Lead Investigators:

Steve Margulis, UCLA; Jose Saez (Loyola Marymount University); Thomas Harmon (UC Merced)

Overview

The main purpose of this project is to explore the use of data assimilation (DA) methods in designing and maximizing the information content (including uncertainty) of sensor networks deployed in a complex environmental system.  The application is a test bed for wastewater reuse for irrigation in Palmdale, CA.   Based on preliminary results, the approach appears to work well in characterizing soil moisture throughout the rootzone using limited sensor measurements near the surface.

Approach

Data Assimilation methodology
The research project focuses on implementing a stochastic unsaturated flow and transport model in conjunction with the EnKF to form the basis of a real-time algorithm for ongoing state and parameter estimation at the Palmdale site.  The EnKF uses a Monte Carlo approach to describe how the conditional probability density of the state evolves over time (between measurements) and how it changes when new measurements are incorporated.  The result is an estimate based on all available model predictions and measurements that has implicitly incorporated input and measurement errors via the stochastic formulation.

In Year 1 of the project the vadose zone soil moisture and temperature models were developed to include forcing by meteorological and irrigation data from the site.  This model forms the basis for the prior estimates of the soil states that are then conditioned on sensor data.  At the Palmdale site, soil moisture sensors were set up at the depths of 20, 40, 60 and 80 cm below the surface.  The EnKF explicitly accounts for the correlation between these measurements and states throughout the domain to update soil moisture estimates on a regular (half-hourly) basis. Based on this updating framework, better state predictions (and more importantly difficult to observe subsurface flux estimates) can be expected if accurate measurements are available.

Systems/Experiments

3.1 Soil moisture estimation experiments.  In this initial phase, soil moisture is estimated spatially and temporally during a 10-day irrigation simulation under various scenarios of uncertainty (i.e. uncertainty of initial state condition, irrigation rate, time-invariant parameters, etc.). Through these synthetic experiments, we were able to examine the feasibility of estimating the distributed states using sparse measurements in conjunction with the EnKF.  The tests were performed using a synthetic truth with corrupted measurements (additive error with known error statistics).  Both an “open-loop” (no assimilated measurements) and EnKF estimate were obtained and compared to the known true states.  For brevity, results are shown only for the case with uncertainty in the initial conditions, irrigation input, and soil hydraulic parameters.

For these tests, the true values of initial soil moisture/temperature were assumed to be uniform with depth, i.e.and , the true irrigation rate is equal to 7.5 cm/hr with 10-minute duration every 6 hours, and the true van Genuchten soil hydraulic properties correspond to a loam soil.  For the estimation problem these inputs are assumed to be random variables along with the states themselves.  Both the open-loop and EnKF assume the initial conditions are normally distributed with use mean values of and soil temperature (with a specified covariance of 0.08 and 9 K2 respectively).  The irrigation rate was assumed to be normally distributed, with each replicate having a constant irrigation rate over simulation period.  To provide a realistic test, the mean irrigation rate of the ensemble is 5cm/hr (as opposed to 7.5 cm/hr for the true) with a covariance (2.5 cm/hr)2.  Hence this provides a more rigorous test to the EnKF to assess whether it can overcome biased irrigation inputs.  Soil hydraulic properties are generally highly uncertain and difficult to estimate.  To generate an ensemble of physically meaningful parameter sets, the joint distribution of the parameters must be specified.  Based on literature values, a joint distribution can be chosen to model the parameters as random variables with non-negativity constraints.  The soil for the open-loop and EnKF simulations was assumed to be different than the true soil (loamy sand).

Figure 1 shows the comparison of the true soil moisture evolution to the open-loop and EnKF ensemble means. Note that the influences of the initial condition and irrigation errors are evident.  The primary result of the erroneous soil hydraulic properties (namely a much higher saturated hydraulic conductivity) is that the irrigation error propagates to a much deeper level in the soil.  The open-loop mean soil moisture is actually smaller (more erroneous) near the surface since more moisture is transported downward.  For the EnKF, the generally high soil moisture is captured as a result of the updates.  However, the error in soil hydraulic parameters is more difficult to overcome than in the other uncertainty experiments (not shown).  As incremental updates are added to the soil near the surface, this soil moisture is quickly redistributed to deeper layers, resulting in the more striated pattern between updates.  This type of error would not necessarily be corrected if more measurements were taken and indicates the complex and persistent errors that can occur when there is parameter error/uncertainty.  To overcome these types of error, parameter estimation will likely need to be included in the framework (discussed further below). 

An example of the ensemble profile plots is shown in Figures 2.  Note that both the open-loop and EnKF ensembles have considerable uncertainty across all depths as a result of the multitude of input uncertainties considered in this experiment.  Overall estimation error results are shown in Table 1 in terms of the time-averaged open-loop and EnKF bias and root mean squared error (RMSE).  It is clear that the EnKF significantly outperforms the open-loop.  An additional major benefit of the approach is that by estimating the states from sensor data, it is possible to additionally gain significant improvements in difficult-to-measure fluxes in the system.  Results from this test (not shown for brevity) indicate an improvement of several hundred percent in the cumulative subsurface water flux toward the ground water table as a result of the improved state estimates. 

 

Table 1.   Soil moisture estimation error.

 

Figure 1 (right).   True states (top panel), and ensemble mean of open-loop and EnKF (middle and lower panels respectively).

Figure 2 (below).   The ensemble soil moisture profiles for the test case at the first update time (black line represents the true soil moisture profile; dark blue lines represent the mean of open-loop & EnKF;  dashed light blue lines represent soil moisture profiles of each replicate; and red, open circles represent measurements).

3.2 Monte Carlo version of mass transport model.  Based on the successful implementation of the soil moisture estimation experiments described above, we have begun the implementation of the nitrogen transport model.  This will not only allow for predictions of nitrate and other constituents, but also allow for the assimilation of nitrate sensor data.  The current version of the model being implemented includes mass balances for the solid and liquid phases of ammonium, the liquid phase of nitrate, and the gaseous phase of nitrogen and includes the key processes involved in nitrogen transport including advection, dispersion, and biological and physical processes in the main nitrogen pathway ().  

Preliminary open-loop ensemble simulation results at the end of a 240-hour simulation using influent concentration for both ammonia and nitrate of 20 ppm are shown in Figure 3.  For ammonia, due to nitrification, the concentration is always below the concentration of influent; but for nitrate, the concentration in the transport process can be higher than the injected concentration via the transformation from ammonia.  The large uncertainty in these results is the result of uncertainty in the irrigation rate as well as in the reaction rate constants.  It is hypothesized that assimilation of nitrate sensor measurements could lead to significant improvements in these estimates as seen in the soil moisture problem.        

Figure 3.   The concentration of nitrogen species with depth at the end of a 240-hour simulation.

Accomplishments

Through the data assimilation framework, the initial synthetic experiments under various uncertainty scenarios to estimate soil moisture have been explored.  Based on the experiment results, it appears that the performance of assimilation framework can efficiently reduce the uncertainty (compared to open-loop results) due to uncertain inputs (i.e., initial condition, irrigation rate, and even time-invariant soil parameters), and describe the time-space evolution of the soil moisture realistically.  More recently we have extended the model to include nitrogen mass transport and the initial implementation of the Monte Carlo version has begun.  The mass transport model will be incorporated into the assimilation scheme so that we may then incorporate the entire suite of sensor data (soil moisture/temperature and nitrate concentrations).

Future Directions

Initial experiments show significant promise of the approach in characterizing complex soil states by merging models and sparse sensor data.  Several tasks are necessary as part of the next stage of the research.  A brief description of future tasks is shown below:

Task 1: Online forward calibration.  The largest errors in the scenarios explored so far occurred in the when soil hydraulic properties are incorrectly specified.  Therefore, we plan to test if time-invariant soil parameters can be estimated online using the EnKF (or the ensemble Kalman smoother).  It is expected that if soil parameters can be estimated accurately, soil moisture propagation can be tracked more realistically in space and time, and uncertainty of soil states can be efficiently reduced. 

Task 2: Online measurement model calibration.  The synthetic experiments mentioned above use only a simple measurement model (direct measurements of soil moisture with additive error). In the future, a more realistic measurement model (i.e. an expression transforming direct milliVolt measurements to volumetric soil moisture) will be incorporated, and we will attempt to estimate the measurement calibration parameters online.

Task 3: Online sensor monitoring.  In task 2, the parameters estimation of each sensor is based on their most robust state in the early life of the network, i.e. we assume the measurements error structure of the sensors can be well characterized during an initial relatively stable period.  However, as the sensor network ages it would be desirable to use the estimation framework to monitor the network itself and gain potential insight into sensor degradation or failure. This type of objective has not yet been examined via data assimilation frameworks, and we will explore this problem and test its feasibility.

People

Faculty:

• Steve Margulis (UCLA Civil & Environmental Engineering)
• Jose Saez (Loyola Marymount Civil & Environmental Engineering)
• Tom Harmon (UC Merced School of Engineering)

Graduate Students:

• Che-Chuan Wu (UCLA Civil & Environmental Engineering)
• Yeonjeong Park (UCLA Civil & Environmental Engineering)