Prediction Error Filter Research Articles

Streaming prediction-error filters

Prediction-error filters (PEFs) are essential for seismic deconvolution and other geophysical estimation problems. We find that nonstationary multidimensional PEFs can be computed in a “streaming” manner, wherein the filter is updated incrementally by accepting one new data point at a time. The computational cost of estimating a streaming PEF is reduced to the cost of a single convolution. In other words, the cost of the PEF design while filtering is equivalent to the cost of applying the filter. Moreover, the nonlinear operation of finding and applying a streaming PEF is invertible at a similar cost, which enables a fast approach for missing data interpolation.

Multichannel deconvolution with a high-frequency structural regularization

The resolution of seismic data determines the ability to characterize stratigraphic features from observed seismic records. Sparse spike inversion, as an important processing method, can effectively improve the band-limited property of the seismic data. However, this approach ignores the spatial information along seismic traces, which causes the unreliability of the reconstructed high-resolution data. In this paper, we develop a high-frequency structure-constrained multichannel deconvolution (HFSC-MD) to alleviate this issue. This method allows the cost function to incorporate high-frequency spatial information in the form of a prediction-error filter (PEF) to regularize the components of the result beyond the original frequency. The PEF, also called a high-frequency reflection structural characterization operator, is estimated from the mapping relationship of low- and high-frequency components. We adopt the alternating direction method of multipliers to solve the cost function in HFSC-MD. Synthetic and field data demonstrate that our method recovers more reliable high-resolution data and enriches reflective structures.

Active control method for fluid borne noise in aerospace fluid systems of variable operation statuses

In the fluid system of aerospace vehicles, the fluid-borne noise (FBN) may induce vibration and damage to components. The FBN may limit the performance of the entire onboard system. To achieve a stable active elimination for the FBN is not easy, for many variable operation statuses exist in practical systems, such as varying-frequency and varying-load status. In this work, an improved feed-forward active control method based on narrowband adaptive notch filters, with an online identification module, is proposed. The algorithm includes three adaptive filters: an adaptive notch filter for producing cancelling signal by using filtered-X least mean square (FXLMS) algorithm, one for online identification to model the real-time secondary path, and an error prediction filter for extracting the precise error for online identification. In the aircraft hydraulic system of high pressure, a so-called relief scheme, also known as the bypass structure, is used for the FNB, controlling an active control valve in the branched line to overflow the main fluid line. The system is modelled with software AMESim for the fluid part and Matlab for the control algorithm. Simulation results show that the control method can performance well in the situations of varying frequency and load. The test rig of a hydraulic system is established, which could change pump speed and fluid pressure. The test results show that more than 90% reduction for a certain frequency can be achieved for stable pressure statuses. And more than 50% reduction is achieved for varying frequency with step changes. As for the load-varying status, the reduction of 82.7% and 65.8% are obtained before and after pressure change. The method proposed is validated to be effective with both numerical and experimental tests.

A distortionless convolution beamformer design method based on the weighted minimum mean square error for joint dereverberation and denoising

This paper designs a weighted minimum mean square error (WMMSE) based distortionless convolution beamformer (DCBF) for joint dereverberation and denoising. By effectively using WMMSE with the constraint of distortionless, a DCBF is deduced, where the outputs of the weighted prediction error (WPE) filter and the WPE-based minimum variance distortionless response (MVDR) beamformer are combined to initialize target signal for balancing signal distortion, residual reverberation and residual noise. In addition, two optimization factors are introduced to reduce the reverberation and noise when the initialized target signal is used for the solution of beamformer. As a result, the designed beamformer is presented as a linear combination of the WMMSE-based convolution beamformer (CBF) and weighted power minimization distortionless response (WPD) filter. The experimental results demonstrate the superior performance of the designed beamformer for joint dereverberation and denoising compared to the reference methods.

High-resolution sequence stratigraphy research based on continuous wavelet transform and INPEFA - A case study in the Kalashayi formation, Tarim Basin, China

The Carboniferous Kalashayi Formation, situated in the Lungu-Sangtamu area within the Tarim Basin, exhibits protracted sedimentation periods, intricate sand-mud depositional sequences, and scant paleontological and core datasets. Precision in delineating high-resolution sequences using conventional core and logging data poses a considerable challenge. To enhance the stratigraphic accuracy of the Kalashayi Formation in the Tarim Basin and facilitate quantitative analysis, this study employs continuous wavelet transform on the gamma ray (GR) curves obtained from core wells within the study area. Subsequently, various sequence boundaries are discerned by integrating the resulting wavelet coefficient curves with time-frequency energy maps. Discrimination and subdivision of base-level cycle structures of varying orders are achieved through temporal trend analyses of integrated prediction error filter analysis (INPEFA) curves. Integration of drilling, logging, lithofacies, and core data enables the identification and subdivision of high-resolution sequence stratigraphy using wavelet analysis and INPEFA techniques. Ultimately, the Carboniferous Kalashayi Formation in the Lungu-Sangtamu region is classified into 2 long-term, 5 medium-term, and 14 short-term base-level cycles, establishing a meticulously delineated isochronous stratigraphic framework. This framework serves as a fundamental basis for subsequent discussions on reservoir prediction within the study area.

A tensor decomposition based multichannel linear prediction approach to speech dereverberation

Dereverberation technology is needed in a wide range of speech applications as reverberation often greatly degrades the quality and intelligibility of the speech signal of interest captured by microphones. The commonly used weighted-prediction-error method generally requires long prediction-error filters to remove the reverberation components, which makes it computationally expensive. To deal with this issue, this paper proposes a computationally efficient dereverberation algorithm based on tensor decomposition in which the long prediction-error filter is decomposed into a group of short sub-filters through multiple Kronecker products. Consequently, the high dimensional cross-correlation matrix that needs to be inverted in the dereverberation algorithm is then converted into a set of low dimensional matrices, which leads to significant reduction in the computational complexity. Simulation results demonstrate the properties of the proposed algorithm.

Algorithms for Sparse Multichannel Blind Deconvolution

In this paper we present two algorithms for sparse multichannel blind deconvolution. The first algorithm is based on a cascade of a forward and a backward prediction error filter (C-PEF). The second consists in an alternating minimization algorithm for estimating both the reflectivity series and the seismic wavelet (AM-SMBD). We also compare the algorithms with other state-of-the-art sparse blind deconvolution algorithms. Simulation results with synthetic data for different SNR levels showed that the AM-SMBD was outperformed (in terms of the Pearson Correlation Coefficient and the Gini Correlation Coefficient) other estimation methods, like the reduced SMBD, the TS-sparseTLS (Toeplitz-structured sparse total least square) and the SMBD-SPG (Sparse multichannel blind deconvolution via spectral projected-gradient). For the same data, the C-PEF was able to provide better results (in terms of the Gini Correlation Coefficient, visual inspection and frequency gain) when compared with the F-SMBD (Fast sparse multichannel blind deconvolution). In a simulation considering reflectivities with different levels of sparsity, the C-PEF seems to be more robust for less sparse data when compared with AM-SMBD and SMBD-SPG (up to a certain degree of sparsity). Finally, simulations considering a real land acquisition show that both algorithms were able to greatly improve the resolution of the seismic data.

Integration of novel uncertainty models construction of college art and art design teaching system based on big data analysis

In this paper to explore the improvement direction of college art and art design teaching system, this paper combines big data technology and intelligent data processing model to construct college art and art design system, and proposes a new blind equalization algorithm of online echo state network based on prediction principle. Moreover, this paper adopts the complex ESN as the prediction error filter, and adds the phase rotation factor after the amplitude adjustment, which solves the limitation that the existing blind equalization algorithm based on the prediction principle cannot handle complex signals. In addition, this paper uses the Kalman filter algorithm to update the parameters of the network online to achieve the purpose of tracking the channel changes in real time. The experimental research results show that the college art and art design teaching system proposed in this paper has a good teaching effect and can effectively improve the effect of art and art design education in colleges and universities.

Prediction of petrophysical classes and reservoir beds through microfacies and pore types characterization, Tahe Ordovician naturally fractured vuggy carbonates

Carbonates commonly present a stacking-pattern architecture determined by the depositional processes despite the diagenetic modifications. This study integrates spectral cyclostratigraphy and petrophysical analysis to locate prolific reservoir beds to optimize reservoir development. This involves subdividing the reservoir succession into different hierarchical units to understand the variation of microfacies and pore-structure types. The variation trends and discontinuity surfaces (chronostratigraphic and lithostratigraphic boundaries) are determined by conjointly interpreting the prediction error filter analysis (PEFA) and integrated PEFA (INPEFA) and synthetic seismic trace and seismogram. We define the rock texture types (microfacies) through self-consistent approximation based on the INPEFA log while considering the variation of depositional sequence and grain size, core/thin-section petrography, and borehole electrical image interpretation. We establish lithology-depth and petrophysical classes-depth profiles using largely well-log data and present new petrophysical classes for naturally fractured vuggy reservoirs. We predict porosity and permeability from the distribution of high-frequency cycles and lithofacies while considering the implications of diagenesis. The results indicate that the depositional processes and diagenesis largely control the quality of the reservoir beds in response to the relative sea-level variations. The reservoir beds comprise the reservoir units with connected-vugs- and fractures-dominated pore systems and the horizons with microporosity-dominated pore systems that are candidates for secondary recovery processes. The study proposes a total porosity variation model that describes an external cardiac compression-like depth profile, revealing alternation between low/tight and high-porosity horizons, with the highly porous intervals associated with the solution-enlarged porosity zones. Permeability varies with the pore type mixing and rock texture type rather than the total porosity. The method well applies to distinguish flow conduits from baffles and barriers in complex carbonate reservoirs.

Forensics Analysis of Resampling via ConvNeXt Block

Images play an important role in transmitting visual information in our life. It could lead to severe consequences if images are manipulated or tampered maliciously. Digital forensics is an important research area to secure multimedia information. Many forensics technologies are applied to protect our community from the abuse of digital information. In many cases, after tampering, attackers could apply operations such as resampling, JPEG compression, blurring, etc. to cover the traces of tampering. Therefore, it is necessary to detect these manipulations in image forensics before exposing forgeries. In this paper, we propose to employ the prediction error filters, ConvNeXt blocks and convolution modules to classify images with different compression quality factors and resampling rates. By tracing the inconsistencies of resampling rates and compression quality factors, it could provide supplementary information for forensics researchers to expose possible forgeries. The proposed method could achieve great classification performance regardless of the interpolation algorithms. Also, it is highly robust against JPEG compression. In addition, the proposed method can be applied for estimating quality factors of JPEG compression.

Sequence Stratigraphy, Depositional Environment and Associated Lithofacies of Lacustrine Shale: A Case From the Upper Fourth Member of Shahejie Formation, Dongying Depression, Bohai Bay Basin

The lithofacies assemblages in the lacustrine shale of the Shahejie Formation in the Dongying Depression have different compositions due to the complex depositional climate, lake water conditions, and sediment supply. Hence, the lithofacies types and depositional conditions were analyzed based on parasequence division and lithofacies classification. According to gamma ray integrated prediction error filter analysis (INPEFA), handheld x-ray fluorescence (XRF) measurements, mineral composition, total organic carbon (TOC) content, and sedimentary structures, six parasequences (P1–P6 from bottom to top) were identified, and seven lithofacies are recognized: (1) organic-poor thin-bedded calcareous mudstone, (2) organic-poor laminated calcareous mudstone, (3) intermediate-organic laminated calcareous mudstone, (4) organic-rich laminated calcareous mudstone, (5) organic-rich calcareous shale, (6) organic-rich laminated carbonate-bearing mudstone, and (7) laminated silty mudstone. When the climate became warm and wet, the water level of the lake increased with a decrease in salinity, and the corresponding lithofacies assemblages in the P1 to P2 parasequences gradually changed from organic-poor thin-bedded and laminated calcareous mudstone to intermediate-organic and organic-rich calcareous shale. When the climate became humid and warmer, resulting in an increasing water level and terrigenous input, the lithofacies assemblages in the P3–P4 parasequences changed from organic-rich calcareous shale and laminated carbonate-bearing mudstone to laminated silty mudstone and calcareous mudstone. At the end, the climate became slightly drier and cooler with a decrease in the water level, and the P5 and P6 parasequences were dominated by intermediate-organic and organic-poor laminated calcareous mudstones. Through elemental analysis and lithofacies observation, we found that the redox conditions of the water may have changed over time. Among the parasequences, the water column of the P2 parasequence was mainly stratified by salinity, that of the P4 parasequence was mainly stratified by temperature, and that of the P3 parasequence was in transition. The synergistic effect of various factors played a key role in the enrichment of organic matter. This study highlights the depositional processes of the lithofacies assemblages in a lacustrine shale and contributes to the understanding of shale oil accumulation in lacustrine shale.

Nonstationary pattern-based signal–noise separation using adaptive prediction-error filter

Abstract Complex field conditions always create different interferences during seismic data acquisition, and there exist several types of noise in the recorded data, which affect the subsequent data processing and interpretation. To separate an effective signal from the noisy data, we adopted a pattern-based method with a two-step strategy, which involves two adaptive prediction-error filters (APEFs) corresponding to a nonstationary data pattern and noise pattern. By introducing shaping regularization, we first constructed a least-squares problem to estimate the filter coefficients of the APEF. Then, we solved another constrained least-square problem corresponding to the pattern-based signal–noise separation, and different pattern operators are adopted to characterize random noise and ground-roll noise. In comparison with traditional denoising methods, such as FXDECON, curvelet transform and local time–frequency (LTF) decomposition, we examined the ability of the proposed method by removing seismic random noise and ground-roll noise in several examples. Synthetic models and field data demonstrate the validity of the strategy for separating nonstationary signal and noise with different patterns.

Paleoenvironment and chemostratigraphy heterogenity of the Cretaceous organic-rich shales

The Cretaceous Qingshankou Formation in the Songliao Basin is rich in shale oil resources, which has become one of the most important exploration targets of lacustrine shale oil in China. Based on X-ray fluorescence element analysis, X-ray diffraction analysis, total organic carbon, rock pyrolysis, scanning electron microscope and nitrogen adsorption, the Paleoenvironment was reconstructed by comprehensive utilization of integrated prediction error filter analysis of chemical stratigraphy, and its relationship with organic geochemistry, mineralogy and pore structure was discussed. The results indicated that the Qingshankou Formation was deposited in the environment with fresh water-brackish water, semi-deep/deep water and strong reduction. The evolution of Paleoenvironment during the deposition of Qingshankou Formation changed from bottom to top, with increasing water depth, decreasing salinity and oxygen content. Paleosalinity was positively correlated with total organic carbon, residual hydrocarbon and carbonate mineral content. From bottom to top, the contents of carbonate and chlorite decreased, while the contents of plagioclase and clay minerals increased slightly. The pores were dominated by intra-illite pores, intra-I/S mixed-layer pores and intra-pyrite pores. Some intra-plagioclase pores and calcite dissolution pores were developed, and the organic matter pores are slightly few. Nitrogen adsorption data showed that the dominate pore size was 40-53 nm. This study clarifies the Paleoenvironmental evolution of the Qingshankou Formation, and may shed lights on lacustrine shale oil accumulation and sweet-spotting. Cited as : Guan, M., Wu, S., Hou, L., Jiang, X., Ba, D., Hua, G. Paleoenvironment and chemostratigraphy heterogenity of the Cretaceous organic-rich shales. Advances in Geo-Energy Research, 2021, 5(4): 444-455, doi: 10.46690/ager.2021.04.09

Seismic data interpolation without iteration using a t-x-y streaming prediction filter with varying smoothness

Although there is an increase in the amount of seismic data acquired with wide-azimuth geometry, it is difficult to achieve regular data distributions in spatial directions owing to limitations imposed by the surface environment and economic factors. To address this issue, interpolation is an economical solution. The current state-of-the-art methods for seismic data interpolation are iterative methods. However, iterative methods tend to incur high computational costs, which restricts their application in cases of large, high-dimensional data sets. Hence, we have developed a two-step noniterative method to interpolate nonstationary seismic data based on streaming prediction filters (SPFs) with varying smoothness in the time-space domain and we extend these filters to two spatial dimensions. Streaming computation, which is the kernel of the method, directly calculates the coefficients of nonstationary SPF in the overdetermined equation with local smoothness constraints. In addition to the traditional streaming prediction-error filter, we adopt a similarity matrix to improve the constraint condition in which the smoothness characteristics of the adjacent filter coefficient change with the varying data. We also design noncausal-in-space filters for interpolation by using several neighboring traces around the target traces to predict the signal; this is performed to obtain more accurate interpolated results than those from the causal-in-space version. Compared to the Fourier projection onto a convex sets interpolation method, our method has the advantages of fast computational speed and nonstationary event reconstruction. Application of our method on synthetic and nonstationary field data finds that it can successfully interpolate high-dimensional data with low computational cost and reasonable accuracy even in the presence of aliased and conflicting events.

Multichannel adaptive deconvolution based on streaming prediction-error filter

Abstract Deconvolution mainly improves the resolution of seismic data by compressing seismic wavelets, which is of great significance in high-resolution processing of seismic data. Prediction-error filtering/least-square inverse filtering is widely used in seismic deconvolution and usually assumes that seismic data is stationary. Affected by factors such as earth filtering, actual seismic wavelets are time- and space-varying. Adaptive prediction-error filters are designed to effectively characterise the nonstationarity of seismic data by using iterative methods, however, it leads to problems such as slow calculation speed and high memory cost when dealing with large-scale data. We have proposed an adaptive deconvolution method based on a streaming prediction-error filter. Instead of using slow iterations, mathematical underdetermined problems with the new local smoothness constraints are analytically solved to predict time-varying seismic wavelets. To avoid the discontinuity of deconvolution results along the space axis, both time and space constraints are used to implement multichannel adaptive deconvolution. Meanwhile, we define the parameter of the time-varying prediction step that keeps the relative amplitude relationship among different reflections. The new deconvolution improves the resolution along the time direction while reducing the computational costs by a streaming computation, which is suitable for handling nonstationary large-scale data. Synthetic model and field data tests show that the proposed method can effectively improve the resolution of nonstationary seismic data, while maintaining the lateral continuity of seismic events. Furthermore, the relative amplitude relationship of different reflections is reasonably preserved.

Denoising for full-waveform inversion with expanded prediction-error filters

Low-frequency data of less than 5 Hz are essential to the convergence of full-waveform inversion (FWI) toward a useful solution. They help to build the velocity model low wavenumbers and reduce the risk of cycle skipping. In marine environments, low-frequency data are characterized by a low signal-to-noise ratio (S/N) and can lead to erroneous models when inverted, especially if the noise contains coherent components. Often, field data are high-pass filtered before any processing step, sacrificing weak but essential signal for FWI. We have denoised the low-frequency data using prediction-error filters that we estimate from a high-frequency component with a high S/N. The constructed filter captures the multidimensional spectrum of the high-frequency signal. We expand the filter’s axes in the time-space domain to compress its spectrum toward the low frequencies and wavenumbers. The expanded filter becomes a predictor of the target low-frequency signal, and we incorporate it in a minimization scheme to attenuate noise. To account for data nonstationarity while retaining the simplicity of stationary filters, we divide the data into nonoverlapping patches and linearly interpolate stationary filters at each data sample. We apply our method to synthetic stationary and nonstationary data, and we find that it improves the FWI results initialized at 2.5 Hz using the Marmousi model. We also demonstrate that the denoising attenuates nonstationary shear energy recorded by the vertical component of ocean-bottom nodes.

5D dealiased seismic data interpolation using nonstationary prediction-error filter

The prediction-error filter (PEF) assumes that seismic data can be destructed to zero by applying a convolutional operation between the target data and the prediction filter in either the time-space or frequency-space domain. We have extended the commonly known PEF in 2D or 3D problems to its 5D version. To handle the nonstationary property of the seismic data, we formulate the PEF in a nonstationary way, which is called the nonstationary prediction-error filter (NPEF). In NPEF, the coefficients of a fixed-size PEF vary across the whole seismic data. In NPEF, we aim at solving a highly ill-posed inverse problem via the computationally efficient iterative shaping regularization. NPEF can be used to denoise multidimensional seismic data and, more importantly, to restore the highly incomplete aliased 5D seismic data. We compare our NPEF method with the state-of-the-art rank-reduction method for the 5D seismic data interpolation in cases of irregularly and regularly missing traces via several synthetic and real seismic data. The results show that although our NPEF method is less effective than the rank-reduction method in interpolating irregularly missing traces especially in the case of a low signal-to-noise ratio, it outperforms the rank-reduction method in interpolating an aliased 5D data set with regularly missing traces.

Dealiased seismic data interpolation using a deep-learning-based prediction-error filter

Deep-learning (DL) technology has emerged as a new approach for seismic data interpolation. DL-based methods can automatically learn the mapping between regularly subsampled and complete data from a large training data set. Subsequently, the trained network can be used to directly interpolate new data. Therefore, compared with traditional methods, DL-based methods reduce the manual workload and render the interpolation process efficient and automatic by avoiding the selection of hyperparameters. However, two limitations of DL-based approaches exist. First, the generalization performance of the neural network is inadequate when processing new data with a different structure compared to the training data. Second, the interpretation of the trained networks is very difficult. To overcome these limitations, we have combined the deep neural network and classic prediction-error filter (PEF) methods, proposing a novel seismic data dealiased interpolation framework called prediction-error filters network (PEFNet). The PEFNet designs convolutional neural networks to learn the relationship between the subsampled data and the PEFs. Thus, the filters estimated by the trained network are used for the recovery of missing traces. The learning of filters enables the network to better extract the local dip of seismic data and has a good generalization ability. In addition, PEFNet has the same interpretability as traditional PEF-based methods. The applicability and the effectiveness of our method are demonstrated here by synthetic and field data examples.

Intelligent identification of sequence stratigraphy constrained by multipopulation genetic algorithm and dynamic time warping technique: A case study of Lower Cretaceous Qingshuihe Formation in hinterland of Junggar Basin (NW China)

AbstractThe division and correlation of sequence stratigraphy are one of the essential steps in petroleum exploration. To comprehensively utilize the sequence stratigraphic information implied in multiple wireline well logs and improve the accuracy and reliability of sequence stratigraphic division, based on the edge detection, optimal partition, and natural evolution theories, and realizes the information fusion (formed one ‘chromosome’) of multiple wireline well logs (named as ‘genes’) which can reflect the feature information of sequence stratigraphy by using multipopulation genetic algorithm (MPGA) based on Fisher algorithm, in this study, geophysical data (wireline well logs and 3D seismic data) were used for spectral trend attribute analyses (wavelet transform, maximum entropy spectrum, time‐frequency analyses, and integrated prediction error filter). Since the seismic reflection characteristics of K1q were difficult to identify and considering subjective factors in conventional methods, a MPGA was also used to intelligently identify the sequence stratigraphy framework of the Lower Cretaceous Qingshuihe (K1q) Formation. Since the MPGA is an optimization algorithm, the code was run 10 times for accuracy with changes in the dividing layers. The optimal parameters in the process of data fusion (gene combination process) are selected. The sequence boundaries (SBs) searched by the MPGA matched those of the spectral trend attribute analyses, confirming that the MPGA was an effective, fast, and user‐friendly tool for defining the main SBs. MPGA can provide an optimal layering scheme (‘elite population’) through simultaneous searching at multiple depth ranges (‘multipopulation’), recording the optimal SBs (‘elite individuals’) of each population and making the results more convinced. Although simulation solutions may not be unique, the MPGA still provides us a chance to gain some insights into the details of the stratigraphy. Based on this intelligent identification, the maximum flooding surface, two first flooding surfaces (FFS), and three SBs were defined in K1q. The cyclic patterns (prograding, aggrading, and retrograding) were identified. They showed the relationship between accommodation space (related to tectonic subsidence of the Chemo paleo‐uplift) and sediment supply. In addition, dynamic time warping (DWT) technique was used for sequence discrimination and correlation on the integrated prediction error filter analyses curve. A massive pebbly sandstone set, named ‘bottom sandstone,’ was identified in the lowstand system tract of SQ1 (K1q1), constrained by SB1 and FFS1, which was a favourable hydrocarbon exploration target.

Cascade of Linear Predictors for Deconvolution of Non-Stationary Channels in Sparse and Antisparse Scenarios

This work deals with adaptive predictive deconvolution of non-stationary channels. In particular, we investigate the use of a cascade of linear predictors in the recovering of sparse and antisparse original signals. To do so, we first discuss the behavior of the Lp Prediction Error Filter (PEF), with p different of 2, showing that it has a superior ability to deal with non-minimum phase channels in comparison with the classical L2 PEF, although it still presents intrinsic limitations due to its direct linear structure. The cascade structure emerges as a possible solution to circumvent this issue. We apply the proposed cascade structure in the deconvolution of non-stationary channels, with minimum-, maximum- , mixed- and variable-phase response, and also noise scenarios. From the simulation results we observed that, besides the duality relation between the Lp norms, they present different algorithmic behavior: the L1 norm attains a fast convergence, enhancing the cascade tracking capacity, but is more sensible to noise. The L4 norm, on the other hand, is more robust to noise, but presents slower convergence and tracking capability.