Prior Probability Density Research Articles

Bayesian-Based Approach in Soil Characterization for Tophole Design

Summary This paper addresses Bayesian-based data-driven site characterization (DDSC) methods for estimating soil parameters used in tophole casing design. Different models are applied to data sets of piezocone tests (CPTu) conducted in Brazilian fields, and their performance is compared to some characterization techniques currently used by the oil and gas industry. Regression models are crucial for soil characterization for tophole casing design. Data-driven methods consist of a powerful tool for this purpose, allowing handling uncertainties originating from soil variability. This paper addresses regression models devoted to sparse data, namely geotechnical lasso (Glasso) and Gaussian process regression (GPR) from a Bayesian perspective considering prior knowledge of site information. This approach provides statistical information on soil parameters like undrained shear strength, supporting structural analysis of wellhead systems, and conductor/surface casing strings. Data sets were collected from in-situ CPTu tests conducted in the Campos basins in eastern Brazil. The first case study addressed primary CPTu data, including cone tip resistance, friction sleeves, and total pore pressure, to characterize undrained shear resistance as a random variable. In this context, the parameter was modeled using Phoon’s modified Bartlett test to calculate sample sizes that ensure stationarity. This approach presented relevant results in assessing the probability of failure for conductor and surface casing design based on the operator’s internal design criteria. In these applications, more robust techniques were used to improve data characterization. Glasso and GPR models are used to model parameter tendencies and evaluate the random data. These methods differ in prior probability density functions adopted—Laplace and Gaussian, respectively—and they are compared with regression techniques widely used in design practice. All the models were trained to estimate undrained shear resistance. Preliminary results confirm that the technique improves the characterization of soil strata and undrained shear strength, with a beneficial effect on the analysis of offshore tophole structural design cases. Some metrics, such as mean absolute error (MAE) and root mean squared error (RMSE), indicate that GPR outperforms other methods. This is an innovative methodology applied in real-case scenarios with data ceded from the partner operator. The formulation evaluates uncertainties associated with the spatial heterogeneity of the material, continuously improving robustness with each new project data. This enables a better understanding of soil behavior in specific oil fields and can assist the decision-making process in well design, improving operational safety. Furthermore, the results of the statistical modeling support reliability-based analysis to deliver probability-based indicators for well integrity design.

Full-field experiment-aided virtual modelling framework for inverse-based stochastic prediction of structures with elastoplasticity

Forecasting of stochastic ductile failures in fabrication and service stages are challenging tasks of advanced structures in practical engineering. Due to the prohibitive costs of repetitive experimental tests to quantify optimal failure-related parameters, numerous studies have turned to simulation-based uncertainty quantification. However, the credibility of these approaches is frequently doubted by the function-generated distribution of random data involved. Thus, an accurate assessment of the material parameters ascertains the appropriate estimation of uncertain parameters, which is essential for the risk/safety assessment of structures in different stages. In this paper, a full-field experiment-aided virtual modelling framework for inverse-based prediction of stochastic elastoplastic failure (EVMF-ISP) is proposed to deliver precise insights regarding the effective distribution range of mechanical parameters. To this end, all available experiment observations serve as the reference for assessing the prior and posterior probability density of the unknown parameters through the real-time inverse uncertainty quantification (UQ) module. The framework can be divided into three parts, where initially a pre-virtual model (PRVM) is formulated, and Bayesian inference is implemented to propagate the experiment observations backwards to ascertain the uncertain parameters. Then, an advanced multidimensional slice sampling method is developed to deal with the derived complex posterior probability density function (PDF) of mechanical parameters. In the end, a reliable stochastic elastoplastic analysis can be conducted with the revised uncertain samples and finalised with post-virtual models (POVMs) for the concerned structures. Such that, accurate and efficient determination of nonlinear response of structures can be directly predicted. The EVMF-ISP framework is logically presented and thoroughly illustrated with practical applications.

Bayesian inference of the mean power of several Gaussian data

The uniform prior probability density for the means of normal data leads to inconsistent Bayesian inference of their mean power and jeopardizes the possibility of selecting among different models that explain the data. We reinvestigated the problem avoiding delivering unrecognised information and looking at it in a novel way. Namely, to consider a finite power, we used a normal prior minimally diverging from the uniform one, hyperparameterised by the mean and variance, and left the data to choose the most supported parameters. We also obtained an extended James–Stein estimator averaging the hyper-parameters and avoiding empirical Bayes techniques.Graphical abstract

Probabilistic Prediction of Floating Offshore Wind Turbine Platform Motions via Uncertainty Quantification and Information Integration

The accurate prediction of short-term platform motions in a real environment is crucial for the safe design, operation, and maintenance of floating offshore wind turbines (FOWTs). Numerical simulations of motions are typically associated with high uncertainties due to abstracted theoretical models, empirical parameters, initial environment parameters, etc. Therefore, it is necessary to integrate other sources of information associated with less uncertainty, e.g., monitoring data, for accurate predictions. In this paper, we propose a probabilistic prediction based on the Bayesian approach that logically integrates motion monitoring data with simulated motion predictions of FOWTs, considering uncertainties in the environment model, structural properties, motion prediction method, monitoring data, etc. The approach consists of constructing a prior probability density function (PDF) of a random variable (which characterizes the largest value of the initial motion response) via numerical simulations and a likelihood function based on platform motion monitoring data and deriving a posterior PDF of the random variable by Bayesian updating. Then, posterior distributions of short-term extreme motion responses are derived using the posterior PDF of the random variable, representing lower uncertainty and improved accuracy. A Metropolis–Hastings algorithm is adopted to obtain PDFs of complex probability distributions. The effectiveness of the approach is demonstrated on a real FOWT platform in Scotland. The proposed probabilistic prediction approach results in posterior distributions of short-term extreme platform motions associated with less uncertainty and higher accuracy, which is attributed to integrating prior knowledge with monitoring data.

Stochastic PDE representation of random fields for large-scale Gaussian process regression and statistical finite element analysis

The efficient representation of random fields on geometrically complex domains is crucial for Bayesian modelling in engineering and machine learning, including Gaussian process regression and statistical finite element analysis. Today’s prevalent random field representations are either intended for unbounded domains or are too restrictive in terms of possible field properties. Because of these limitations, new techniques leveraging the historically established link between stochastic PDEs (SPDEs) and random fields have been gaining interest in the statistics and engineering literature. The SPDE representation is especially appealing for engineering applications with complex geometries which already have a finite element discretisation for solving the physical conservation equations. In contrast to the dense covariance matrix of a random field, its inverse, the precision matrix, is usually sparse and equal to the stiffness matrix of an elliptic SPDE. In this paper, we use the SPDE representation to develop a scalable framework for large-scale statistical finite element analysis and Gaussian process (GP) regression on geometrically complex domains. The statistical finite element method (statFEM) introduced by Girolami et al. (2022) is a novel approach for synthesising measurement data and finite element models. In both statFEM and GP regression, we use the SPDE formulation to obtain the relevant prior probability densities with a sparse precision matrix. The properties of the priors are governed by the parameters and possibly fractional order of the SPDE so that we can model on bounded domains and manifolds anisotropic, non-stationary random fields with arbitrary smoothness. We use for assembling the sparse precision matrix the same finite element mesh used for solving the physical conservation equations. The observation models for statFEM and GP regression are such that the posterior probability densities are Gaussians with a closed-form mean and precision. The expressions for the mean vector and the precision matrix do not contain dense matrices and can be evaluated using only sparse matrix operations. We demonstrate the versatility of the proposed framework and its convergence properties with one and two-dimensional Poisson and thin-shell examples.

Simplifying asteroseismic analysis of solar-like oscillators

Context.The asteroseismic analysis of stellar power density spectra is often computationally expensive. The models used in the analysis may require several dozen parameters to accurately describe features in the spectra caused by the oscillation modes and surface granulation. Many of these parameters are often highly correlated, making the parameter space difficult to quickly and accurately sample. They are, however, all dependent on a much smaller set of parameters, namely the fundamental stellar properties.Aims.We aim to leverage this to develop a method for simplifying the process of sampling the model parameter space for the asteroseismic analysis of solar-like oscillators, with an emphasis on mode identification.Methods.Using a large set of previous observations, we applied principal component analysis to the sample covariance matrix to select a new basis on which to sample the model parameters. Selecting the subset of basis vectors that explains the majority of the sample variance, we then redefined the model parameter prior probability density distributions in terms of a smaller set of latent parameters.Results.We are able to reduce the dimensionality of the sampled parameter space by a factor of two to three. The number of latent parameters needed to accurately model the stellar oscillation spectra cannot be determined exactly but is likely only between four and six. Using two latent parameters, the method is able to produce models that describe the bulk features of the oscillation spectrum, while including more latent parameters allows for a frequency precision better than ≈10% of the small frequency separation for a given target.Conclusions.We find that sampling a lower-rank latent parameter space still allows for accurate mode identification and parameter estimation on solar-like oscillators over a wide range of evolutionary stages. This allows for the potential to increase the complexity of spectrum models without a corresponding increase in computational expense.

Efficient reliability updating methods based on Bayesian inference and sequential learning Kriging

Reliability updating is an effective tool for reappraising reliability level of system when new observation information is obtained. The adaptive Kriging based reliability updating method (RUAK) inserts the adaptive Kriging into traditional simulation method to improve the efficiency of reliability updating. However, an identical candidate sampling pool is used to simultaneously estimate the prior failure probability and the posterior one in RUAK, which leads to a waste of computational resources in case of significant difference between the importance regions in estimation of prior and posterior failure probabilities. To overcome this disadvantage, an efficient reliability updating framework based on Bayesian inference and sequential learning Kriging is proposed in this paper. In the proposed method, two candidate sampling pools respectively for estimating the prior and posterior failure probabilities are separately constructed by prior probability density function (PDF) and posterior PDF obtained by Bayesian inference. Then, the Kriging model is established and sequentially refined in these two candidate sampling pools to accurately estimate the corresponding failure probabilities. Through combining different simulation methods with the proposed framework, the Monte Carlo simulation based and importance sampling based sequential learning Kriging methods are respectively developed for reliability updating.

Generative models and Bayesian inversion using Laplace approximation

The Bayesian approach to solving inverse problems relies on the choice of a prior. This critical ingredient allows expert knowledge or physical constraints to be formulated in a probabilistic fashion and plays an important role for the success of the inference. Recently, Bayesian inverse problems were solved using generative models as highly informative priors. Generative models are a popular tool in machine learning to generate data whose properties closely resemble those of a given database. Typically, the generated distribution of data is embedded in a low-dimensional manifold. For the inverse problem, a generative model is trained on a database that reflects the properties of the sought solution, such as typical structures of the tissue in the human brain in magnetic resonance imaging. The inference is carried out in the low-dimensional manifold determined by the generative model that strongly reduces the dimensionality of the inverse problem. However, this procedure produces a posterior that does not admit a Lebesgue density in the actual variables and the accuracy attained can strongly depend on the quality of the generative model. For linear Gaussian models, we explore an alternative Bayesian inference based on probabilistic generative models; this inference is carried out in the original high-dimensional space. A Laplace approximation is employed to analytically derive the prior probability density function required, which is induced by the generative model. Properties of the resulting inference are investigated. Specifically, we show that derived Bayes estimates are consistent, in contrast to the approach in which the low-dimensional manifold of the generative model is employed. The MNIST data set is used to design numerical experiments that confirm our theoretical findings. It is shown that the approach proposed can be advantageous when the information contained in the data is high and a simple heuristic is considered for the detection of this case. Finally, the pros and cons of both approaches are discussed.

Fatigue Reliability Based on Predicted Posterior Stress Ranges Determined from Strain Measurements of Wind Turbine Support Structures

In the present paper, an approach for updating the continuous stress range distribution of a welded connection of a wind turbine support structure with predicted information from strain measurements is presented. Environmental conditions, such as wind or, in offshore fields, waves and currents, in combination with rotor excitations generate cyclic stresses affecting the reliability of welded joints of the support structure over the service life. Using strain measurements, these conditions can be monitored, and the resulting stress ranges, under consideration of measurement, mechanical and material uncertainties, can be reconstructed. These stress ranges can be used as an input for updating the prior probability density function (PDF) of the stress ranges predicted by the overall dynamics and a detailed design analysis. Applying Bayesian probability theory and decision theoretical implications, the predicted posterior probability density of the stress ranges is calculated based on the design information and uncertainties. This approach is exemplified, and it is shown how the predicted stress ranges and the design stress ranges are distributed. The prior and the predicted posterior stress ranges are used for a reliability calculation for potentially entering a pre-posterior decision analysis.

Prediction of reservoir physical parameters based on adaptive step size stochastic inversion

Reservoir physical parameter prediction is the ultimate goal of the quantitative seismic interpretation. Rock physics inversion is a method to obtain reservoir physical parameters from seismic elastic properties and the accuracy of predicted physical parameters depends largely on the inversion algorithm. We proposed an adaptive step size stochastic algorithm. In the algorithm, the step size of the parameters to be inverted are not fixed, but adaptive. Firstly, the parameters to be inverted are sampled according to their prior probability density distribution, and the error of each sampling point is calculated. Then select the sampling points with small errors and calculate their probability density distribution, and then encrypt the sampling point according to the new probability density distribution. Repeat the sampling several times, and finally get the optimal solution to the parameters to be inverted. Compared with the conventional given step size algorithm, the adaptive step size algorithm is more accurate. The prediction of reservoir physical parameters based on adaptive step size stochastic inversion is applied to sandstone reservoir, obtained reservoir physical parameters, such as porosity, water saturation, clay content, and their uncertainty distribution.

Meta-analysis of the experimental coefficient of variation in wheat using the Bayesian and Frequentist approaches

A large set of variables is assessed for progeny selection in a plant-breeding program and other agronomic fields. The meta-analysis of the coefficient of variation (CVe) produces information for researchers and breeders on the experimental quality of trials. This analysis can also be applied in the decision-making process of the experimental plan regarding the experimental design, the number of repetitions, and the treatments and plants/progenies to be measured. In this study, we evaluated the dataset distribution and the descriptive statistics of CVe through the Frequentist and Bayesian approaches, aiming to establish the credibility and confidence intervals. We submitted CVe data of ten wheat (Triticum aestivum L.) traits reported in 1,068 articles published to the Bayesian and Frequentist analyses. Sample data were analyzed via Gamma and normal models. We selected the model with the lowest Akaike Information Criterion (AIC) value, and then we tested three link functions. In the Bayesian analysis, uniform distributions were used as non-informative priors for the Gamma distribution parameters with three ranges of q~U (a,b,). Thus, the prior probability density function was given by: pθ =1β-α, θ ∈ α,β. The Bayesian and Frequentist approaches with the Gamma model presented similar results for CVe; however, the range Bayesian credible intervals was narrower than the Frequentist confidence intervals. Gamma distribution fitted the CVe data better than the normal distribution. The credible and confidence intervals of CVe were successfully applied to wheat traits and could be used as experimental accuracy measurements in other experiments.

An improved GNSS/INS navigation method based on cubature Kalman filter for occluded environment

Global navigation satellite systems and inertial navigation systems (GNSSs/INSs) based on the cubature Kalman filter (CKF) can provide an effective land vehicle navigation and positioning solution. The accuracy of CKF depends mainly on the prior probability density function (PDF), which is inaccurate when the GNSS signal is temporarily blocked in an urban environment. Although the improved CKF (ICKF) can reduce cubature point errors by using the historical information obtained iteratively, the generation of cubature points is still related to measurement noise. To obtain a more accurate PDF, adjustment off the sensor measurement noise is necessary. In this paper, an adaptive ICKF for GNSS/INS system based on the Mahalanobis distance called MD-ICKF is proposed, which expands the usability of the micro-electronic mechanical system based integrated navigation system in an urban environment. Two lower triangular matrices obtained by Cholesky decomposition are used to adjust the cubature point errors during each iteration. Furthermore, the measurement noise will be inflated to obtain a more accurate PDF when it exceeds the threshold calculated by the MD and chi-square test. The field experiment results show that the proposed method has better positioning accuracy and performance than CKF and ICKF in an occluded environment

A probabilistic model on streamwise velocity profile in open channels using Tsallis relative entropy theory

The present study uses the concept of information entropy to derive a velocity distribution model in open channel flow. The Tsallis relative entropy theory is explored for that purpose, where the prior probability density function (PDF) is chosen from the maximum Tsallis entropy distribution. Both the cases of the fixed and varying entropy index are considered from the literature. The velocity equations are derived analytically using the homotopy analysis method (HAM) together with the Padé approximation technique in a general framework. The approximations and the HAM-based series solution are verified against laboratory and field data sets. Also, the prediction accuracies of the proposed models are assessed through the measures of statistical errors. It is seen that the model corresponding to the varying index is superior to the other model. The entropy index of Tsallis relative entropy is considered a variable, and the non-unity values of this index justify the applicability of the entropy function. Moreover, it is observed that one of the Lagrange multipliers corresponding to the velocity model with a varying index becomes negligible subject to the data sets considered, and hence, it simplifies the model development mathematically. The proposed approach can be further extended to develop models for estimating the suspended sediment concentration and shear stress distribution.

Improved unscented Kalman filter for state of charge estimation of lithium-ion battery with one-step randomly measurement loss and inaccurate noise covariance matrices

Accurate state of charge estimation is of great significance for optimizing the management of lithium batteries. Due to the influence of actual environmental factors, random measurement loss and inaccurate noise covariance matrices will appear in the estimation process. Especially when the two situations occur at the same time, the estimation of SOC will be seriously affected. Thus, an improved unscented Kalman filter is presented to solve the above problems in this paper. First, the state space model of nonlinear system is constructed, which can fully reflect one-step random measurement loss, and the nominal noise covariance matrix is used to reflect the uncertainty. Second, as the prior probability density function of the parameters, the Beta prior and inverse Wishart prior are used in the derivation of the joint probability density function. Finally, these parameters are derived based on variational Bayesian theory, the updated state vector is obtained. Simulation results shows that the proposed method can deal with random measurement loss and inaccurate noise covariance matrices and the high accuracy in dynamic stress test and federal urban driving schedule.

Plate‐Locking, Uncertainty Estimation and Spatial Correlations Revealed With a Bayesian Model Selection Method: Application to the Central Chile Subduction Zone

AbstractInversions of geodetic data are regularly used to estimate interseismic locking in subduction zones. However, the ill‐posed nature of these problems motivates us to include prior information, physically consistent with processes of the subduction seismic cycle. To deal with model instabilities, we present an inversion method to estimate both plate‐locking and model uncertainties by inverting Global Navigation Satellite System derived velocities based on a Bayesian model selection scheme. Our method allows us to impose positivity constraints via a multivariate folded‐normal distribution, with a specified covariance matrix. Model spatial correlations are explored and ranked to find models that best explain the observed data and for a better understanding of locking models. This approach searches for hyperparameters of the prior joint multivariate probability density function (PDF) of model parameters that minimize the Akaike Bayesian Information Criterion (ABIC). To validate our approach, we invert synthetic displacements from analytic models, yielding satisfactory results. We then apply the method to estimate the plate‐locking in Central Chile (28°–39°S) and its relation to the coseismic slip distribution of earthquakes with magnitudes Mw > 8.0, on the subduction zone since 2010. We also search among different prior PDFs for a single ductile‐fragile limit depth. Our results confirms a spatial correlation between locked asperities and the 2010 Mw 8.8 Maule and 2015 Mw 8.3 Illapel earthquake rupture zones. The robustness of our locking model shows potential to improve future seismic and tsunami hazard estimations.

Explosive source signature determination: Two unequal shots in the same hole

AbstractBuried explosive charges generate seismic waves with unrivalled bandwidth. The source time function is not always repeatable and is difficult to measure, but is required for subsequent data processing including reverse time migration and full waveform inversion. I present a new method for estimating the explosive seismic source time function for every shot, using two unequal shots in the same shot hole. The measured data for each charge are deconvolved for the modelled monopole seismic source time function to recover estimated earth impulse responses or Green's functions. The source model parameters are adjusted to maximize the likeness of the estimated Green's functions from the two shots, applying physical constraints to obtain the prior probability density functions for each parameter. Ten parameters specify the two source models, but only five of these are independent and a grid search is used to find them. The method is tested on an experimental data set, obtained for a different purpose, consisting of a line of single geophones and three in‐line shot holes, each with two unequal charges, the larger one at twice the depth of the smaller. The method works well where the charge size ratio is at least 2: the estimated source time functions are recovered and the estimated Green's functions are more similar than the seismograms before deconvolution. To apply the method in practice, the smaller charge should be deeper than the larger. If the small charge is small enough, the vertical separation between the charges can be reduced to less than 1 m. The Green's functions for the two charges would then be almost identical for frequencies up to 250 Hz, and the differences in the measured data from the two charges would be caused principally by the differences in the two source time functions.

Bayesian Multi‐Model Estimation of Fault Slip Distribution for Slow Slip Events in Southwest Japan: Effects of Prior Constraints and Uncertain Underground Structure

AbstractWe consider a Bayesian multi‐model fault slip estimation (BMMFSE), which incorporates many candidates of the underground structure (Earth structure and plate boundary geometry) model characterized by a prior probability density function (PDF). The technique is used to study long‐term slow slip events (L‐SSEs) that occurred beneath the Bungo Channel, southwest Japan, in around 2010 and 2018. We here focus on the two advantages of BMMFSE: First, it allows for estimating slip distribution without introducing relatively strong prior information such as smoothing constraints, by combining a fully Bayesian inference and better consideration of model uncertainty to avoid overfitting. Second, the posterior PDF for the underground structure is also obtained during the fault slip estimation, which can be used as priors for the estimation of slip distribution for recurring events. The estimated slip distribution obtained using BMMFSE agreed better with the distribution of deep tectonic tremors at the down‐dip side of the main rupture area than those based on stronger prior constraints when the corresponding Coulomb failure stress changes are compared. This finding suggests a mechanical relationship between the L‐SSE and the synchronized tremors. The use of the posterior PDF of the underground structure estimated for the 2010 L‐SSE as prior PDF for the 2018 event resulted in more consistent estimation with the data, indicated by a smaller value of an information criterion.

Machine learning‐accelerated gradient‐based Markov chain Monte Carlo inversion applied to electrical resistivity tomography

ABSTRACTExpensive forward model evaluations and the curse of dimensionality usually hinder applications of Markov chain Monte Carlo algorithms to geophysical inverse problems. Another challenge of these methods is related to the definition of an appropriate proposal distribution that simultaneously should be inexpensive to manipulate and a good approximation of the posterior density. Here we present a gradient‐based Markov chain Monte Carlo inversion algorithm that is applied to cast the electrical resistivity tomography into a probabilistic framework. The sampling is accelerated by exploiting the Hessian and gradient information of the negative log‐posterior to define a proposal that is a local, Gaussian approximation of the target posterior probability. On the one hand, the computing time to run the many forward evaluations needed for both the data likelihood evaluation and the Hessian and gradient computation is decreased by training a residual neural network to predict the forward mapping between the resistivity model and the apparent resistivity value. On the other hand, the curse of dimensionality issue and the computational effort related to the Hessian and gradient manipulation are decreased by compressing data and model spaces through a discrete cosine transform. A non‐parametric distribution is assumed as the prior probability density function. The method is first demonstrated on synthetic data and then applied to field measurements. The outcomes provided by the presented approach are also benchmarked against those obtained when a computationally expensive finite‐element code is employed for forward modelling , with the results of a gradient‐free Markov chain Monte Carlo inversion, and also compared with the predictions of a deterministic inversion. The implemented approach not only guarantees uncertainty assessments and model predictions comparable with those achieved by more standard inversion strategies, but also drastically decreases the computational cost of the probabilistic inversion, making it similar to that of a deterministic inversion.

Empirical Bayes identification of stationary processes and approximation of Toeplitz spectra

We study Statistical Consistency of an approximate subspace identification procedure for the infinite dimensional a posteriori model of a frequency estimation problem in an Empirical Bayesian framework. By first imposing a natural uniform prior probability density on the unknown frequencies, the estimation of the hyperparameters of the a priori distribution can be accomplished by a sequence of subspace identification techniques. These techniques exploit the special structure of the covariance matrix of the a posteriori process which is discovered by making a connection with classical results on energy concentration in deterministic signal processing. The convergence of the spectrum of the subspace estimates to the (nonrational) spectral density of the a posteriori process has an analytic counterpart in the approximation of symmetric positive definite Toeplitz matrices by submatrices of finite rank. This is proven by a weak-sense convergence theorem for Toeplitz spectra.

A Quantile-Conserving Ensemble Filter Framework. Part I: Updating an Observed Variable

Abstract A general framework for deterministic univariate ensemble filtering is presented. The framework fits a continuous prior probability density function (PDF) to the prior ensemble. A functional representation for the observation likelihood is combined with the prior PDF to get a continuous analysis (posterior) PDF. Cumulative distribution functions for the prior and analysis are also required. The key innovation is that an analysis ensemble is computed so that the quantile of each ensemble member is the same as its prior quantile. Many choices for the prior PDF family and the likelihood function are described. A choice of normal prior with normal likelihood is equivalent to the ensemble adjustment Kalman filter. Some other choices for the prior include gamma, inverse gamma, beta, beta prime, lognormal, and exponential distributions. Both prior distributions and likelihoods can be defined over a set of intervals giving additional flexibility that can be used to implement methods like a Huber likelihood for observations with occasional outliers. Priors and likelihoods can also be defined as sums of distributions allowing choices like bivariate normals or kernel filters. Empirical distributions, for instance piecewise linear approximations to arbitrary PDFs and functions can be used. Another empirical choice leads to the rank histogram filter. Results here are univariate and can be used to compute increments for observed variables or marginal distributions for any variable for a reanalysis. Linear regression of increments can be used to update state variables in a serial filter to build a comprehensive data assimilation system. Part 2 will discuss other methods for extending the framework to multivariate data assimilation. Significance Statement Data assimilation is used to combine information from model forecasts with subsequent observations to obtain better estimates of the current state of the atmosphere or other parts of the Earth system. Ensemble data assimilation uses a number of forecasts to get more information about uncertainty. A new method allows much more flexibility in the assumptions that must be made when doing ensemble data assimilation. As an example, the method can be better for quantities that are bounded like the amount of an atmospheric trace pollutant.