOTHING ERRORS, SHARPENING DETAILS

Abstract. We have used high-precision, high-resolution digital terrain models (DTMs) of the NASA Mars Science Laboratory and Mars 2020 rover landing sites based on mosaicked images from the Mars Reconnaissance Orbiter High Resolution Imaging Science Experiment (MRO HiRISE) camera as a reference data set to evaluate DTMs based on Mars Express High Resolution Stereo Camera (MEX HRSC) images. The Next Generation Automatic Terrain Extraction (NGATE) matcher in the SOCET SET/GXP® commercial photogram- metric system produces DTMs with relatively good (small) horizontal resolution but high error, and results are terrain dependent, with poorer resolution and smaller errors on smoother surfaces. Multiple approaches to smoothing the NGATE DTMs give very similar tradeoffs between resolution and error. Smoothing the NGATE DTMs with a 5x5 lowpass filter is near optimal in terms of both combined resolution-error performance and local slope estimation, but smoothing with a single pass of an area-based matcher, which has been the standard approach for generating planetary DTMs at the U.S. Geological Survey to date results in similar errors and only slightly worse resolution. DTMs from the HRSC team processing pipeline fall within this same trade space but are less sensitive to terrain roughness. DTMs produced with the Ames Stereo Pipeline also fall in this space at resolutions intermediate between NGATE and the team pipeline. Although DTM resolution and error each vary by a factor of 2, the product of resolution and error is much more consistent, varying by ≤20% across multiple image sets and matching algorithms. Refinement of the stereo DTM by photoclinometry can yield significant quantitative improvement in resolution and some improvement in error (improving their product by as much as a factor of 2), provided that albedo variations over distances smaller than the stereo DTM resolution are not too severe.

A deep learning network for estimation of seismic local slopes

The local slopes contain rich information of the reflection geometry, which can be used to facilitate many subsequent procedures such as seismic velocities picking, normal move out correction, time-domain imaging and structural interpretation. Generally the slope estimation is achieved by manually picking or scanning the seismic profile along various slopes. We present here a deep learning-based technique to automatically estimate the local slope map from the seismic data. In the presented technique, three convolution layers are used to extract structural features in a local window and three fully connected layers serve as a classifier to predict the slope of the central point of the local window based on the extracted features. The deep learning network is trained using only synthetic seismic data, it can however accurately estimate local slopes within real seismic data. We examine its feasibility using simulated and real-seismic data. The estimated local slope maps demonstrate the successful performance of the synthetically-trained network.

Robust Nonstationary Local Slope Estimation

The plane-wave destruction (PWD) method has been a widely used local slope estimation method in the seismic community. It is based on the discretization of the plane-wave partial differential equation (PDE) and the linearization of the PDE with respect to the local slope. Solving the linearized inverse problem for the slope perturbation is equivalent to solving a smoothness constrained optimization based on a shaping regularization method. The smoothness constraint in the shaping regularization is important in that it not only controls the stability and smoothness of the solution, i.e., slope perturbation, but also affects the accuracy and resolution of the solution. The traditional PWD algorithm is not easy to compromise between the smoothness and resolution of the estimated local slope because it uses a stationary triangle smoothing operator as the shaping operator. Here, we propose to improve the robustness of the PWD algorithm by introducing a nonstationary triangle smoothing operator into the shaping regularization framework in order to adaptively constrain the solution according to the local signal reliability. The smoothing is weak in areas with a higher probability of signals and is strong in areas with a higher likelihood of noise. The smoothing radius can be adaptively estimated based on an optimization model, which is solved by a line-search method. The proposed new slope estimation method is referred to as a nonstationary method compared with the traditional stationary one. The effectiveness and benefits of the new slope estimation method are validated via several synthetic and field data examples.

Estimating local slope in the time-frequency domain: Velocity-independent seismic imaging in the near surface

Seismic reflection processing for multicomponent data is very time consuming. To automatically streamline and shorten this process, a new approach for estimating the local event slope (local static shift) in the time-frequency domain is proposed and tested. The seismic event slope is determined by comparing the local phase content of Stockwell transformed signals. This calculation allows for noninterfering arrivals to be aligned by iteratively correcting trace by trace. Alternatively, the calculation can be used in a velocity-independent imaging framework with the possibility of exporting the determined time and velocities for each common midpoint gather, which leads to a more robust moveout correction. Synthetic models are used to test the robustness of the calculation and compare it directly to an existing method of local slope estimation. Compared to dynamic time warping, our method is more robust to noise but less robust to large time shifts, which limits our method to shorter geophone spacing. We apply the calculation to near-surface shear-wave data and compare it directly to semblance/normal-moveout processing. Examples demonstrate that the calculation yields an accurate local slope estimate and can produce sections of better or equal quality to sections processed using the conventional approach with much less user time input. It also serves as a first example of velocity-independent processing applied to near-surface reflection data.

Fractal-Based Local Range Slope Estimation from Single SAR Image with Applications to SAR Despeckling and Topographic Mapping

In this paper, we propose a range slope estimation procedure from single synthetic aperture radar (SAR) images with both methodological and applicative innovations. The retrieval algorithm is based on an analytical linearized direct model, which relates the SAR intensity data to the range local slopes and encompasses both a surface model and an electromagnetic scattering model. Scene topography is described via fractal geometry, whereas the Small Perturbation Method is adopted to represent the scattering behavior of the surface. The range slope map is then used to estimate the surface topography and the local incidence angle map. For topographic mapping applications, also referred to as shape from shading, a regularization procedure is derived to recover the azimuth local slope and reduce distortions. Then we present a new intriguing application of the inversion procedure in the field of SAR despeckling. Proposed techniques and high-level products are tested in a wide series of experiments, where the algorithms are applied to both simulated (canonical) and actual SAR images. It is proved that the proposed range slope retrieval technique can (1) provide an estimate of the surface shape, with overall better performance w.r.t. typical models used in this field and (2) be useful in advanced despeckling techniques.

An Events Rearrangement Strategy-Based Robust Principle Component Analysis

Random noise in seismic data can affect the performance of reservoir characterization and interpretation, which makes denoising become an essential procedure. This letter focuses on suppressing random noise in poststack seismic data while preserving the edges of desired signals. Due to the lateral continuity of seismic data, polynomial fitting (PF) method can be a good alternative in attenuating random noise. However, discontinuities exist widely in poststack seismic data, which might be damaged by the PF filter. By contrast, principle component analysis (PCA)-based filters have better performance in edge preserving, but there appear artifacts in the denoised results using the PCA-based filters. Thus, we propose an edge-preserving polynomial PCA filter which combines advantages of the PF and PCA methods by optimizing a PCA problem with a weighted polynomial constraint. The weight coefficient is determined adaptively according to the signal-to-noise ratio estimation and the energy proportion in the selected analysis window, which can help distinguish the horizontal continuous events and the edges effectively. To deal with the complicated slopes which make the local linear hypothesis invalid, we introduce a robust local slope estimation method and apply the slope estimation-based event tracing strategy to horizontally align the data set. Synthetic and field data examples show that the proposed method has a better performance in noise attenuation and edge preserving, compared with the edge-preserving PF method. In addition, the denoised results are free from artifacts.

Compression of local slant stacks by the estimation of multiple local slopes and the matching pursuit decomposition

Because local slant stacking increases the data dimension in beam migration, the volume of local slant stacks can be enormous and can obstruct efficient data processing. In addition, a proper beam compression algorithm can reduce the computation of ray tracing and beam mapping. Thus, compressing the local slant stacks with high fidelity can improve the efficiency of beam migration. A new approach is proposed to efficiently compress the local slant stacks. This approach combines the estimation of multiple local slopes based on the structure tensor to reduce the number of slopes, and the sparse representation for the slant stacked data via the matching pursuit decomposition to reduce the number of temporal samples. Furthermore, a new algorithm to estimate multiple local slopes based on the second-order structure tensor is proposed to handle the intersecting events efficiently. Several data examples indicated that the new compression algorithm required much less storage. Meanwhile, the new algorithm can restore the significant events and tolerate some random noise. The migration results determined that this compression algorithm does not obviously degrade the quality of the beam migration result, and it even makes the migration result more clear by suppressing the random noise smearing.

Enhancing seismic reflections using empirical mode decomposition in the flattened domain

Due to different reasons, the seismic reflections are not continuous even when no faults or no discontinuities exist. We propose a novel approach for enhancing the amplitude of seismic reflections and making the seismic reflections continuous. We use plane-wave flattening technique to provide horizontal events for the following empirical mode decomposition (EMD) based smoothing in the flattened domain. The inverse plane-wave flattening can be used to obtain original curved events. The plane-wave flattening process requires a precise local slope estimation, which is provided by the plane-wave destruction (PWD) algorithm. The EMD based smoothing filter is a non-parametric and adaptive filter, thus can be conveniently used. Both pre-stack and post-stack field data examples show tremendous improvement for the data quality, which makes the following interpretation easier and more reliable.

An improved coherence algorithm with robust local slope estimation

During recent years, dip scanning technique has been applied to traditional coherence algorithms to bring higher stability and robustness. In our research, we find that the slope information used for coherence estimation should trace layers correctly but contain few fluctuations and details about discontinuities. The second demand is useful for coherence algorithms to detect small discontinuities of structures. However, dip scanning technique cannot trace layers correctly if the slope variation of structures is large. Furthermore, estimated slopes may contain fluctuations and details which make small discontinuities invisible in coherence cube. In addition, dip scanning leads to huge computation cost when applied to seismic data with large slope variation. We introduce complex seismic trace analysis combined with L1 norm minimization to slope estimation to improve the coherence estimation. Slopes estimated with complex seismic trace analysis contain fewer fluctuations and details about discontinuities and can be used to trace layers more accurately compared with dip scanning. We introduce the L1 norm minimization to overcome the drawbacks of low stability and low robustness of Hilbert transform. By estimating slopes with supertrace data, we enhance the algorithm's ability to detect small scale discontinuities and furthermore, its robustness to noise. In addition, the computational efficiency is also improved by reducing the dip scan range. Applications on synthetic and real data sets show that our coherence can highlight faults and channels effectively and reveal more details than the original cross-correlation-based coherence algorithm with dip scanning with lower runtime overhead does.

Prestack time imaging algorithm with simultaneous velocity estimation in hard rock environments

Reflection seismic imaging faces several difficulties in hard rock environments. One of them is the estimation of the propagation velocity of seismic waves. Therefore, imaging algorithms that do not require prior construction of a velocity model seem promising for such environments. In this paper we illustrate an application of prestack time migration, which does not require an input velocity model, to hard rock conditions, and we demonstrate its effectiveness on synthetic data. This approach is based on an estimation of local event slopes (horizontal slownesses) in common-shot and common-receiver gathers and a subsequent calculation of the migration attributes (migration velocity, vertical traveltime and horizontal coordinates of the migrated reflection point). These attributes allow us to derive all the information needed to construct a time-migrated image. We also use the obtained migration velocities as an input velocity model for Kirchhoff prestack time migration (PSTM) and compare the results of the proposed approach with a conventional Kirchhoff migration using as an input the picked NMO velocity model. This application to a hard rock synthetic model illustrates the potential of the presented migration algorithm for imaging in hard rock seismic exploration. We believe that this approach can be used in hard rock seismic processing workflows as an automatic tool to obtain an input velocity model for the Kirchhoff PSTM.

On local slope estimation in partial linear models under Gaussian subordination

We address estimation of trend functions and time dependent slope in a partial linear model when the errors are unknown time-dependent functionals of latent Gaussian processes. Asymptotic results are derived under short-memory and long-memory correlations in the data.

Multi-baseline phase unwrapping algorithm for INSAR

A novel multi-baseline phase unwrapping algorithm based on the unscented particle filter for interferometric synthetic aperture radar (INSAR) technology application is proposed. The proposed method is not constrained by the nonlinearity of the problem and is independent of noise statistics, and performs noise eliminating and phase unwrapping at the same time by combining with an unscented particle filter with a path-following strategy and an omni-directional local phase slope estimator. Results obtained from multi-baseline synthetic data and single-baseline real data show the performance of the proposed method.