## Modeling

**Hydrological Modeling**

The hydrological modelling uses MOBIDIC, a fully-distributed physics-based hydrologic model. MOBIDIC was originally developed by Fabio Castelli at the University of Florence in Italy and has undergone recent improvements at MIT. It simultaneously solves mass and energy balance, and provides simulations of soil moisture at hourly intervals over large spatial areas. Calibration and validation of the model is done against available observations of stream flow and soil moisture, while keeping model parameters within realistic values.

**Landscape Simulation**

To analyse the spatial spatial aggregation techniques for scaling up measurements from a heterogeneous landscape to km scale SMAP pixels a physics based radar landscape simulator was developed.

## Control Systems

The sensor placement problem can be stated as determining the optimal locations of a set of available sensors given the temporal and spatial statistics on soil moisture in the area of interest, the measurement models of ground sensors, the physical limitations of the sensors’ hardware, and the sensors’ energy constraints. The optimal (or near-optimal) solution is the one that minimizes an objective function that consists of the error associated with the estimate of the true mean of moisture in the area, and the cost of communication associated with the structure and configuration of the sensor placement.

This problem has been studied for a variety of application scenarios; examples include underwater sensing (Shastri and Diwekar 2006; Berry et al. 2006), structural fault detection (Worden and Burrows 2001), and detecting landslides (Terzis et al. 2006). Terzis et al. (2006) considered a 3-D underground sensor placement problem to detect movement of the soil to predict a landslide. However, the main concern there is to figure out the change in location of these sensors, rather than the mean of the field.Sensor placement studies have used a variety of objective functions. One commonly used objective is to provide coverage and connectivity, often assuming certain geometric properties on a single sensor’s coverage area; see for example Bai et al. (2006), Dhillon and Chakrabarty (2003), and Kar and Banerjee (2003). This objective is very different from our objective of minimizing the error in the estimation of the true mean of the sensor field.

Another commonly used approach is to build a model for the underlying process of interest and optimize certain utility functions, including information theoretic criteria like mutual information and entropy; see for instance Krause et al. (2006). This approach belongs to the broader theory of experimental design. Details on this literature may be found in a thorough survey by Krause el al. (2007) and the references therein. While correlated, information theoretic objective functions are also different from ours; specifically, a mutual-information maximizing sensor deployment may not minimize the error in the estimation of the true mean.

Most of the work on sensor placement has not considered the cost of communication associated with the placement. A notable exception is the work of Krause et al. (2006), where any possible sensor placement is assigned a “sensing quality” and a communication cost. The philosophy in our problem is similar to that of Krause et al. (2006). However, the sensing quality associated with any sensor placement in our problem is the error of the estimate of the true mean of soil moisture in the area of interest; in Krause et al. (2006), the sensing quality is quantified by an information theoretic measure. This difference in the sensing quality assessment has significant implications on the solution methodology and computational approaches that we will use to solve our sensor placement problem.

To effectively address our sensor placement problem, we first need to discretize the continuous sensing field into a finite number of regions, each corresponding to a possible location to place a sensor. The resulting problem is highly challenging when the number N of possible sensor locations and the number M of available sensors (M<<N) are large, and a brute-force enumeration method is not an option. It is also non-trivial due to the special statistical features of soil moisture data, which are state dependent and dynamic over time.

To be able to formulate the aforementioned optimization problem, one approach is to derive the steady-state statistics of soil moisture using probability distribution of the surface state. Alternatively, we can try to formulate and solve individual optimization problems, one for each surface state (i.e., with steady-state soil moisture statistics given that state). These result in potentially different sensor placement solutions, and will need to be combined, e.g., through some type of weighted average using surface state distribution. In this project we will explore both approaches.