In precision agriculture, monitoring of soil moisture plays an essential role in correct decision making. In practice, regular mesh installation, or large random deployment of moisture sensors over a large field is not possible due to cost and maintenance prohibitions. Consequently, direct measurement of moisture is possible at only a few points in the field. A value for the moisture may then be estimated for the remaining areas using a variety of algorithms.

It is shown that although soil moisture varies spatially, the values are typically spatially correlated. Consequently, they have a sparse representation in the frequency domain. For such signals, compressive sensing (CS) has proven to be an effective tool in estimating the missing variable values, from sensed values.

CS theory is based on a l0-norm optimization problem which is non-deterministic polynomial-time (NP) hard problem and requires an exhaustive search over all possible locations of the nonzero entries in the corresponding sparse signal. For most real-life applications, this optimization translates into a very large-scale problem which takes substantial time and computing resources to solve. This is usually circumvented by instead using an approximation of the l0-norm.

The l0-norm presents two challenges when incorporated into an optimization problem. It is both non-smooth and non-linear.  Smooth approximations of the zero norm exist in various linear and non-linear forms. The nature of each approximation makes it more apt for a different type of application (with respect to size of the problem, nonconvexity of the original problem, and the requisite computational speed). In this paper, some different approximations of the zero norm are compared to determine which type is more suited to soil moisture application problems.

The data set that is used for numerical experiments is described. It is extracted from the simulation of a simple field using the state-of-the-art TIN-based Real-time Integrate Basin Simulator (tRIBS). The problem is then solved for different approximations of l0-norm and a detailed comparative study is presented.