The ensemble transform Schmidt–Kalman filter: A novel method to compensate for observation uncertainty due to unresolved scales

Abstract

Data assimilation is a mathematical technique that uses observations to improve model predictions through consideration of their respective uncertainties. Observation error due to unresolved scales occurs when there is a difference in scales observed and modeled. To obtain an optimal estimate through data assimilation, the error due to unresolved scales must be accounted for in the algorithm. In this work, we derive a novel ensemble transform formulation of the Schmidt–Kalman filter (ETSKF) to compensate for observation uncertainty due to unresolved scales in nonlinear dynamical systems. The ETSKF represents the small-scale variability through an ensemble sampled from the representation error covariance. This small-scale ensemble is added to the large-scale forecast ensemble to obtain an ensemble representative of all scales resolved by the observations. We illustrate our new method using a simple nonlinear system of ordinary differential equations with two timescales known as the swinging spring (or elastic pendulum). In this simple system, our novel method performs similarly to another method of compensating for uncertainty due to unresolved scales. Indeed, the use of small-scale ensemble statistics has potential as a new approach to compensate for uncertainty due to unresolved scales in nonlinear dynamical systems but will need further testing using more complicated systems.