Applications of TLS and related methods in the environmental sciences

Rainfall-Runoff and Signal Separation Problems: The process of converting rainfall into runoff is a highly nonlinear problem due to the soil-water interaction that starts when rainfall reaches the ground. Additional variables to consider are evaporation, transpiration, losses due to vegetation and land use, and the different flow processes that take place in a watershed. For instance, baseflow is a much slower process than groundwater and surface flow. Given records of rainfall and runoff data, one can build an accurate state-space model such as xk+1 = Axk + Buk + wk yk = Cxk + Duk + vk, where at time k, uk, yk, and xk are, respectively, the rainfall, runoff, and the state of the system. Such models have been used in real-time forecasting scenarios for flood control purposes [12]. However, the above model does not take into account the nonlinearities of the rainfall-runoff process. Most lumped rainfall-runoff models separate the baseflow and groundwater components from the measured runoff hydrograph in an attempt to model these as linear hydrologic reservoir units. Similarly, rainfall losses due to infiltration as well as other abstractions are separated from the measured rainfall hyetograph, which are then used as inputs to the linear hydrologic reservoir units. This data pre-processing is in essence a nonlinear signal separation problem that separates rainfall into infiltration and excess rainfall, and the measured hydrograph into surface flow and groundwater flow. These are then used to build separate linear models such as x g k+1 = Agx g k + Bgu g k y g k = Cgx g k + Dgu g k, xsk+1 = Asx s k + Bsu s k y k = Csx s k + Dsu s k, where uk = u g k + u s k yk = y g k + y s k. In the separation process, a TLS approach is used since the infiltration process is an exponential signal. Thus, the classical NMR fitting techniques [2, 6, ?, 17] are used. Physical Parameter Extraction Problems: When modeling physical processes such as infiltration, where water flows into different compartments, one is faced with a physical parameter extraction problem. This is quite evident in black-box system identification where an unknown similarity transformation matrix destroys the physical meaning of the problem. Here we show that such similarity trsnsformation can be recovered as a post identification TLS problem. That is, suppose the identified state-space system matrices are {Ā, B̄, C̄, D̄}, while the physical parameter matrices are those of a mass-spring-damper system with mass m, spring constant k, and damping coefficient b. The table below shows the parameter matrices. 1 Physical Model Identified Model A = [ 0 1 − k m − b m ]