State space models are well-suited for time series in which the evolution of variables cannot be well represented by deterministic basis functions. A key challenge in using state space models is the selection of the noise variance parameters. To better understand their impact on the model's filtering behavior, we derive the frequency response of several commonly used state space models by utilizing the connection between the Kalman smoother and regularized least squares problems. We show that the frequency response reveals distinguishing spectral features between competing state space models and explains the flattening of the log-likelihood function when the ratio of observation to disturbance variance becomes large. Using Dutch tide gauge data, we illustrate how the frequency response can be used to make informed decisions about the variance parameters, which could in turn support more reliable interpretations of time series data.

Sea surface currents are of significant importance in various scientific and maritime applications. There are several measurement techniques available to study surface currents, however, they have limitations in spatial coverage and resolution. This study presents a proof-of-concept for a new measurement principle that relies on the difference between a ship's speed relative to water and land. The approach involves estimating the ship speed vector relative to water from optical satellite imagery of Kelvin wakes. This ship speed vector is subtracted from the ship speed over ground, which is determined from Automatic Identification System (AIS) data, to estimate the surface current. A case study in the Strait of Gibraltar was performed using two months of Sentinel-2 imagery, which yielded 81 visible Kelvin wakes over 25 images. Surface currents were estimated in directions parallel and perpendicular to the ship's sailing line for each Kelvin wake. The estimated currents were validated with respect to surface currents derived from High-Frequency Radars (HFRs) and modelled currents from the Copernicus Marine Environmental Monitoring Service (CMEMS). The uncertainty in the two surface current components was estimated using triple collocation. After removing 12 data points with large ship course variability, standard deviations of 0.14 and 0.16 m s⁻¹ were estimated for the surface currents along and across the sailing line, respectively. Despite limitations in measurement frequency due to satellite revisit times, cloud cover and Kelvin wake visibility, this new method can provide accurate estimates of sea surface currents in regions with high vessel density.