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Two-Sweep Discharge Interpolation based on Inverse Streamflow Routing

Discharge Interpolation

Discharge, a.k.a, streamflow, fundamentally differs from other geophysical variables because it's value at a gauging point reflects the amount of water integrated over its entire contributing basin and contributing time. In other words, it's not a locally defined quantity and has no spatial continuity. As neighboring values provide no direct information about the discharge at a point, regular interpolation methods (all dependent on spatial continuity) do not work. To interpolate discharge across the entire river network, one has to resolve the relationship between discharges at any two points on the network, which essentially translates to the need to resolve the spatially distributed runoff everywhere in the basin. Here the Inverse Streamflow Routing (ISR), the only existing techniques to infer spatially distributed runoff, is used to perform discharge interpolation in two steps: (1) runoff inversion and (2) discharge reconstruction.

How it works

The method tries to propagate the observed discharge information across all reachable parts of the river network (up/downstream from gauging point) and all reachable times (before/after observation time) using a two-sweep procedure that first propagates information backward in time to the furthest possible upstream (inverse routing) and then propagates it forward in time to the furthest possible downstream (forward routing). Figure below provides a detailed illustration of the proposed scheme. The first sweep of the procedure, known as the Inverse Streamflow Routing (ISR) to generate spatially distributed runoff fields, plays the key role here (left side of Figure). The ISR helps to guarantee an exhaustive propagation of observed information by updating the boundary influx (runoff) at pixel level (the furthest possible upstream) throughout the entire spatial and temporal domains. The second sweep simply re-runs the same routing model forward using the runoff fields derived from the first sweep to reconstruct continuous discharge values everywhere (right side of Figure). Since ISR does not require an initial guess of discharge from the routing model (Pan and Wood, 2013), the proposed method works for both data assimilation (if an initial guess exists) and pure interpolation of observations (without an initial guess). Since the discharge records are ultimately created by a routing model, this approach preserves all the physical consistencies embodied by the chosen routing model and its parameters such as the flow confluence relationship on the river network and the resulting mass balance, wave velocity and diffusivity (if a diffusive wave routing model is used). Such a strong physical consistency can hardly be implemented by methods based on statistical correlations between different gauging points or different state variables in the routing model, for example, the river kriging method. When used as an interpolator, the proposed method can also exactly reproduce the input observations at gauging locations/times (Pan and Wood, 2013).

Propagate information beyond basin boundaries

River network does not go beyond its own drainage basin and discharge information at neighboring basins can't directly contribute to the discharge estimation in another basin. However, ISR does allow us to infer runoff fields, whose values are spatially correlated across neighboring basins since different basins may experience a same storm. This allows us to build in spatially correlation in runoff (errors) across basin boundaries. Algorithmically, such cross-basin inference is realized through correlated first-guess runoff errors in the ISR procedure. In the study Yang et al. 2019, the runoff error correlation is a gradually decaying function of geographic distance and distance in time.


The method is an application of ISR and thus it inherits all its the benefits:

  • It's a complete propagation of observed information in space and time within the drainage basin;

  • Fully coupled interpolation/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.

Additionally, it offers:

  • Interpolate across all locations on the river network;

  • Exact reproduction of discharge values at observed gauging locations;

  • Allow spatially correlated storm information to propagate across basin boundary - much stronger gap-filling capability.

Sample code for 30 global basins

  • Download link here


Discharge interpolation papers:

Fisher, C. K., M. Pan, and E. F. Wood, 2020: Spatiotemporal Assimilation/Interpolation of Discharge Records through Inverse Streamflow Routing. Hydrol. Earth Syst. Sci.,

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,

The ISR paper:

Pan, M.. and E. F. Wood, 2013: Inverse Streamflow Routing. Hydrol. Earth Syst. Sci., 17, 4577-4588,

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