Reach Scale Hydrology
Inverse Streamflow Routing
Inverse Streamflow Routing (ISR) tries to estimate the spatially distributed runoff and its temporal evolution from streamflow observations over discrete gauging locations. A most simple example would be to watch a flood wave passing by a river and to estimate where/when the storm water could come from, given our knowledge on how water flows (e.g. from where to where and how fast).
How it works
The streamflow --> runoff estimation problem is framed as the inverse of the well studied runoff --> streamflow estimation problem. The latter one (a forward model) is referred as "streamflow routing" and we use river routing models to do that. For an easy inversion, we look for a simple and linear forward routing model only, i.e. a 1-D diffusive wave model. Details can be found in Pan and Wood (2013). There are two major steps:
(1) Wrap the entire diffusive wave routing model into one (gigantic) linear operator H such that we can write the routing process in the form y = Hx, where x is runoff state vector and y is the streamflow output vector at time t. Given that the streamflow is the integrated runoff arriving at the gauging location from the entire contribution area and time, x, y, and H need to be augmented in time.
(2) Invert the linear operator using a fixed-interval Kalman Smoother (an augmented filter). Here the observation error covariance is usually set to zero and that forces the inverted runoff to exactly reproduce the in observed streamflow at gauging locations. This zero observation error essentially creates a Constrained Kalman Filter (CKF) (click for more details) that produces the exact reproduction of observed streamflow.
Algorithmic considerations for the ISR implementation
ISR is implemented through a fixed-interval Kalman smoother and the smoothing is performed window by window in time (interval by interval). However, streamflow calculation will always lag behind runoff by the maximum travel time k of the basin. Thus a smoothing window (fixed interval) can only completely update the runoff in that window except the last k steps because the runoff water in that time range hasn't fully reached the most downstream (outlet) and future streamflow data is needed to completely update it. So, the next smoothing window can't start immediately after the current one and it needs to be rewound by k steps (i.e. overlapped by k steps) to make sure these k steps will be completely updated.
What ISR actually does and its unique benefits
It's a lot of hassles to do ISR. In plain but less precise words, it distributes streamflow water observed at the gauge back across its contributing area/time. It essentially propagates information observed at the gauges back in time and space until the very beginning of routing process - runoff generated at pixel level. As a data assimilation (DA) technique, it offers many unique features:
It's a complete propagation of observed information in space and time;
Fully coupled assimilation across all gauges - all observations within the reachable space and reachable time are assimilated together and the runoff estimates reflect the collective influence of all gauge data;
Compared to statistical DA, ISR builds in all our physical knowledge about the routing process in the form of a routing model, including the how flood wave travels in the channel and the river network geometry/topology.
The main ISR paper:
Pan, M.. and E. F. Wood, 2013: Inverse Streamflow Routing. Hydrol. Earth Syst. Sci., 17, 4577-4588, https://doi.org/10.5194/hess-17-4577-2013.
Some of the application papers are as follows:
Fisher, C. K., M. Pan, and E. F. Wood, 2020: Spatiotemporal Assimilation/Interpolation of Discharge Records through Inverse Streamflow Routing. Hydrol. Earth Syst. Sci., https://doi.org/10.5194/hess-24-293-2020.
Yang, Y., P. Lin, C. K. Fisher, M. Turmon, J. Hobbs, C. M. Emery, J. T. Reager, C. H. David, H. Lu, K. Yang, Y. Hong, E. F. Wood, and M. Pan, 2019: Enhancing SWOT Discharge Interpolation through Spatio-temporal Correlations. Remote Sensing of Environment, https://doi.org/10.1016/j.rse.2019.111450.