A Reduced‐Order Successive Linear Estimator for Geostatistical Inversion and its Application in Hydraulic Tomography

Zhang

et al. 2019

Vadose Zone Journal

Self Cite

Groundwater level data require EnRML for better interpretation. Different methods perform similarly when assimilating surface soil moisture data. Soil water pressure head data are the most valuable in terms of parameter estimation. MCMC performs well in homogeneous soil but degrades in heterogeneous soil. The EnKF method relies more on a relatively large number of ensembles than does EnRML. In the past few decades, different data assimilation methods have been proposed to estimate soil parameters. It is not clear whether a straightforward sampling approach is sufficient or whether a linear filter or even an advanced nonlinear filter is needed to interpret the potential information carried by different data types (e.g., pressure head and water content data from multiple depths, easily available surface soil moisture data, and groundwater level data). In this study, three classical data assimilation methods, i.e., the ensemble Kalman filter (EnKF), the ensemble randomized maximum likelihood filter (EnRML), and the Markov chain Monte Carlo (MCMC), were investigated numerically in terms of the utility to cope with three different types of observations. Results show that, compared with the EnKF approach, EnRML is a superior method to extract the parameter information from observations. The MCMC approach performs well in homogeneous soil but not in heterogeneous soil. Regardless of the data assimilation methods and the soil heterogeneity, point‐scale soil water pressure head data are the most valuable in terms of soil parameter estimation, followed by groundwater level data, which require a nonlinear filter to interpret. A smaller observation error for groundwater level data leads to obviously improved parameter estimates by using EnRML with a slight improvement by using EnKF. The stable performance of the EnKF method relies more heavily on a relatively large number of ensembles than the EnRML method, whereas only a few ensemble members are needed for EnRML in a homogeneous soil column.

“…For the heterogeneous soil column, the logarithmic parameters were assumed to be second‐order stationary with a covariance function defined by the exponential form:

C_{Y} (h) = {normalσ}_{Y}^{2} \exp (- \frac{| h |}{λ}) = {normalσ}_{Y}^{2} \exp (- \frac{| z_{1} - z_{2} |}{λ})

Section: Methodsmentioning

confidence: 99%

Investigation of Data Assimilation Methods for Soil Parameter Estimation with Different Types of Data

Zhang

et al. 2019

Vadose Zone Journal

Self Cite

Water Resources Research

“…This update is to reflect the improvement (i.e., reduction of the uncertainty) of the estimate due to the inclusion of the measured head data. Note that at iteration r = 0,

R_{ff}^{(0)}

( N × N ) is the unconditional covariance of parameters, which could be characterized by an exponential spatial covariance function (or other covariance functions as well, see Zha et al, ):

R_{ff} ((),, P, P^{'}) = σ_{f}^{2} exp ([], - \sqrt{{(x - x^{'})}^{2} / λ_{x}^{2} + {(z - z^{'})}^{2} / λ_{z}^{2}})

That implies that the correlation relationship between the K s at the point P ( x , z ) and that at the point P ′( x ′, z ′) decreases as the separation distance between the two points normalized by the correlation length increases, in which the variance of ln K s is denoted by

σ_{f}^{2}

. The parameters, λ x and λ z are the correlation scales in x and z directions tabulated in Table , respectively.…”

Section: Data Collection and Ifamentioning

confidence: 99%

“…The responses at these locations are then simultaneously analyzed through a stochastic estimation model-SLE, which is briefly described below. For details, please refer to Yeh and Liu (2000), Xiang et al (2009), Zha et al (2018, and Zhu and Yeh (2005).…”

Section: Hydraulic Tomographymentioning

confidence: 99%

“…This update is to reflect the improvement (i.e., reduction of the uncertainty) of the estimate due to the inclusion of the measured head data. Note that at iteration r = 0, R 0 ð Þ ff (N × N) is the unconditional covariance of parameters, which could be characterized by an exponential spatial covariance function (or other covariance functions as well, see Zha et al, 2018):…”

Section: Hydraulic Tomographymentioning

confidence: 99%

See 1 more Smart Citation

Fusion of Hydraulic Tomography and Displacement Back Analysis for Underground Cavern Stability Investigation

Gao

Yeh

et al. 2018

Self Cite

This paper first elucidates the importance of hydraulic heterogeneity of a rock mass on the design of an underground water‐sealed storage cavern. It then introduces an information fusion approach that integrates hydraulic tomography, displacement back analysis, and a stochastic successive linear estimator to map spatially varying hydraulic conductivity (Ks), Young's modulus (E′), cohesion (c′), and internal friction angle (ϕ′). This algorithm yields statistically unbiased estimate of the Ks, E′, c′, and ϕ′ fields and their uncertainty due to insufficient samples of their heterogeneity. Afterward, this uncertainty is translated into the probability of failure distribution around the cavern for the risk assessment. Results show that heterogeneity of Ks plays an important role in the deformation and failure of the cavern. They also show that the proposed approach can reveal more detailed parameter distributions than kriging approach, based on a large number of samples of the parameters before excavations. Further, the predicted probability of failure distribution based on the proposed approach can effectively reflect the actual failure locations.

“…Thus, for better management, conservation and restoration of riverine ecosystems, it is essential to accurately characterize the heterogeneous streambed (Cardenas 2015;Jiménez et al 2015). Due to difficulties in measuring the hydraulic conductivity directly, there are increasing interests in applying inverse modeling methods to estimate it from indirect measurements of state variables in groundwater hydrology (Zhu et al 2017;Lan et al 2018;Liao et al 2018;Zha et al 2018). …”

Section: Introductionmentioning

confidence: 99%

Bayesian Monitoring Design for Streambed Heat Tracing: Numerical Simulation and Sandbox Experiments

Zhang

et al. 2018

Groundwater

Heat tracing methods have been widely employed for subsurface characterization. Nevertheless, there were very few studies regarding the optimal monitoring design for heat tracing in heterogeneous streambeds. In this study, we addressed this issue by proposing an efficient optimal design framework to collect the most informative diurnal temperature signal for Bayesian estimation of streambed hydraulic conductivities. The data worth (DW) was measured by the expected relative entropy between the prior and posterior distributions of the conductivity field. An adaptively refined Gaussian process surrogate was employed to alleviate the computational burden, resulting in at least three orders of magnitude of speed-up. The applicability of the optimal experimental design framework was evaluated by both numerical and sandbox experimental cases. Results showed that the most informative locations centered in the transition zones among the main patterns of the hydraulic conductivity field, while the most informative times centered in a short period after the minimum/maximum temperature appeared. With the fixed number of measurements, extending the calibration period was more beneficial than increasing the monitoring frequency in improving the estimation results. To our best knowledge, this work is the first study on Bayesian monitoring design for streambed characterization with the heat tracing method. The method and results can provide guidance on selecting monitoring strategies under budget-limited conditions.