Optimal Regularization Parameter Research Articles

Bootstrap selection of ridge regularization parameter: a comparative study via a simulation study

In multiple linear regressions, it is known that least-squares estimates of the parameters are likely to be too large in absolute value and possibly of wrong sign, if explanatory variables are correlated. To reduce the undesirable effects of collinearity, the ridge estimator has been proposed as an alternative method to the least squares estimator. The biggest debate with the ridge estimator is the selection of the regularization parameter. Several methods for the selection of the regularization parameter have been discussed. Although most of these methods are based on minimizing the mean square error of the ridge estimator, bootstrap resampling method is also proposed to provide an optimal regularization parameter which gives an estimate of mean square error of prediction. The purpose of this paper is to provide a comprehensive study of regularization parameter selection methods which consider bootstrap approach as well as estimation and prediction approaches. The paper concludes with application of these procedures to simulation studies and real data.

A two-level method for image denoising and image deblurring models using mean curvature regularization

The mean curvature (MC)-based image denoising and image deblurring models are used to enhance the quality of the denoised images and deblurred images respectively. These models are very efficient in removing staircase effect, preserving edges and other nice properties. However, high order derivatives appear in the Euler–Lagrange equations of the MC-based models which create problems in developing an efficient numerical algorithm. To overcome this difficulty, we present a robust and efficient Two-Level method for MC-based image denoising and image deblurring models. The Two-Level method consists of solving one small problem and one large problem. The small problem is a nonlinear system, having high order derivative, on Level I (image having small number of pixels). The large problem is one less expensive system, having low order derivative, on Level II (image having large number of pixels). The derivation of the optimal regularization parameter of Level II is studied and formula is presented. Numerical experiments on digital images are presented to exhibit the performance of the Two-Level method.

Compound fault diagnosis of rolling bearing using PWK-sparse denoising and periodicity filtering

Since the noise in signals influences the diagnosis and separation of bearing compound faults, this study proposes a new method combining periodicity weighted Kurtosis (PWK)-sparse denoising (SD) with periodicity filtering (PF) to extract repetitive impulses of compound faults. The proposed method involves fault identification and fault separation. Firstly, since Kurtosis only measures transient feature and neglects periodic feature when evaluating repetitive impulses, to overcome this drawback, a new index, PWK, is proposed to adaptively select optimal regularization parameter in SD. Then, with PWK measuring repetitive impulses extracted by SD, PWK-SD is used to identify fault types. Subsequently, based on the fault types, the proposed PF separates the compound bearing faults. During separation processing, PF uses periodic feature of sparse coefficients selecting the relevant impulse atoms to separate mixed repetitive impulses. Simulation and experimental results indicate the effectiveness of the proposed method in diagnosing and separating compound bearing faults.

Reweighted generalized minimax-concave sparse regularization for duct acoustic mode detection with adaptive threshold

Acoustic mode detection is attached great significance for providing guidance to noise reduction design of commercial aero-engine with high-bypass ratio. Compressive sampling method has been creatively employed in this field due to its notable performance on reducing the number of microphones in acoustic mode measurements. However, the classical ℓ1-norm regularized compressive sampling model tends to underestimate the dominant mode amplitudes of interest. Moreover, the traditional regularization parameter selection strategy with fixed threshold brings out inefficient and cumbersome work. In this paper, we propose a nonconvex penalized compressive sampling model with adaptive threshold, to seek the sparse and accurate solution of acoustic mode detection from limited measurements, and provide a sufficiently efficient way to adaptively seek the optimal regularization parameter. Firstly, the reweighted generalized minimax-concave (ReGMC) regularization is employed to improve the accuracy of acoustic pressure reconstruction, which feasibly enhances sparsity with maintaining the convexity of the cost function. Secondly, the k-sparsity strategy is introduced to set regularization parameters adaptively. Finally, the applicability of the proposed approach is verified on a multi-stage aero-engine fan test rig. Experimental results demonstrate that the nonconvex ReGMC regularized method outperforms the classical ℓ1-norm, producing more accurate results in mode detection with fewer measurements and being more robust towards background noise.

Ill-Posedness Determination of Moving Force Identification and Parameters Selection for Regularization Methods

Moving force identification (MFI) from dynamic responses of bridges is a typical inverse problem with ill-posedness. Under the efforts of researchers, some regularization methods have been presented to solve the ill-posed problem, but there still lacks an effective index to reveal the ill-posedness of the vehicle–bridge dynamic system such that it can be utilized as a guidance for the regularization parameter selection. In this paper, an ill-posedness indicator (IPI) defined as the ratio of the Fourier coefficient to the singular value is adopted to reveal the ill-posedness in the MFI problem. Simulation results show that the larger the IPI value is, the more obvious the ill-posedness of the vehicle–bridge system equation, namely, the intrinsic factor of ill-posedness in MFI is attributed to very large IPI value. The maximum IPI value increases with the increasing noise level, which leads directly to the ill-posedness of the vehicle–bridge system equation. In addition, a relative percentage error (RPE) is used to select the optimal regularization parameters, while evaluating the ill-posedness existing in the MFI. Using the proposed IPI value, the influence of ill-posedness on identified results is evaluated in this study, which can assist qualitatively and quantitatively in selecting optimal regularization parameters and proper regularization methods.

IMAGE REGISTRATION WITH OPTIMAL REGULARIZATION PARAMETER SELECTION BY LEARNED AUTO ENCODER FEATURES.

In this paper, we propose a method that optimizes a regularization parameter for the regularized Free Form Deformation (FFD) non-rigid image registration. The developed process utilizes autoencoder generated image representations to assess image data generalization quality by the regularization parameter. Both pixel intensity and learned features are used to improve the overall accuracy and regularity of the resulting inverse problem solution. We implement the new selection criterion with its use in the non-rigid image FFD registration based on multi-level Bspline with L2-regularization, and validate the method with synthetic and real histopathology image datasets. Both qualitative and quantitative results suggest the efficacy of our developed method for fine-tuning histopathology microscope images.

Stone weir scour modelling in curved canals using a weighted regularized extreme learning machine*

AbstractIn this paper, a novel weighted regularized extreme learning machine (WRELM) was used for the first time to simulate the scour depth around J‐, I‐ and W‐shaped stone weirs in curved canals. In the first step, dimensionless parameters affecting the scour depth around stone weirs were detected. Then, six WRELM models were developed using these parameters. It is worth mentioning that three different experimental models were utilized for verifying the WRELM models. Moreover, 70% of the experimental data were used to train the models and the remaining 30% to test them. After that, the number of neurons existing within the hidden layer was chosen to be 13. Then the best activation function of the WRELM model was selected. The optimal regularization parameter was also found for this activation function. Conducting a comprehensive sensitivity analysis in the next step detected the best WRELM model, as well as the most influencing input variables. The results of the superior model were compared with the regularized extreme learning machine (RELM) and extreme learning machine (ELM) models to prove the noticeable superiority of the WRELM. Furthermore, a formula was proposed to compute the scour gap around stone weirs with different shapes situated in curved canals. Finally, the sensitivity of the input variables on scour values was examined.

Assimilation of Doppler radar radial wind data in the GRAPES mesoscale model with observation error covariances tuning

AbstractThe accurate estimation of observation error statistics for high‐density remote‐sensing data is a fundamental step toward optimally utilizing such data, including radar data, for operational numerical weather prediction. Based on the high‐resolution (3 × 3 km) GRAPES model together with the three‐dimensional variational assimilation system (GRAPES_3DVAR) developed by the China Meteorological Administration, a set of assimilation processes for radar radial wind data is designed. First, a scheme is proposed to guarantee the quality of radar radial wind data by correcting the velocity ambiguity for the data. The observation error covariances of the radar data in the GRAPES_3DVAR assimilation system are re‐estimated by using a diagnostic method based on analysis residuals, showing that the re‐estimated radial wind observation error covariances increase with altitude. An optimal regularization parameter is selected by the L‐curve method to constrain the radar observations that were assimilated using inaccurate observation error covariances. Furthermore, to verify the assimilation effects of the refined radial velocity observation error covariances and the constrained observations by the regularization parameter on the analysis and forecasting, a numerical study is carried out on typhoonYunque, which occurred on 3 August 2018. The results show that the modified Doppler radar wind data assimilation using the re‐estimated radial velocity observation error covariances enhances the wind analysis field and the water vapour transport, and accordingly obtains better short‐term precipitation estimates. The radar wind data assimilation results of using the constrained observations also indicate an enhanced wind analysis field and water vapour transport, which leads to better short‐term precipitation estimates.

Two-dimensional magnetotelluric data inversion using Lanczos bidiagonalization method with active constraint balancing

The magnetotelluric (MT) technique is an electromagnetic geophysical method, which is widely used as a complementary to seismic surveys for exploration of hydrocarbon reservoirs. In the inversion process, the method of matrix inverse calculation has a considerable effect on the speed of the inversion and the quality of obtained models. Lanczos Bidiagonalization (LB) method has been reported to be a fast and efficient approach for solving the inversion problems. In this study, we employ LB method for inverting large-scale 2D MT data. In LB algorithm, the full set of equations is replaced by a dimensionally reduced system of equations. As a result, the speed of the solution procedure is increased, while the original problem is solved with a high accuracy. In addition, we employ active constraint balancing approach for determining the optimum regularization parameter. The advantage of the method is that for highly resolvable parameters, a small value of the Lagrangian multiplier is assigned, and vice versa. The results of the synthetic data inversion show that both methods require equal computer memory but LB method is faster and more reliable than conjugate gradient method. The proposed approach is also applied to inverse real MT data collected from the Kashan area. The Kashan area is the most interesting area for oil and gas exploration of the Central Iran Basin. The inversion results obtained by LB are in a good agreement with the geological structure of the study area and the drilling data.

An adaptive boundary algorithm for the reconstruction of boundary and initial data using the method of fundamental solutions for the inverse Cauchy–Stefan problem

In this paper, a recent algorithm, based around the method of fundamental solutions (MFS), for reconstructing boundary data in inverse Stefan problems is extended and applied to inverse Cauchy–Stefan problems, wherein initial data must also be reconstructed. A key feature of the algorithm is that it is adaptive and iterates to find the optimal locations of the source points that are required by the method. Tikhonov regularization is used to take care of the ill-conditioned matrix that the MFS generates, with the algorithm being able to determine the optimal regularization parameter automatically. The effects of accuracy and random noise on the optimal location and number of source points are also evaluated. In addition, we consider a nonlinear variant of the inverse problem where one has to identify the moving boundary along with the missing initial data. Numerical experiments, carried out on five different benchmark examples, show promising results.

Machine-learning-based prediction of regularization parameters for seismic inverse problems

Regularization parameter selection (RPS) is one of the most important tasks in solving inverse problems. The most common approaches seek the optimal regularization parameter (ORP) from a sequence of candidate values. However, these methods are often time-consuming because they need to conduct the estimation process on all candidate values, and they are always restricted to solve certain problem types. In this paper, we propose a novel machine learning-based prediction framework (MLBP) for the RPS problem. The MLBP first generates a large number of synthetic data by varying the inputs with different noise conditions. Then, MLBP extracts some pre-defined features to represent the input data and computes the ORP of each synthetic example by using true models. The pairs of ORP and extracted features construct a training set, which is used to train a regression model to describe the relationship between the ORP and input data. Therefore, for newly practical inverse problems, MLBP can predict their ORPs directly with the pre-trained regression model, avoiding wasting computational resources on improper regularization parameters. The numerical results also show that MLBP requires significantly less computing time and provides more accurate solutions for different tasks than traditional methods. Especially, even though the MLBP trains the regression model on synthetic data, it can also achieve satisfying performance when directly applied to field data.

A machine learning approach to optimal Tikhonov regularization I: Affine manifolds

Despite a variety of available techniques, such as discrepancy principle, generalized cross validation, and balancing principle, the issue of the proper regularization parameter choice for inverse problems still remains one of the relevant challenges in the field. The main difficulty lies in constructing an efficient rule, allowing to compute the parameter from given noisy data without relying either on any a priori knowledge of the solution, noise level or on the manual input. In this paper, we propose a novel method based on a statistical learning theory framework to approximate the high-dimensional function, which maps noisy data to the optimal Tikhonov regularization parameter. After an offline phase where we observe samples of the noisy data-to-optimal parameter mapping, an estimate of the optimal regularization parameter is computed directly from noisy data. Our assumptions are that ground truth solutions of the inverse problem are statistically distributed in a concentrated manner on (lower-dimensional) linear subspaces and the noise is sub-gaussian. We show that for our method to be efficient, the number of previously observed samples of the noisy data-to-optimal parameter mapping needs to scale at most linearly with the dimension of the solution subspace. We provide explicit error bounds on the approximation accuracy from noisy data of unobserved optimal regularization parameters and ground truth solutions. Even though the results are more of theoretical nature, we present a recipe for the practical implementation of the approach. We conclude with presenting numerical experiments verifying our theoretical results and illustrating the superiority of our method with respect to several state-of-the-art approaches in terms of accuracy or computational time for solving inverse problems of various types.

Target Adaptive Differential Iterative Reconstruction (TADI): A Robust Algorithm for Real-Time Electrical Impedance Tomography

Electrical impedance tomography (EIT) is a noninvasive functional diagnostic technique that has been successfully applied to human lung and brain. EIT reconstruction is an ill-conditioned problem: the regularization parameters establish a trade-off between acceptable data fitting and acceptable solution stability. This trade-off is necessary to select the optimal regularization parameters for each data frame to continuously obtain high-quality images in real-time monitoring. However, using the objective regularization parameter selection method before each image reconstruction may reduce the reconstruction efficiency, and using heuristic selection for all images may cause significant quality degradation, thereby creating challenges in long-term clinical EIT monitoring. This study explores a robust EIT target adaptive differential iterative reconstruction algorithm that does not require the advance selection of the optimal regularization parameter to facilitate the clinical application of EIT long-term bedside monitoring. Specifically, second-order Taylor approximation expansion and Euler–Lagrange theorem were used to derive the conductivity differential iteration relationship. Furthermore, the reconstruction quality and generalization ability of the proposed algorithm were validated based on comparisons with the L-curve and the generalized cross-validation parameter selection methods through simulations, phantom experiments, and human lung recruitment data. Compared with the L-curve and generalized cross-validation methods, the TADI algorithm reduces the position error by 33.07% and 47.17%, the ringing by 26.49% and 32.69%, and the shape deformation by 18.34% and 19.40% on average, respectively. The results revealed that the TADI algorithm could achieve a significant improvement over the existing state-of-the-art methods in terms of stability, consistency, and efficiency.

Unsupervised Evaluation and Optimization for Electrical Impedance Tomography

Electrical impedance tomography (EIT) is a radiation-free, fast, and convenient tomographic technology that uses only surface electrodes. EIT has the potential to provide continuous bedside monitoring of patients, for example, during mechanical ventilation. However, the soft-field nature of EIT leads to an ill-posed reconstruction problem that often results in multiple image solutions. This is exacerbated when EIT systems are used in variable measurement setups and scenarios, as few adjustment methods or EIT calibration approaches work reliably or require a reference phantom. The goals of this work are to design quality metrics to provide feedback for the EIT process to guide the system in adjusting the algorithm and parameters, for example, to alert the system in the event of poor image quality due to electrode connection variability. This then may be dealt with by physical electrode adjustment. In this article, the metrics, OAD and AL, representing object and artifact discreteness and artifact level, were designed to evaluate EIT image quality in an unsupervised way. Based on the quality index feedback, an image optimization process is proposed. The developed metrics and optimization methods were validated through data simulated in typical circular domain and thorax scenarios, and clinical data from human respiration. The conventional supervised evaluation metric, i.e., the correlation coefficient (COR) between the reference image and the reconstructed image, was used for comparison. The accuracy of new metrics was represented as the correlation coefficient of the new metric and COR. From the results, OAD and AL could improve the image quality by optimizing regularization parameters. The accuracy results of OAD and COR in the circular domain are 0.9429 and 0.9328, and those in thorax scenarios are 0.9741 and 0.8393. OAD and AL provide the evaluation of EIT images without using any prior knowledge or reference conductivity distribution, providing feedback to improve the reliability of EIT systems and reduce the risk of erroneous diagnosis.

Multi-parameter Tikhonov regularization-based OTPA with application to ship-radiated noise evaluation

The purpose of this study is to reduce the errors caused by the inversion of the transfer function (TF) matrix when evaluating ship-radiated noise by operational transfer path analysis. The singular value decomposition (SVD), generalized cross validation (GCV), and L-curve methods are separately introduced to evaluate the TF matrix, and the performances are compared. In order to overcome the shortcomings of the aforementioned methods and further reduce the errors, the optimized multi-parameter (M-P) Tikhonov regularization method based on the criterion of condition number is proposed to create an optimal regularization parameter to evaluate the TF matrix herein. The feasibility is verified with a double-layer cylindrical shell model experiment in Thousand Islets Lake. The obtained results indicate that the average error of M-P Tikhonov regularization is reduced by up to 0.38 dB compared with that of the L-curve, 0.68 dB compared with that of the GCV, and 1.34 dB compared with that of the SVD under various combinations of noise levels, which can provide guidance for ship-radiated noise evaluation in engineering applications.

Robust Symbol Detection in Large-Scale Overloaded NOMA Systems

We a framework for the design of low-complexity and high-performance receivers for multidimensional overloaded non-orthogonal multiple access (NOMA) systems. The framework is built upon a novel compressed sensing (CS) regularized maximum likelihood (ML) formulation of the discrete-input detection problem, in which the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> -norm is introduced to enforce adherence of the solution to the prescribed discrete symbol constellation. Unlike much of preceding literature,.g., (Assaf et al., 2020, Yeom et al., 2019, Nagahara, 2015, Naderpour and Bizaki, 2020, Hayakawa and Hayashi, 2017, Hayakawa and Hayashi, 2018, and Zeng et al., 2020), the method is not relaxed into the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm, but rather approximated with a continuous and asymptotically exact expression without resorting to parallel interference cancellation (PIC). The objective function of the resulting formulation is thus a sum of concave-over-convex ratios, which is then tightly convexified via the quadratic transform (QT), such that its solution can be obtained via the iteration of a simple closed-form expression that closely resembles that of the classic zero-forcing (ZF) receiver, making the method particularly suitable to large-scale set-ups. By further transforming the aforementioned problem into a quadratically constrained quadratic program with one convex constraint (QCQP-1), the optimal regularization parameter to be used at each step of the iterative algorithm is then shown to be the largest generalized eigenvalue of a pair of matrices which are given in closed-form. The method so obtained, referred to as the Iterative Discrete Least Square (IDLS), is then extended to address several factors of practical relevance, such as noisy conditions, imperfect channel state information (CSI), and hardware impairments, thus yielding the Robust IDLS algorithm. Simulation results show that the proposed art significantly outperforms both classic receivers, such as the linear minimum mean square error (LMMSE), and recent CS-based state-of-the-art (SotA) alternatives, such as the sum-of-absolute-values (SOAV) and the sum of complex sparse regularizers (SCSR) detectors. It is also shown via simulations that the technique can be integrated with existing iterative detection-and-decoding (IDD) methods, resulting in accelerated convergence.

Novel Adaptive Sparse-Spike Deconvolution Bearing Fault Detection Method Based on Curvelet Transform

This paper has proposed a novel bearing fault detection method about adaptive Sparse-spike Deconvolution based on Curvelet Transform (CTSSD), where the novel technique about adaptive Sparse-spike Deconvolution names after ASSD. Its purpose is to recover the pulse sequence from a vibration signal including complex noise, and to evaluate the periodic pulse position and pulse amplitude. Firstly, in order to make the results sparse and improves the stability of the result, the L1 norm regularization method is proposed in this paper, which is used to constrain the signal pulse sequence sparsely. Secondly, considering that regularization parameters are not adaptive, the Quantum behavior Particle Swarm Optimization (QPSO) algorithm is proposed to determine the optimal regularization parameters, adaptively. Finally, considering that the periodic features of ASSD extraction are not continuous, the Curvelet transform is further introduced. The fault signal is transformed into the Curvelet domain, and the Curvelet coefficient is used to characterize the fault signal pulse sequence. This method proposed in this paper is applied to the simulation signal and the vibration signal of rolling bearing fault, and is compared with the ASSD and the minimum entropy deconvolution (MED) to verify the reliability and effectiveness.

Predictor complexity and feature selection affect Maxent model transferability: Evidence from global freshwater invasive species

AbstractAimEcological niche models (ENMs) are widely used to address urgent real‐world problems such as climate change effects or invasive species; however, the generality of models when projected through space and/or time, that is transferability, remains a key challenge. Here, we explored the effects of complex predictors and feature selection on ENM transferability in a widely employed algorithm, Maxent, using five globally invasive freshwater species as case studies.LocationGlobal.MethodsWe modelled the global distributions of five notorious freshwater invasive species (African sharptooth catfish Clarias gariepinus, Mozambique tilapia Oreochromis mossambicus, American bullfrog Lithobates catesbeianus, red swamp crayfish Procambarus clarkii, and Australian redclaw crayfish Cherax quadricarinatus), using three predictor datasets of varying complexities derived from two commonly used climatic data sources (WorldClim and IPCC) and three methods of model tuning that differentially incorporated feature selection. Spatially explicit transferability assessments were then conducted using a suite of evaluation metrics previously used to quantify Maxent model performance.ResultsWe show that in the absence of detailed biological knowledge of focal species, simpler predictor datasets produce models that are more accurate than those calibrated using comprehensive “bioclimatic” datasets. Additionally, we find that tuning models for both optimal regularization parameters as well as feature‐class combinations led to the greatest increases in transferability and geographic niche conservatism. Results indicate a tenuous link between model transferability and Akaike's information criterion corrected for small sample sizes (AICc), suggesting that the indiscriminate use of AICc as an estimate of model parsimony may lead to erratic model performance.Main conclusionsOur findings demonstrate that methodological considerations can drastically affect the reliability of spatial and possibly temporal projections, which has severe implications when ENMs are used to infer species’ niches, and quantify ecological or evolutionary change across impacted landscapes.

Optimal parameters for image reconstruction in ghost imaging via sparsity constraints

Ghost imaging via sparsity constraints (GISC) is an advanced imaging technique. The reconstruction quality of GISC is affected by the sparse ratio of the object, the regularization parameter, and the iteration number. Influences of these parameters on the peak signal-to-noise ratio (PSNR) of the reconstructed image are discussed and evaluated. The optimal regularization parameter and iteration number at different sparse ratios are given. Then the reconstructed images of GISC using the optimal parameters at different sparse ratios are shown. The improvement of the reconstruction quality of GISC utilizing the optimal parameters is confirmed through comparison with normalized ghost imaging. Finally, the reconstruction quality of GISC with random noise is analyzed, and a method to obtain the sparse ratio of the object by analyzing the signal of the bucket detector is discussed.

Error propagation in regional geoid computation using spherical splines, least-squares collocation, and Stokes\u2019s formula

Current International Association of Geodesy efforts within regional geoid determination include the comparison of different computation methods in the quest for the “1-cm geoid.” Internal (formal) and external (empirical) approaches to evaluate geoid errors exist, and ideally they should agree. Spherical radial base functions using the spline kernel (SK), least-squares collocation (LSC), and Stokes’s formula are three commonly used methods for regional geoid computation. The three methods have been shown to be theoretically equivalent, as well as to numerically agree on the millimeter level in a closed-loop environment using synthetic noise-free data (Ophaug and Gerlach in J Geod 91:1367–1382, 2017. https://doi.org/10.1007/s00190-017-1030-1). This companion paper extends the closed-loop method comparison using synthetic data, in that we investigate and compare the formal error propagation using the three methods. We use synthetic uncorrelated and correlated noise regimes, both on the 1-mGal (=10^{-5}~{mathrm {m s}}^{-2}) level, applied to the input data. The estimated formal errors are validated by comparison with empirical errors, as determined from differences of the noisy geoid solutions to the noise-free solutions. We find that the error propagations of the methods are realistic in both uncorrelated and correlated noise regimes, albeit only when subjected to careful tuning, such as spectral band limitation and signal covariance adaptation. For the SKs, different implementations of the L-curve and generalized cross-validation methods did not provide an optimal regularization parameter. Although the obtained values led to a stabilized numerical system, this was not necessarily equivalent to obtaining the best solution. Using a regularization parameter governed by the agreement between formal and empirical error fields provided a solution of similar quality to the other methods. The errors in the uncorrelated regime are on the level of sim 5 mm and the method agreement within 1 mm, while the errors in the correlated regime are on the level of sim 10 mm, and the method agreement within 5 mm. Stokes’s formula generally gives the smallest error, closely followed by LSC and the SKs. To this effect, we note that error estimates from integration and estimation techniques must be interpreted differently, because the latter also take the signal characteristics into account. The high level of agreement gives us confidence in the applicability and comparability of formal errors resulting from the three methods. Finally, we present the error characteristics of geoid height differences derived from the three methods and discuss them qualitatively in relation to GNSS leveling. If applied to real data, this would permit identification of spatial scales for which height information is preferably derived by spirit leveling or GNSS leveling.