Model Misfit Research Articles

Make Some Noise: Generating Data from Imperfect Factor Models

Researchers simulating covariance structure models sometimes add model error to their data to produce model misfit. Presently, the most popular methods for generating error-perturbed data are those by Tucker, Koopman, and Linn (TKL), Cudeck and Browne (CB), and Wu and Browne (WB). Although all of these methods include parameters that control the degree of model misfit, none can generate data that reproduce multiple fit indices. To address this issue, we describe a multiple-target TKL method that can generate error-perturbed data that will reproduce target RMSEA and CFI values either individually or together. To evaluate this method, we simulated error-perturbed correlation matrices for an array of factor analysis models using the multiple-target TKL method, the CB method, and the WB method. Our results indicated that the multiple-target TKL method produced solutions with RMSEA and CFI values that were closer to their target values than those of the alternative methods. Thus, the multiple-target TKL method should be a useful tool for researchers who wish to generate error-perturbed correlation matrices with a known degree of model error. All functions that are described in this work are available in the fungible R library. Additional materials (e.g., R code, supplemental results) are available at https://osf.io/vxr8d/.

Careless responding detection revisited: Accuracy of direct and indirect measures.

To screen for careless responding, researchers have a choice between several direct measures (i.e., bogus items, requiring the respondent to choose a specific answer) and indirect measures (i.e., unobtrusive post hoc indices). Given the dearth of research in the area, we examined how well direct and indirect indices perform relative to each other. In five experimental studies, we investigated whether the detection rates of the measures are affected by contextual factors: severity of the careless response pattern, type of item keying, and type of item presentation. We fully controlled the information environment by experimentally inducing careless response sets under a variety of contextual conditions. In Studies 1 and 2, participants rated the personality of an actor that presented himself in a 5-min-long videotaped speech. In Studies 3, 4, and 5, participants had to rate their own personality across two measurements. With the exception of maximum longstring, intra-individual response variability, and individual contribution to model misfit, all examined indirect indices performed better than chance in most of the examined conditions. Moreover, indirect indices had detection rates as good as and, in many cases, better than the detection rates of direct measures. We therefore encourage researchers to use indirect indices, especially within-person consistency indices, instead of direct measures.

Simultaneous prediction of petrophysical properties and formation layered thickness from acoustic logging data using a modular cascading residual neural network (MCARNN) with physical constraints

Predicting petrophysical properties from logging data using deep neural networks (DNNs) improves the efficiency of subsurface exploration. Traditional methods, however, heavily rely on data pre-processing. Therefore, a novel Modular Cascading Residual Neural Network (MCARNN) is proposed to simultaneously retrieve layered thickness and petrophysical properties of porous media directly from raw acoustic logging data without pre-processing. The MCARNN mainly comprises two cascading modules: the first is tasked with retrieving layered thickness, while the second one further reconstructs petrophysical properties based on the thickness information. To more accurately simulate the uncertainties of the real world, we incorporate some known geophysical knowledge, including the distribution range of actual petrophysical parameters, physical constraints among porous medium parameters, configurations of logging instruments, and stratified geological models. During this process, all Biot's porous parameters are independently randomized to cover a wide range of geological rock types. Furthermore, to enhance MCARNN's generalization ability under different random noise distributions, we independently generate the training and testing datasets using distinct numerical algorithms, thus preventing the “inverse crime” phenomenon. Additionally, varying levels of Gaussian white noises are added to the training dataset, and the transmitter-receiver configuration of the logging instrument is varied to enhance the training dataset. Two numerical experiments are implemented to validate the reliability and superiority of the MCARNN. It is shown that the proposed MCARNN can not only accurately predict porous petrophysical parameters when the background layer configurations are known but also be able to simultaneously reconstruct layered thickness and petrophysical parameters with low model misfit. Notably, MCARNN successfully predicts the porous velocity dispersion curves within a broad frequency band based on limited frequency points—a task typically requiring auxiliary equations in traditional inversion methods, demonstrating the advanced predictive capability of the proposed MCARNN. Through this research, we not only address the deficiencies in existing real-world logging datasets in terms of sample volume and label reliability but also effectively integrate rock physics models with data-driven neural networks, providing a novel and effective pathway for reservoir parameter prediction in the field of geophysical logging.

Global Predicted Bathymetry Using Neural Networks

AbstractA coherent portrayal of global bathymetry requires that depths are inferred between sparsely distributed direct depth measurements. Depths can be interpolated in the gaps using alternate information such as satellite‐derived gravity and a mapping from gravity to depth. We designed and trained a neural network on a collection of 50 million depth soundings to predict bathymetry globally using gravity anomalies. We find the best result is achieved by pre‐filtering depth and gravity in accordance with isostatic admittance theory described in previous predicted depth studies. When training the model, if the training and testing split is a random partition at the same resolution as the data, the training and testing sets will not be independent, and model misfit is underestimated. We solve this problem by partitioning the training and testing set with geographic bins. Our final predicted depth model improves on old predicted depth model RMSE by 16%, from 165 to 138 m. Among constrained grid cells, 80% of the predicted values are within 128 m of the true value. Improvements to this model will continue with additional depth measurements, but predictions at higher spatial resolution, being limited by upward continuation of gravity, should not be attempted with this method.

Deep-learning-based airborne transient electromagnetic inversion providing the depth of investigation

We develop an integrated workflow that uses deep-learning (DL)-based approaches for processing and inverting airborne transient electromagnetic (ATEM) data. Our novel workflow automates these preprocessing steps and enables real-time inversion in the field. Thus, we develop an entire inversion workflow using three DL networks that cover all steps from preprocessing to imaging. The preprocessing DL network performs interpolation to discard data that are severely noise contaminated and suppress the effects of noise in a late-time channel. We use an inversion DL network and a depth of investigation (DOI) network to generate images of subsurface resistivities exclusively within the DOI range where reliable predictions can be made. To optimize the inversion process, our approach focuses on designing the inversion DL network to simultaneously minimize data misfit and model misfit. By addressing these two aspects, we ensure a more robust outcome in the final resistivity images. The practical applicability of the workflow is verified by comparing the imaging results of the field data with those of conventional inversion and geologic interpretation. Each workflow is nearly automatic and very fast; we expect that our workflow will contribute to the development of real-time imaging software for the ATEM survey, which expands the applications of the ATEM survey in various fields.

High Subset Homogeneity Impairs Structural Validity of the Barratt Impulsiveness Scale

Abstract: Problems in providing evidence of structural validity of the Barratt Impulsiveness Scale (BIS) are addressed. As the German translation of a short version of the scale (BIS-15) included pairs of items with highly similar item statements, like the original English version, we hypothesized high subset homogeneity (HSH) as the source of these problems. HSH denotes a situation in which a subset of items shows a larger degree of homogeneity than the remainder of the items of the scale. In a sample of 287 university students, we investigated BIS data by means of confirmatory factor analysis (CFA) models including BIS factors only and models additionally including HSH factors. Whereas the models without HSH factors yielded model misfit, good model fit was observed for the models with HSH factors. These results suggested that BIS items basically showed structural validity, but this validity was impaired by HSH.

An advanced inverse modeling framework for efficient and flexible adjoint-based history matching of geothermal fields

In this study, we present an efficient and flexible adjoint-based framework for history matching and forecasting geothermal energy extraction at a large scale. In this framework, we applied the Principal Component Analysis to reduce the parameter space for representing the complex geological model. The adjoint method is implemented for gradient calculation to speed up the history-matching iteration process. Operator-based linearization (OBL) used in this framework makes the calculation of the physical state and its derivatives very efficient and facilitates the matrix assembly in the adjoint method. This study primarily focuses on history matching based on combined observation of well production and in-situ electromagnetic measurements to predict the temperature front. However, different types of misfit terms can be added to the objective function based on practical considerations. For example, our history-matching case studies include model misfit terms applied for regularization purposes. The measurement data is extracted from the true model, and realistic measurement errors are considered. Also, in this work, we propose an optimal weighting strategy for the terms of the objective function to balance their sensitivity with respect to the model control variables. The high efficiency of the framework is demonstrated for the geothermal doublet model implemented at the heterogeneous reservoir with multiple realizations. The framework allows for generating posterior Randomized Maximum Likelihood (RML) estimates of the entire ensemble of the realizations with a reasonable computational cost. Results show that the framework can achieve reliable history-matching results based on the doublets production data and the reservoir electromagnetic measurement.

The Effects of Aberrant Responding on Model-Fit Assuming Different Underlying Response Processes.

Aberrant responding on tests and surveys has been shown to affect the psychometric properties of scales and the statistical analyses from the use of those scales in cumulative model contexts. This study extends prior research by comparing the effects of four types of aberrant responding on model fit in both cumulative and ideal point model contexts using graded partial credit (GPCM) and generalized graded unfolding (GGUM) models. When fitting models to data, model misfit can be both a function of misspecification and aberrant responding. Results demonstrate how varying levels of aberrant data can severely impact model fit for both cumulative and ideal point data. Specifically, longstring responses have a stronger impact on dimensionality for both ideal point and cumulative data, while random responding tends to have the most negative impact on data model fit according to information criteria (AIC, BIC). The results also indicate that ideal point data models such as GGUM may be able to fit cumulative data as well as the cumulative model itself (GPCM), whereas cumulative data models may not provide sufficient model fit for data simulated using an ideal point model.

A reduced basis ensemble Kalman method

In the process of reproducing the state dynamics of parameter dependent distributed systems, data from physical measurements can be incorporated into the mathematical model to reduce the parameter uncertainty and, consequently, improve the state prediction. Such a data assimilation process must deal with the data and model misfit arising from experimental noise as well as model inaccuracies and uncertainties. In this work, we focus on the ensemble Kalman method (EnKM), a particle-based iterative regularization method designed for a posteriori analysis of time series. The method is gradient free and, like the ensemble Kalman filter (EnKF), relies on a sample of parameters or particle ensemble to identify the state that better reproduces the physical observations, while preserving the physics of the system as described by the best knowledge model. We consider systems described by parameterized parabolic partial differential equations and employ model order reduction techniques to generate surrogate models of different accuracy with uncertain parameters. Their use in combination with the EnKM involves the introduction of the model bias which constitutes a new source of systematic error. To mitigate its impact, an algorithm adjustment is proposed accounting for a prior estimation of the bias in the data. The resulting RB-EnKM is tested in different conditions, including different ensemble sizes and increasing levels of experimental noise. The results are compared to those obtained with the standard EnKF and with the unadjusted algorithm.

Sensitivity tests performed on passive low frequency seismic (LFS) data acquired in the Central North Sea to delineate hydrocarbon deposits

In 2018 TenzorGEO carried out a trial in the North Sea to acquire passive seismic data (LFS – low frequency seismic technology) as part of a blind test with an unnamed operator. The result from the trial is a hydrocarbon probability map showing data informed zones with delineated hydrocarbons. To further test the acquired data and the level of sensitivity of the model used in the analysis, a sensitivity test was performed to assess the level of model misfit and validate the vertical resolution of the technology. The results from the sensitivity tests was able to show how the hydrocarbon probability map will respond to: changes in depth, presence of other hydrocarbon layers within the study area, and the exclusion of sensor observation points should the sensors be affected or damaged during installation. Overall, the tests were able to validate the capability and potential of the LFS technology in identifying and delineating multiple hydrocarbon reservoirs with a predetermined interval spacing between the reservoirs across a field. The sensitivity tests provide an opportunity to test and fine-tune the application of LFS in detecting and delineating deeper structures with potential hydrocarbons and also monitoring CO2 storage reservoirs.

Post‐Rifting Relaxation During 2015–2020 Following the Bárðarbunga‐Holuhraun Dike Intrusion and Eruption in Iceland

AbstractPost‐rifting ground deformation may be driven by viscoelastic relaxation of stresses generated by dike intrusions. The single‐dike intrusion of the 2014–2015 Bárðarbunga eruption in Iceland presents an opportunity for a detailed study of this process. We use continuous Global Navigation Satellite System (GNSS) and Interferometric Synthetic Aperture Radar (InSAR) velocity fields to analyze the 2015–2020 post‐rifting deformation, showing uplift on both sides of the dike and horizontal displacement away from the dike after correcting for background deformation. Two GNSS stations experience baseline lengthening at a rate of 19 mm/yr in the direction perpendicular to the strike of the dike. A two‐layer viscoelastic model with a 0.4 × 1019 Pa s viscoelastic half‐space overlain by an 18 km thick elastic layer best explains the observed horizontal and vertical InSAR and GNSS displacements. The model misfit space shows a second regime of good fit, likely driven by deformation near the dike that may result from cooling compaction of the emplaced dike.

Physics-Driven Deep Learning Inversion with Application to Magnetotelluric

Due to the strong capability of building complex nonlinear mapping without involving linearization theory and high prediction efficiency; the deep learning (DL) technique applied to solve geophysical inverse problems has been a subject of growing interest. Currently, most DL-based inversion approaches are fully data-driven (namely standard deep learning), the performance of which largely depends on the training sample sets. However, due to the heavy burden of time and computational resources, it can be challenging to supply such a massive and exhaustive training dataset for generic realistic exploration scenarios and to perform network training. In this work, based on the recent advances in physics-based networks, the physical laws of magnetotelluric (MT) wave propagation is incorporated into a purely data-driven DL approach (PlainDNN) and thus builds a physics-driven DL MT inversion scheme (PhyDNN). In this scheme, the forward operator modeling MT wave propagation is integrated into the network training loop, in the form of minimizing a hybrid loss objective function composed of the data-driven model misfit and physics-based data misfit, to guide the network training. Consequently, the proposed PhyDNN method will take the advantage of the fully data-driven DL and conventional physics-based deterministic methods, allowing it to deal with complex realistic exploration scenarios. Quantitative and qualitative analysis results demonstrate that the PhyDNN can honor the physical laws of the MT inverse problem, and with other conditions unchanged, the PhyDNN outperforms the PlainDNN and the classical deterministic Occam inversion method. When processing field data, the PhyDNN method yields considerably impressive inversion results compared to the Occam method, and the corresponding simulated MT responses agree well with the real measurements, which confirms the effectiveness and applicability of the PhyDNN method.

Examination of the Problem Gambling Severity Index for Use with Older Adults: A Rasch Model Approach.

Research and prevalence studies commonly use the nine-item Problem Gambling Severity Index (PGSI; Ferris and Wynne in The Canadian Problem Gambling Index, 2001) to measure problem gambling severity in older adults. However, the appropriateness of the PGSI for use with older adults must be evaluated to ensure accurate interpretation of scores. This study evaluated the PGSI with older adults that gamble using a Rasch model approach, which has not yet been examined in the literature. Data from the Quinte Longitudinal Study (Wiiliams et al. in The Quinte Longitudinal Study of Gambling and Problem Gambling 2006-2011, Bay of Quinte region, Ontario [Canada] (V28), 2014) were utilized, accessed through the Gambling Research Exchange of Ontario (GREO). Using WINSTEPS.V5, a Rasch rating scale model was applied to an older adult sample from Ontario, Canada (n = 571, Mage = 66, range = 60-80 +) to assess: (1) the dimensionality and fit between individual PGSI items and the underlying latent construct of problem gambling; (2) utility of items and response options; and (3) potential differential item functioning (DIF) between genders. Using a larger sample (N = 3206, Mage = 45, range = 18-80 +), DIF was examined across three age subgroups. Results supported unidimensionality of the PGSI and item polarity (rpm range = 0.42-0.85). Infit and outfit statistics showed mixed model misfit for three items. The use of three response options were deemed productive for measurement, but almost always was not frequently endorsed. Model-person separation (1.00) and reliability (0.50) were poor, and model-item separation (6.12) and reliability (0.97) were excellent. There was no significant DIF between older adults by gender. Three items showed significant DIF between older and younger adults. Possible modifications to improve the PGSI for use with older adults are discussed.

Validity of the Central Sensitization Inventory (CSI) through Rasch analysis in patients with knee osteoarthritis.

Central sensitization (CS) is a known contributor to chronic pain in people with knee osteoarthritis (KOA) and is commonly measured by psychophysical testing or patient-reported methods such as the Central Sensitization Inventory (CSI). However, previous studies have shown a weak association between the two. We therefore sought to evaluate the validity of the CSI through Rasch analysis in patients with KOA. We performed a secondary analysis of a multicenter cohort study with patients with KOA consulting orthopedic surgeons. Rasch analysis was conducted considering person factors of age, sex, BMI, pain intensity, pain catastrophizing, and quantitative sensory test findings using pressure pain thresholds and temporal summation to assess how the CSI fits to the Rasch model (supporting validity). We used RUMM2030 software to model fit estimates, making adjustments as required to achieve model fit (P > 0.05). Data from 293 patients were included (58.7% female, mean age 63.6years, 49.1% obese) Initial evaluation with Rasch modelling indicated misfit. Eleven of 25 items on the CSI displayed disordered thresholds which were rescored by collapsing response categories until the thresholds demonstrated sequential progression. Reanalysis demonstrated persistent model misfit so a subtest was developed to address local dependency of 6 items. Thereafter, model fit was achieved (P = 0.071, indicating not differing from Rasch model) and acceptable unidimensionality (P = 0.068 with 95% CI 0.043-0.093). The CSI was able to be fit to the Rasch model after rescoring while retaining all 25 items. The unidimensionality validates CS as measured by the CSI as a singular construct. Key Points • The Central Sensitization Inventory (CSI) was able to be fit to the Rasch model after rescoring while retaining all 25 items. • The unidimensionality of the CSI validates CS as a singular construct. • Our results suggest rescoring of the CSI for people with KOA, but it should be confirmed and replicated in larger samples prior to clinical use.

Combined Use of Glucose-Specific Model Identification and Alarm Strategy Based on Prediction-Funnel to Improve Online Forecasting of Hypoglycemic Events.

Advanced decision support systems for type 1 diabetes (T1D) management often embed prediction modules, which allow T1D people to take preventive actions to avoid critical episodes like hypoglycemia. Real-time prediction of blood glucose (BG) concentration relies on a subject-specific model of glucose-insulin dynamics. Model parameter identification is usually based on the mean square error (MSE) cost function, and the model is usually used to predict BG at a single prediction horizon (PH). Finally, a hypo-alarm is raised if the predicted BG crosses a threshold. This work aims to show that real-time hypoglycemia forecasting can be improved by leveraging: a glucose-specific mean square error (gMSE) cost function in model's parameters identification, and a "prediction-funnel," that is, confidence intervals (CIs) for multiple PHs, within the hypo-alarm-raising strategy. Autoregressive integrated moving average with exogenous input (ARIMAX) models are selected to illustrate the proposed solution (use of gMSE and prediction-funnel) and its assessment against the conventional approach (MSE and single PH). The gMSE penalizes the model misfit in unsafe BG ranges (e.g., hypoglycemia), and the prediction-funnel allows raising an alarm by monitoring if the CIs cross a suitable threshold. The algorithms were evaluated by measuring precision (P), recall (R), F1-score (F1), false positive per day (FP/day), and time gain (TG) on a real dataset collected in 11 T1D individuals. The best performance is achieved exploiting both the gMSE and the prediction-funnel: P = 65%, R = 88%, F1 = 75%, FP/day = 0.29, and mean TG = 15 minutes. The combined use of a glucose-specific metric and an alarm-raising strategy based on the prediction-funnel allows achieving a more effective and reliable hypoglycemia prediction algorithm.

Measurement properties of the Swedish clinical outcomes in routine evaluation outcome measures (CORE-OM): Rasch analysis and short version for depressed and anxious out-patients in a multicultural area

IntroductionThe Swedish version of the patient-reported Clinical Outcomes in Routine Evaluation Outcome Measures (CORE-OM) has demonstrated high reliability and acceptable convergent validity in explanatory factor analyses. However, the fundamental scale properties have not yet been validated according to item response theory. The aim of this study was to analyze the measurement properties of the Swedish CORE-OM in a cohort of psychiatric out-patients with depression and anxiety in a multicultural area and to explore combinations of items based on shorter versions of the scale (CORE-10, CORE-6D) to improve measurement properties.MethodsData from CORE-OM assessments of 337 patients were analyzed using Rasch analysis. The patients had a mean age of 30 ± 14 years, the majority were women (72%). Requirements for measurement properties were checked: overall model fit, item fit residuals, targeting, internal consistency, differential item functioning and thresholds. Sensitivity to change was also analyzed.ResultsThe CORE-OM showed high internal consistency (person separation index = 0.947) and adequate targeting, but there was overall model misfit (item trait interaction χ2 = 917.53, p < 0.001), indication of local dependency, and differential item functioning in 9 items. The risk items showed problems with disordered thresholds. The emotional component of the shorter CORE-6D showed the best fit for our sample. Adding 3 items to include depressive and trauma-related content resulted in a unidimensional 8-item set with acceptable reliability, model fit, targeting and sensitivity to change.ConclusionFor out-patients with diagnosed depression or anxiety in a multicultural area, the Swedish CORE-OM showed high internal consistency, but also validity problems. Based on the shorter CORE-6D version, a unidimensional 8-item set could be an alternative brief measure of psychological distress for this population, but further validity studies are required. Qualitative studies exploring the CORE-OM items in non-native speakers are also warranted.

A Model Comparison Approach to Posterior Predictive Model Checks in Bayesian Confirmatory Factor Analysis

ABSTRACT Posterior Predictive Model Checking (PPMC) is frequently used for model fit evaluation in Bayesian Confirmatory Factor Analysis (BCFA). In standard PPMC procedures, model misfit is quantified by comparing the location of an ML-based point estimate to the predictive distribution of a statistic. When the point estimate is far from the center posterior predictive distribution, model fit is poor. Not included in this approach, however, is the variability of the Maximum Likelihood (ML)-based point estimates. We propose a new method of PPMC based on comparing posterior predictive distributions of a hypothesized and saturated BCFA model. The method uses the predictive distribution of the saturated model as a reference and the Kolmogorov-Smirnov (KS) statistic to quantify the local misfit of hypothesized models. The results of the simulation study suggest that the saturated model PPMC approach was an accurate method of determining local model misfit and could be used for model comparison. A real data example is also provided in this study.

Empirical Bayes Model Averaging with Influential Observations: Tuning Zellner’s [formula omitted] Prior for Predictive Robustness

The behavior of Bayesian model averaging (BMA) for the normal linear regression model in the presence of influential observations that contribute to model misfit is investigated. Remedies to attenuate the potential negative impacts of such observations on inference and prediction are proposed. The methodology is motivated by the view that well-behaved residuals and good predictive performance often go hand-in-hand. Focus is placed on regression models that use variants on Zellner's g prior. Studying the impact of various forms of model misfit on BMA predictions in simple situations points to prescriptive guidelines for “tuning” Zellner's g prior to obtain optimal predictions. The tuning of the prior distribution is obtained by considering theoretical properties that should be enjoyed by the optimal fits of the various models in the BMA ensemble. The methodology can be thought of as an “empirical Bayes” approach to modeling, as the data help to inform the specification of the prior in an attempt to attenuate the negative impact of influential cases.

A Spherical Harmonic Martian Crustal Magnetic Field Model Combining Data Sets of MAVEN and MGS

AbstractThis study presents a new spherical harmonic (SH) model of the crustal magnetic field of Mars, based on the magnetic field data set measured by the Mars Global Surveyor (MGS) and the Mars Atmosphere and Volatile EvolutioN (MAVEN) spacecraft. To minimize the influence of external fields due to solar wind interaction with Mars, we rejected data that were observed dayside and above an altitude of 500 km. The data points of MAVEN were reduced by using a proxy of solar wind activity that identified and rejected any data measured during magnetically disturbed intervals. We used a conventional least squares technique to estimate the Gauss coefficients fitted to the reduced data set and made a compromise between model misfit and model roughness by truncating the SH model at degree 110. This model is capable of representing crustal fields with a spatial resolution approaching ∼200 km at 120 km altitude and ∼260 km at the Martian surface. Since our model fits MAVEN's observational data better than previous models, especially the data obtained during MAVEN's low altitude periapsis passes, we conclude that it may more accurately approximate the low‐altitude crustal field. We calculate the crustal field power spectrum of various models and find that small‐scale fields at low altitudes were underestimated by most previous models. This new model could benefit future studies associated with the Martian crustal field and its interaction with the solar wind.

Neuro‐Fuzzy Kinematic Finite‐Fault Inversion: 2. Application to the Mw6.2, August/24/2016, Amatrice Earthquake

AbstractIn this article, we validate the neuro‐fuzzy kinematic finite‐fault inversion method by studying the rupture process of the , Aug/24/2016, Amatrice, central Italy, earthquake. We jointly invert three different datasets to infer the spatio‐temporal slip distribution, namely static and high‐rate GNSS data (&lt;= Hz) and strong‐motion data ( Hz). Each data set is used to constrain a different frequency range of the source model, depending on the sensitivity of the data set. The inferred slip shows a slow nucleation phase at shallow depths of 3–4 km, followed by a bilateral rupture that forms two asperities, one to the NW (Norcia) and another to the SE (Amatrice) of the hypocenter. Our inferred slip is compared with those previously obtained using well‐established methods. In order to select an adequate number of fuzzy basis functions, we propose two alternative procedures, which yield the same general slip features. The first approach consists of ensuring that the inverse problem is formally over‐determined and uses the same number of basis functions at all frequencies. The second approach is based on a maximum‐likelihood analysis of the model misfit and selects a different number of basis functions for each frequency. The maximum‐likelihood approach allows for more basis functions at high frequencies, where more detail in the spatial slip distribution is needed. The solution obtained with the maximum‐likelihood approach provides a more physically plausible source time function, which shows less back slip artifacts. The accurate prediction of high‐rate GNSS traces not used in the inversion attests the robustness of the inferred slip model.