Development of Genetic Algorithm based Hidden Markov Model for Root Zone Soil Moisture Predictions

1. Introduction

Root zone soil moisture plays the key role for many hydro-climate processes. Accurate root zone soil moisture data are necessary for efficient water resources management. Many hydrological models such as Soil-Plant-Atmosphere-Water (SWAP) [van Dam et al., 1997], Community Land Model (CLM) [Oleson et al., 2004], Variable Infiltration Capacity (VIC) Macroscale Hydrologic Model [Liang et al., 1996], etc., have been developed for simulating root zone soil moisture dynamics. Furthermore, several other approaches were developed for soil moisture estimations. A sequential assimilation approach was developed [Mahfouf, 1990] and improved [Bouttier et al., 1993a,b] for estimating soil moisture dynamics using atmospheric temperature and relative humidity. However, these schemes have limitations due to the number and complexity of required input parameters in modeling.

In this study, we explored a multivariate framework for predicting root zone soil moisture dynamics with a time series of spatially-distributed rainfall across multiple weather locations under two different hydro-climatic regions. The objective of our research was to develop a genetic algorithm-based hidden Markov model (HMMGA) for predicting root zone soil moisture dynamics using only rainfall and (limited) historical soil moisture measurements at multiple weather stations. Rainfall occurrence probabilities were also reproduced and analyzed.

2. Methods and Materials

            Our proposed approach includes three steps; 1) development of a genetic algorithm based hidden Markov model (HMMGA) for deriving the optimal hidden state sequence (representing daily land surface wetness) using rainfall data, 2) estimations of clustered historical root zone soil moisture statistics corresponding to the land surface wetness, and 3) predictions of root zone soil moisture dynamics using the optimal hidden state sequence and clustered soil moisture statistics (Fig. 1).

2.1 Prediction of root zone soil moisture based on HMMGA

            A hidden Markov model (HMM) is a statistical Markov model (Eq. 1) in which the probability of daily precipitation occurrence is conditioned on a number of hidden states (representing the weather condition) [Kirshner, 2005]. A genetic algorithm was integrated with HMM for searching the optimized initial hidden states (k^*). Based on the optimized parameters (k^*), the HMM can train the HMM parameter set (Q, Eq. 2) for better estimating the optimal hidden state sequence (indicating the land surface wetness comprised of the four-hidden states; state 1-wet condition, state 2-relatively wet condition, state 3-relatively dry condition, and state 4-dry condition) with precipitation data. Genetic algorithms are well known in optimization approach literature. The readers are referred to Ines and Mohanty, [2008] and Shin et al., [2012].

P(O_t|S_1:t, O_1:t-1) = P(O_t|S_t)                                                                                (1)

Q={A, k^*}={ A, B, p_i}                                                                                                (2)

where, O_t: the rainfall occurrence sequences, S_t: the hidden states, A: the transition probability matrix (a_i,j) on row state i and column state j, B: the observation probability, p_i: the first state probability (t=1) on row state i, and t: the time, respectively. 

We categorized historical root zone soil moisture data measured at the soil depths (0-5, 0-10, 10-30, 30-50, and 0-50 cm) based on the four-hidden states with the K-mean clustering algorithm [Hartigan and Wong, 1979] and calculated the statistics of categorized soil moisture data. Using the optimal hidden state sequence and clustered soil moisture statistics, root zone soil moisture dynamics were predicted. The Oklahoma (1995-2009, 130 km x 130 km) and Illinois (1994-2010, 300 km x 500 km) domains were selected for validating this approach. Each domain has seven weather stations.

3. Results and discussions

            The estimated optimal hidden state sequences (R²: 0.945 and 0.912 for the Oklahoma and Illinois domains) identified well with the observed rainfall occurrence probabilities. The yearly reproduced rainfall occurrence probabilities (R²: 0.460~0.676, RMSE: 0.074~0.084) at the weather stations for the Oklahoma domain matched the observations, but the Illinois domain had uncertainties (R²: -0.383~0.436, RMSE: 0.063~0.084). It indicated that the HMMGA processes are considerably affected by the distance between the weather stations in the study domain. The averaged rainfall occurrence predictions (R²: 0.784 and RMSE: 0.012 for the Oklahoma domain, R²: 0.962 and RMSE: 0.006 for the Illinois domain) during the simulation period across the weather stations were more identifiable with the observations. This result showed that the HMMGA performs better across the multiple weather stations than daily predictions.

Our proposed approach estimated well the averaged soil moisture estimates (R²: 0.999 and RMSE: 0.005) at the soil depth of 0-5 cm across the weather stations for the Oklahoma domain compared to the measurements. Also, the averaged root zone soil moisture estimations (R²: 0.675~0.918 and RMSE: 0.024~0.041) at the different soil depths (0-10, 10-30, 30-50, and 0-50 cm) for the Illinois domain matched the observations with some uncertainties. Although the daily root (0-50 cm) zone soil moisture estimates (R²: 0.433 and RMSE: 0.058) for the Illinois domain have uncertainties, this approach performed well in modeling. The root zone soil moisture predictions had slightly different trends compared to the rainfall occurrence probabilities for both domains. Our findings showed that the estimated root zone soil moistures were influenced by not only rainfall, but also the land surface conditions (i.e., soil textures, vegetation covers, topography, etc.). Thus, this approach estimated well root zone soil moisture dynamics with precipitation and historical root zone soil moisture data. The calibrated (R²: 0.877 and RMSE: 0.040) and validated (R²: 0.310 and RMSE: 0.051) results showed the robustness of our proposed approach for the root zone soil moisture estimations as shown in Fig. 2.

4. Conclusions

This result demonstrated that the newly developed approach can estimate well the rainfall occurrence probabilities and root zone soil moisture dynamics at the multiple weather locations under different hydro-climate regions. It indicated that our approach can predict root zone soil moisture dynamics for the future with climate change scenarios (i.e., global climate model-GCM, regional climate model-RCM, etc.) and limited historical root zone soil moisture. Also, our proposed approach can be applied for scaling down remotely sensed (RS) precipitation and soil moisture products. Thus, our approach could provide an attractive alternative for water resources management efficiently.