Water Resources Research
Supporting Information for
A Bayesian Approach to Improved Calibration and Prediction of Groundwater Models With
Structural Error
T. Xu, A. J. Valocchi
Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, USA
Contents of this file
Text S1
Figure S1- S4
Text S1. Generation of recharge and evapotranspiration rates in the synthetic case study
The generation of the spatiotemporally varying recharge rate in the virtual reality and the simple model
(Section 3.1) is described with more details here. First, annual recharge rates for 20 years are generated
from a first order autoregressive AR(1) model, with a long-term mean of 0.1 m/year. Second, the annual
recharge rates were distributed to every month using a fixed set of monthly multiplier. For the recharge
rate used in the virtual reality, a spatially varying factor field was simulated for each month, using
SGeMS, with a mean of 1, a variance of 1 and an isotropic spherical variogram with range equal to 1 km.
The spatial varying recharge field for each month was calculated by multiplying the factor field with the
recharge rate of that month. For the recharge rate in the simple model, on the other hand, the spatial factor
fields were first contaminated with noise, sampled at five virtual climate stations, and then extrapolated
throughout the whole domain to obtain "smoothed" factor fields. This introduces input error induced by,
e.g. limited sampling locations for precipitation, coarse resolution for soil type map. The new factor fields
were then multiplied by the monthly recharge rates to calculate the spatiotemporally varying recharge in
the simplified model. The EVT fields were generated in a similar way.
Figure S1. Up: Drawdown calibration error of the standard LSR method plotted versus drawdown
computed by the LSR calibrated model. Bottom: Drawdown calibration error of the Bayesian
approach plotted versus posterior mean of the Bayesian approach. The Bayesian calibration error
is calculated as the difference between calibration targets and Bayesian posterior mean. It can be
seen that the Bayesian approach resulted in error with smaller magnitude and more evenly spread
around 0.
Figure S2. Stream gain-and-loss (𝚫𝑸) calibration error time series of the standard LSR method
(upper) and the Bayesian approach (bottom). The Bayesian calibration error is calculated as the
difference between calibration targets and Bayesian posterior mean. An increasing trend can be
observed from the LSR calibration error. The GP posterior mean captured this trend; therefore the
remaining residual is more evenly distributed around 0.
Figure S3. Autocorrelation function (ACF) of calibration error of the conventional LSR (blue) and
the Bayesian approach (red). The dash-dotted lines enclose 95% confidence interval that the true
correlations were 0. One lag equals three month as drawdowns and stream gain-and-loss are
observed quarterly. Strong temporal correlation can be observed for LSR calibration error within
the time span of one year, leading the use of a diagonal error covariance matrix dubious. On the
other hand, the Bayesian residual has significantly weaker temporal correlation, mostly falling
within the 95% confidence bound. This is because the GP error model captures the correlation
structure in model structural error.
Figure S4. Correlation coefficients among the calibration error for the seven drawdown locations
and stream gain-and-loss, resulted from the conventional LSR (a) and the Bayesian approach (b).
Strong correlation can be observed among drawdown locations and between drawdowns and
stream gain-and-loss. Similarly with temporal correlation, the presence of such correlation makes
the use of a diagonal error covariance matrix dubious. For the Bayesian calibration error, the
correlation among calibration targets is significantly smaller, within the range of (-0.5, 0.5),
indicating that the GP error model indeed captures the correlation structure in model structural
error and renders nearly white-noise remnant error.