Kalman Filter
The Kalman filter is a digital filter and data assimilation (DA) algorithm that provides 'best' estimates of system state. It recursively estimates state variables in a noisy, linear, dynamical system by leveraging a series of measurements, in conjunction with initial state predictions from a forward model, to generate estimates of unknown variables. It requires a linear model of system state and a Gaussian-like distribution of measurement errors. Its estimates, or updates, combine a model prediction with a measurement using a weighted average. More weight is allocated to estimates that have greater certainty. The result is generation of estimates that tend to be more accurate than estimates based on a single measurement or simulation. As part of the update process, the joint probability distribution over the variables for each assimilation time are estimated. The Kalman filter is widely used in many technical and quantitative fields and can often be implemented in real time.
Dynamic Integration of AI-ML Predictions with Process-Based Model Simulations
Data assimilation (DA) is used to integrate artificial intelligence including machine learning (AI-ML) and process-based models to produce a dynamic operational water balance tool for groundwater management. The management tool is a three-step calculation. In the first step, a traditional process-based water budget model provides forward model predictions of aquifer storage from meteorological observations, estimates of pumping and diversion discharge, and estimates of recharge. A Kalman filter-based DA approach is the second step and generates updated storage volumes by combining a trained AI-ML model, providing replacement 'measurements' for missing observations, with forward model predictions. The third 'correction' step uses modified recharge and pumping, adjusted to account for the difference between Kalman update storage and forward model predicted storage, in forward model re-simulation to approximate updated storage volume. Use of modified inputs in the correction provides a mass conservative water budget framework based on AI-ML predictions. Pumping and recharge values are uncertain and unobserved in the study region and can be adjusted without contradicting measurements.