Fransdimensional Bayesian inversion of timedomain airborne EM data

To reduce the dependence of EM inversion on the choice of initial model and to obtain the global minimum, we apply transdimensional Bayesian inversion to time-domain airborne electromagnetic data. The transdimensional Bayesian inversion uses the Monte Carlo method to search the model space and yields models that simultaneously satisfy the acceptance probability and data fitting requirements. Finally, we obtain the probability distribution and uncertainty of the model parameters as well as the maximum probability. Because it is difficult to know the height of the transmitting source during flight, we consider a fixed and a variable flight height. Furthermore, we introduce weights into the prior probability density function of the resistivity and adjust the constraint strength in the inversion model by changing the weighing coefficients. This effectively solves the problem of unsatisfactory inversion results in the middle high-resistivity layer. We validate the proposed method by inverting synthetic data with 3% Gaussian noise and field survey data.

Adaptive Bayesian inversion of pore water pressures based on artificial neural network : An earth dam case study

Most earth-dam failures are mainly due to seepage,and an accurate assessment of the permeability coefficient provides an indication to avoid a disaster.Parametric uncertainties are encountered in the seepage analysis,and may be reduced by an inverse procedure that calibrates the simulation results to observations on the real system being simulated.This work proposes an adaptive Bayesian inversion method solved using artificial neural network(ANN)based Markov Chain Monte Carlo simulation.The optimized surrogate model achieves a coefficient of determination at 0.98 by ANN with 247 samples,whereby the computational workload can be greatly reduced.It is also significant to balance the accuracy and efficiency of the ANN model by adaptively updating the sample database.The enrichment samples are obtained from the posterior distribution after iteration,which allows a more accurate and rapid manner to the target posterior.The method was then applied to the hydraulic analysis of an earth dam.After calibrating the global permeability coefficient of the earth dam with the pore water pressure at the downstream unsaturated location,it was validated by the pore water pressure monitoring values at the upstream saturated location.In addition,the uncertainty in the permeability coefficient was reduced,from 0.5 to 0.05.It is shown that the provision of adequate prior information is valuable for improving the efficiency of the Bayesian inversion.

Bayesian Markov chain Monte Carlo inversion for anisotropy of PP-and PS-wave in weakly anisotropic and heterogeneous media

A single set of vertically aligned cracks embedded in a purely isotropic background may be con- sidered as a long-wavelength effective transversely iso- tropy （HTI） medium with a horizontal symmetry axis. The crack-induced HTI anisotropy can be characterized by the weakly anisotropic parameters introduced by Thomsen. The seismic scattering theory can be utilized for the inversion for the anisotropic parameters in weakly aniso- tropic and heterogeneous HTI media. Based on the seismic scattering theory, we first derived the linearized PP- and PS-wave reflection coefficients in terms of P- and S-wave impedances, density as well as three anisotropic parameters in HTI media. Then, we proposed a novel Bayesian Mar- kov chain Monte Carlo inversion method of PP- and PS- wave for six elastic and anisotropic parameters directly. Tests on synthetic azimuthal seismic data contaminated by random errors demonstrated that this method appears more accurate, anti-noise and stable owing to the usage of the constrained PS-wave compared with the standards inver- sion scheme taking only the PP-wave into account.

Geoacoustic Inversion for Bottom Parameters via Bayesian Theory in Deep Ocean

We develop a new approach to estimating bottom parameters based on the Bayesian theory in deep ocean. The solution in a Bayesian inversion is characterized by its posterior probability density （PPD）, which combines prior information about the model with information from an observed data set. Bottom parameters are sensitive to the transmission loss （TL） data in shadow zones of deep ocean. In this study, TLs of different frequencies from the South China Sea in the summer of 2014 are used as the observed data sets. The interpretation of the multidimensional PPD requires the calculation of its moments, such as the mean, covariance, and marginal distributions, which provide parameter estimates and uncertainties. Considering that the sensitivities of shallow- zone TLs vary for different frequencies of the bottom parameters in the deep ocean, this research obtains bottom parameters at varying frequencies. Then, the inversion results are compared with the sampling data and the correlations between bottom parameters are determined. Furthermore, we show the inversion results for multi- frequency combined inversion. The inversion results are verified by the experimental TLs and the numerical results, which are calculated using the inverted bottom parameters for different source depths and receiver depths at the corresponding frequency.

Complex spherical-wave elastic inversion using amplitude and phase reflection information

Unlike the real-valued plane wave reflection coefficient(PRC)at the pre-critical incident angles,the frequency-and depth-dependent spherical-wave reflection coefficient(SRC)is more accurate and always a complex value,which contains more reflection amplitude and phase information.In near field,the imaginary part of complex SRC(phase)cannot be ignored,but it is rarely considered in seismic inversion.To promote the practical application of spherical-wave seismic inversion,a novel spherical-wave inversion strategy is implemented.The complex-valued spherical-wave synthetic seismograms can be obtained by using a simple harmonic superposition model.It is assumed that geophone can only record the real part of complex-valued seismogram.The imaginary part can be further obtained by the Hilbert transform operator.We also propose the concept of complex spherical-wave elastic impedance(EI)and the complex spherical-wave EI equation.Finally,a novel complex spherical-wave EI inversion approach is proposed,which can fully use the reflection information of amplitude,phase,and frequency.With the inverted complex spherical-wave EI,the velocities and density can be further extracted.Synthetic data and field data examples show that the elastic parameters can be reasonably estimated,which illustrate the potential of our spherical-wave inversion approach in practical applications.

Calibrate complex fracture model for subsurface flow based on Bayesian formulation

In practical development of unconventional reservoirs,fracture networks are a highly conductive transport media for subsurface fluid flow.Therefore,it is crucial to clearly determine the fracture properties used in production forecast.However,it is different to calibrate the properties of fracture networks because it is an inverse problem with multi-patterns and highcomplexity of fracture distribution and inherent defect of multiplicity of solution.In this paper,in order to solve the problem,the complex fracture model is divided into two sub-systems,namely"Pattern A"and"Pattern B."In addition,the generation method is grouped into two categories.Firstly,we construct each sub-system based on the probability density function of the fracture properties.Secondly,we recombine the sub-systems into an integral complex fracture system.Based on the generation mechanism,the estimation of the complex fracture from dynamic performance and observation data can be solved as an inverse problem.In this study,the Bayesian formulation is used to quantify the uncertainty of fracture properties.To minimize observation data misfit immediately as it occurs,we optimize the updated properties by a simultaneous perturbation stochastic algorithm which requires only two measurements of the loss function.In numerical experiments,we firstly visualize that small-scale fractures significantly contribute to the flow simulation.Then,we demonstrate the suitability and effectiveness of the Bayesian formulation for calibrating the complex fracture model in the following simulation.

Conditional inversion to estimate parameters from eddy-flux observations

Aims Data assimilation is a useful tool to extract information from large datasets of the net ecosystem exchange(NEE)of CO_(2) obtained by eddy-flux measurements.However,the number of parameters in ecosystem models that can be constrained by eddy-flux data is limited by conventional inverse analysis that estimates parameter values based on one-time inversion.This study aimed to improve data assimilation to increase the number of constrained parameters.Methods In this study,we developed conditional Bayesian inversion to maximize the number of parameters to be constrained by NEE data in several steps.In each step,we conducted a Bayesian inversion to constrain parameters.The maximum likelihood estimates of the constrained parameters were then used as prior to fix parameter values in the next step of inversion.The conditional inversion was repeated until there were no more parameters that could be further constrained.We applied the conditional inversion to hourly NEE data from Harvard Forest with a physiologically based ecosystem model.Important Findings Results showed that the conventional inversion method constrained 6 of 16 parameters in the model while the conditional inversion method constrained 13 parameters after six steps.The cost function that indicates mismatch between the modeled and observed data decreased with each step of conditional Bayesian inversion.The Bayesian information criterion also decreased,suggesting reduced information loss with each step of conditional Bayesian inversion.A wavelet analysis reflected that model performance under conditional Bayesian inversion was better than that under conventional inversion at multiple time scales,except for seasonal and half-yearly scales.In addition,our analysis also demonstrated that parameter convergence in a subsequent step of the conditional inversion depended on correlations with the parameters constrained in a previous step.Overall,the conditional Bayesian inversion substantially increased the number of parameters to be constrained by NEE data and can be a powerful tool to be used in data assimilation in ecology.

Adaptive Ensemble Kalman Inversion with Statistical Linearization

The ensemble Kalman inversion(EKI),inspired by the well-known ensemble Kalman filter,is a derivative-free and parallelizable method for solving inverse problems.The method is appealing for applications in a variety of fields due to its low computational cost and simple implementation.In this paper,we propose an adaptive ensemble Kalman inversion with statistical linearization(AEKI-SL)method for solving inverse problems from a hierarchical Bayesian perspective.Specifically,by adaptively updating the unknown with an EKI and updating the hyper-parameter in the prior model,the method can improve the accuracy of the solutions to the inverse problem.To avoid semi-convergence,we employ Morozov’s discrepancy principle as a stopping criterion.Furthermore,we extend the method to simultaneous estimation of noise levels in order to reduce the randomness of artificially ensemble noise levels.The convergence of the hyper-parameter in prior model is investigated theoretically.Numerical experiments show that our proposed methods outperform the traditional EKI and EKI with statistical linearization(EKI-SL)methods.

Government environmental control measures on CO_(2) emission during the 2014 Youth Olympic Games in Nanjing:Perspectives from a top-down approach

Strict air pollution control measures were conducted during the Youth Olympic Games(YOG)period at Nanjing city and surrounding areas in August 2014.This event provides a unique chance to evaluate the effect of government control measures on regional atmospheric pollution and greenhouse gas emissions.Many previous studies have observed significant reductions of atmospheric pollution species and improvement in air quality,while no study has quantified its synergism on anthropogenic CO_(2)emissions,which can be coreduced with air pollutants.To better understand to what extent these pollution control measures have reduced anthropogenic CO_(2)emissions,we conducted atmospheric CO_(2)measurements at the suburban site in Nanjing city from 1^(st) July to 30^(th) September 2014 and 1^(st) August to 31^(st) August 2015,obvious decrease in atmospheric CO_(2)was observed between YOG and the rest period.By coupling the a prioriemission inventory with atmospheric transport model,we applied the scale factor Bayesian inversion approach to derive the posteriori CO_(2)emissions in YOG period and regular period.Results indicate CO_(2)emissions from power industry decreased by 45%,and other categories also decreased by 16%for manufacturing combusting,and 37%for non-metallic mineral production.Monthly total anthropogenic CO_(2)emissions were 9.8(±3.6)×10^(9) kg/month CO_(2) for regular period and decreased to 6.2(±1.9)×10^(9) kg/month during the YOG period in Nanjing city,with a 36.7%reduction.When scaling up to whole Jiangsu Province,anthropogenic CO_(2)emissions were 7.1(±2.4)×10^(10) kg/month CO_(2)for regular period and decreased to 4.4(±1.2)×10^(10)kg/month CO_(2) during the YOG period,yielding a 38.0%reduction.

Probabilistic Fisher discriminant analysis based on Gaussian mixture model for estimating shale oil sweet spots

The delineation of shale oil sweet spots is a crucial step in the exploration of shale oil reservoirs.A single attribute such as total organic carbon(TOC)is conventionally used to evaluate the sweet spots of shale oil.This study proposes a probabilistic Fisher discriminant approach for estimating shale oil sweet spots,in which the probabilistic method and Gaussian mixture model are incorporated.Statistical features of shale oil facies are obtained based on the well log interpretation of the samples.Several key parameters of shale oil are projected to data sets with low dimensions in each shale oil facies.Furthermore,the posterior distribution of different shale oil facies is built based on the classification of each shale oil facies.Various key physical parameters of shale oil facies are inversed by the Bayesian method,and important elastic properties are extracted from the elastic impedance inversion(EVA-DSVD method).The method proposed in this paper has been successfully used to delineate the sweet spots of shale oil reservoirs with multiple attributes from the real pre-stack seismic data sets and is validated by the well log data.

Inverse analysis of coupled carbon-nitrogen cycles against multiple datasets at ambient and elevated CO_(2)

Aims Carbon(C)sequestration in terrestrial ecosystems is strongly regulated by nitrogen(N)processes.However,key parameters that determine the degree of N regulation on terrestrial C sequestration have not been well quantified.Methods Here,we used a Bayesian probabilistic inversion approach to estimate 14 target parameters related to ecosystem C and N interactions from 19 datasets obtained from Duke Forests under ambient and elevated carbon dioxide(CO_(2)).Important FindingsOur results indicated that 8 of the 14 target parameters,such as C:N ratios in most ecosystem compartments,plant N uptake and external N input,were well constrained by available datasets whereas the others,such as N allocation coefficients,N loss and the initial value of mineral N pool were poorly constrained.Our analysis showed that elevated CO_(2)led to the increases in C:N ratios in foliage,fine roots and litter.Moreover,elevated CO_(2)stimulated plant N uptake and increased ecosystem N capital in Duke Forests by 25.2 and 8.5%,respectively.In addition,elevated CO_(2)resulted in the decrease of C exit rates(i.e.increases in C residence times)in foliage,woody biomass,structural litter and passive soil organic matter,but the increase of C exit rate in fine roots.Our results demonstrated that CO_(2)enrichment substantially altered key parameters in determining terrestrial C and N interactions,which have profound implications for model improvement and predictions of future C sequestration in terrestrial ecosystems in response to global change.

VAE-KRnet and its Applications to Variational Bayes

In this work,we have proposed a generative model,called VAE-KRnet,for density estimation or approximation,which combines the canonical variational autoencoder(VAE)with our recently developed flow-based generativemodel,called KRnet.VAE is used as a dimension reduction technique to capture the latent space,and KRnet is used to model the distribution of the latent variable.Using a linear model between the data and the latent variable,we show that VAE-KRnet can be more effective and robust than the canonical VAE.VAE-KRnet can be used as a density model to approximate either data distribution or an arbitrary probability density function(PDF)known up to a constant.VAE-KRnet is flexible in terms of dimensionality.When the number of dimensions is relatively small,KRnet can effectively approximate the distribution in terms of the original random variable.For high-dimensional cases,we may use VAE-KRnet to incorporate dimension reduction.One important application of VAE-KRnet is the variational Bayes for the approximation of the posterior distribution.The variational Bayes approaches are usually based on the minimization of the Kullback-Leibler(KL)divergence between the model and the posterior.For highdimensional distributions,it is very challenging to construct an accurate densitymodel due to the curse of dimensionality,where extra assumptions are often introduced for efficiency.For instance,the classical mean-field approach assumes mutual independence between dimensions,which often yields an underestimated variance due to oversimplification.To alleviate this issue,we include into the loss the maximization of the mutual information between the latent random variable and the original random variable,which helps keep more information from the region of low density such that the estimation of variance is improved.Numerical experiments have been presented to demonstrate the effectiveness of our model.

AN ACCELERATION STRATEGY FOR RANDOMIZE-THEN-OPTIMIZE SAMPLING VIA DEEP NEURAL NETWORKS

Randomize-then-optimize (RTO) is widely used for sampling from posterior distributions in Bayesian inverse problems. However, RTO can be computationally intensive forcomplexity problems due to repetitive evaluations of the expensive forward model and itsgradient. In this work, we present a novel goal-oriented deep neural networks (DNN) surrogate approach to substantially reduce the computation burden of RTO. In particular,we propose to drawn the training points for the DNN-surrogate from a local approximatedposterior distribution – yielding a flexible and efficient sampling algorithm that convergesto the direct RTO approach. We present a Bayesian inverse problem governed by ellipticPDEs to demonstrate the computational accuracy and efficiency of our DNN-RTO approach, which shows that DNN-RTO can significantly outperform the traditional RTO.

Experimental warming shifts coupling of carbon and nitrogen cycles in an alpine meadow

Aims Terrestrial ecosystem carbon(C)uptake is remarkably regulated by nitrogen(N)availability in the soil.However,the coupling of C and N cycles,as reflected by C:N ratios in different components,has not been well explored in response to climate change.Methods Here,we applied a data assimilation approach to assimilate 14 datasets collected from a warming experiment in an alpine meadow in China into a grassland ecosystem model.We attempted to evaluate how experimental warming affects C and N coupling as indicated by constrained parameters under ambient and warming treatments separately.Important Findings The results showed that warming increased soil N availability with decreased C:N ratio in soil labile C pool,leading to an increase in N uptake by plants.Nonetheless,C input to leaf increased more than N,leading to an increase and a decrease in the C:N ratio in leaf and root,respectively.Litter C:N ratio was decreased due to the increased N immobilization under high soil N availability or warming-accelerated decomposition of litter mass.Warming also increased C:N ratio of slow soil organic matter pool,suggesting a greater soil C sequestration potential.As most models usually use a fixed C:N ratio across different environments,the divergent shifts of C:N ratios under climate warming detected in this study could provide a useful benchmark for model parameterization and benefit models to predict C-N coupled responses to future climate change.