Error Propogation Case Study

This case study is based on work carried out by Faisal Hossain and Nitin Katiyar (2006) Sensitivity Analyses of Satellite Rainfall Estimation Error to Open-Book Watershed Models of Varying levels of Conceptualization and Spatial Aggregation PDF.

This study is referred to here as H&K.

Introduction

H&K state that new developments in satellite-based measurements of rainfall (for example the Global Precipitation Measurement (GPM) exercise), require better understanding of the problem of errors in run-off estimation resulting in errors in rainfall estimation; the so called error propagation problem. Given a workable solution to error propagation, satellite-based observations of rainfall could provide an enormous benefit for flood forecasting particularly in medium to large size ungauged catchments common to developing nations. H&K investigate how three types of conceptual rainfall runoff models (each relatively simple but exhibiting increasing levels of conceptual complexity) interact with a typical satellite-like rainfield. The object is to qualify how errors in the rainfall field propagate through various forms of rainfall-runoff model.

The rainfall-runoff models:

H&K use an “open-book” style rainfall runoff model (Yen & Chow 1969) comprised of 20 by 12 grid volumes to represent the Oaklahoma Mesonet region (-100degW -95degW, 37degN -34degN). A schematic of the open book model concept is shown in Figure 1.

Figure 1. open-book rainfall runoff model

The three models differ only in the way they generate rainfall excess for each grid volume. The rainfall excess from each grid cell becomes overland flow and eventually channel flow. For each model overland flow follows a standard Manning’s or Darcy-Weisbach equation; and channel flow routing follows a Manning’s equation scheme. The necessary parameters for overland and channel flow are defined from field data.

The three models are then:

Statistical model

Here the precipitation p(t) is partitioned into infiltration ap(t) and surface runoff (1-a)p(t) (where a is a model parameter and p(t) is precipitation at time t). When the soil water store reaches its maximum all precipitation is converted to surface runoff.

Linear model

This uses a linear differential water balance equation to relate cell storage to overland saturation excess flow and subsurface flow. Therefore, the linear model offers a degree conceptualisation of the "real world" process by explicity incorporating a storage reservoir and maintaining a water balance between cells.

Nonlinear model

Again like the linear model this model assumes an explicit storage mechanism but here a power law parameter is used to govern the subsurface storage to discharge relationship.

By using the above three representations for each model cell, H&K have introduced a scale of complexity and increasing mechanistic conceptualisation. This scale, ranging from a statistical model with very low parametric dimension, through a linear mechanistic representation to a nonlinear process, involves the addition of extra parameters at each stage resulting in the usual trade-off between the ease of parameter specification afforded by simple models and the ability to represent conceptual processes afforded by complex models.

Satellite rainfall error model

Estimates of rainfall fields based on satellite observations are prone to error. H&K simulated the type of error found in satellite estimates using the SREM2D error model. This was achieved by generating a reference rainfall field from a combination of radar and rainguage measurements covering a four month period at one-hourly intervals over a grid of 0.25deg resolution. This reference rainfall field was then corrupted using the SREM2 noise model to generate a new rainfall field i.e., within the simulation (described next) the reference rainfall field represents the true rainfall that fell in the simulated version of the "real world" and the corrupted version represents the best satellite based estimate of this true rainfall field.

Simulation method

• The reference rainfall field was applied to the 3 versions of the open-book rainfall runoff model and the downstream discharge collected. This discharge was the reference timeseries.

• The SREM2D noise model was then applied to the reference rainfall field 15 times to generate 15 Monte Carlo satellite-like versions of the referance rainfall.

• The 15 satellite-like rainfall fields were applied to the 3 open-book model versions.

• The 15 satellite-like rainfall fields were rescaled to 0.5deg and again applied to the 3 open-book models.

For each of the three models a measure of predictive ability and structural validity was dafined according to Equation 1; where EP is exceedance probability and UR is uncertainty ratio.

H&K emphasis the importance of considering the models structural validity as well as just the model fit to data.

Results and comments

The error bounds are largest for the statistical model particularly during peak flow events.

Aggregation of the statistical model to 0.5deg increases the uncertainty bounds but also increases the confidence in the model structure. H&K suggest that the statistical model benefits from an aggregation of rainfall data.

The inclusion of a storage reservoir in the formulation of the linear model improves the characteristics of the hydrograph although peak and low flow show similar errors to the statistical model.

Aggregation increases simulation uncertainty but does not improve the model structure validity. H&K suggest that an optimum level of disaggregation may exist beyond which no further improvement in predictive uncertainty can be attained.

For the nonlinear model, the inclusion of the nonlinear storage-discharge term increased predictive uncertainty and model structural validity. H&K suggest the increase in process complexity may not necessarily warrant improved hydraulic simulation when the scale of the application is large i.e., the model grids are larger than 0.25deg (approximately 25km).

References

• Hossain, F. and N. Katiyar(2006). Sensitivity Analyses of Satellite Rainfall Estimation Error to Open-Book Hydrologic Models of Varying Levels of Conceptualisation and Spatial Aggregation, Hydrol. Sci. Journal. (Revised and in review).

• Yen, B.C., and V.T. Chow. 1969. A laboratory study of surface runoff due to moving rainstorms, Water Resources Research. 5(5): 989-1006.