On representing model inadequacy in Rainfall-Runoff models Paul Smith & Keith Beven Lancaster Environment Centre, Lancaster University, UK
[email protected]
1. Introduction
3. An Example Application Initial results are presented from an example application of the methodology to the Leaf River catchment. This humid catchment is located north of Collins, Mississippi, USA and covers an area of 1944 sq. km.Three years of daily precipitation; potential evapotranspiration and discharge observations are used along the simplified HyMOD hydrological model. HyMOD, along with the Leaf River data set has been used by several authors for the testing of calibration strategies [1, 6, 7, 9, 10, 11]. HyMOD is a conceptually simple hydrological model with non-linear filter component based on the rainfall excess model of Moore [5] followed by linear routing (see Figure 1).
ET
V1 αER
P
C
ER
F.V1
V2
F(c)
F.V3
Q V4
1
V3
F.V2
(1−α)ER
Rainfall - Runoff modelling; predicting river discharge from the prevailing meteorological conditions; is a key aspect of the hydrological sciences. It is of significance in many practical situations such as issuing flood warnings and water resource management. Commonly rainfall-runoff models are of a non-linear state space form with the states, representing stores within the system, evolving deterministically in response to meteorological forcing. Such models (e.g. Section 3) are often highly conceptual with parameters that are only loosely related to physically observable quantities such as soil type and depth. It is common practise to use the rainfall - runoff data to calibrate the parameter values so the hydrological model reproduces the observed rainfall-runoff data in a ‘reasonable’ way. The pattern of the residuals between the discharge proposed by the rainfall-runoff model and that observed may be complex and temporally non-stationary. The analysis of this residual pattern is made more complex by the fact that the observations of the runoff are often poor in the periods of most interest (e.g. flood events). This poster starts to explore the utilisation of a model inadequacy term [3] to improve the prediction of the next significant hydrological event. The representation of the model inadequacy (Section 2) is motivated by considering the consistency of the inadequacy of single time step evolutions; i.e. that when evolving from similar states with similar inputs the hydrological model should have similar levels of inadequacy. The calibration of the hydrological model parameters (Sections 2 and 3) can then be considered the selection of parameter sets for which the pattern of model inadequacy is most consistent (in the above terms).
K.V4
0
Figure 1: Schematic of the HyMOD hydrological model. Excess rainfall ER is generated conditional on the current moisture content of the catchment. The storage capacity distribution function F (c) is related to the maximum storage capacity (C) and a shape parameter (b). The excess rainfall is routed with respect to parameter α either through the three linear reservoirs of the quick pathway (with parameter F ), or through the single linear reservoir of the slow pathway, with parameter K. P is the precipitation and ET the evapotranspiration.
The model inadequacy function is represented using a Bayesian k-nearest neighbour methodology [2]. Let yt be the observed value at time t and yˆt be the corresponding model prediction. The two are related by
In the results shown below the prior distributions for the hydrological model parameters (Table 1) are taken from literature values. The precision parameter κ was given a reference prior P (κ) ∝ κ−1. The prior distribution of k, the number of nearest neighbours, was specified as uniform on 2,...,50. Summation over k allows a slice sampler [8] to be implemented which draws a sample from P (θ, κ |D). Values of k are then drawn from P (k |θ, κ D).
(1) (2)
Table 1: Ranges for independent uniform prior distributions of the HyMOD parameters.
2. Representation of the model inadequacy
y˜t = φtyˆt + t yt = y˜t + ηt.
The multiplicative element of the model inadequacy in (1) is based on the adaptive correction used in [4]. The two additive quantities t and ηt are presumed to be random draws from zero mean Gaussian distributions. Without further information their variances cannot be separated so instead write κ 1 1 P (yt |φt, yˆt, κ) = κ 2 (2π)− 2 exp − (yt − φtyˆt)2 . (3) 2 The gain φt is unknown and ascribed a distribution based upon prior belief and the k nearest neighbours. The k nearest neighbours are the k time steps having the same characteristic (either ‘driven’ with precipitation or ‘non-driven’, no precipitation), which have the closest initial states at the start of the time step as measured by the L2-norm of the standardised states. Let the yt,1:k = yt,i : i = 1, . . . , k and yˆt,1:k = yˆt,i : i = 1, . . . , k denote the observed and modelled values at these time steps. The evaluation of P yt yˆt, yt,1:k , yˆt,1:k , k, κ is based on the conditional probability relationship P yt, yt,1:k yˆt, yˆt,1:k , k, κ P yt yˆt, yt,1:k , yˆt,1:k , k, κ = (4) P yt,1:k yˆt,1:k , k, κ
Figure (3) shows the observed discharge data and the model output for a sample from the posterior distribution. Typically, for the samples generated, the model appears to underestimate high discharges yet respond to rapidly to small inputs (e.g. between day 100 and 150). This suggests that the formulation for the inadequacy may not result in conditioned model parameters consistent with hydrological beliefs.
Parameter C [mm] b [-] α [-] K [days−1] F [days−1]
Range 50 - 700 0.1 - 2 0.4 - 1 0.001 - 0.1 0.1 - 0.99
4. Some Initial Results Figure (2) shows summaries of the posterior distribution of the hydrological model parameters. The posterior distribution is well conditioned compared to the prior distributions, indicating significant information has been extracted from the available data. In both location and scale the posterior parameter distribution differ significantly from those derived from the same data using alternative formulations of stochastic model to describe the discrepancy between yt and yˆt (e.g. [9]).
Figure 3: Observed runoff data (circles) and a typical realisation of the modelled output from a sample of the posterior distribution (line).
5. Conclusions The conclusions drawn from these results must be tempered by the fact that this is a very preliminary, almost cursory, analysis. Subject to this warning it is possible to state: 1. The methodology outlined can be implemented, though sensitivity to implementation of the Gaussian quadrature procedure has not been fully explored; 2. Without considering the predictive capabilities of the error representation it is hard to access the utility of the technique for real life applications; 3. Greater consideration should be paid to the formulation of the prior distributions of both the model inadequacy terms and the parameters of the hydrological model; 4. Investigation of alternative distance metrics in the knearest neighbours algorithm may be desirable. These comments will form the basis of future work. References [1] D. P. Boyle, H. V. Gupta, and S. Sorooshian. Toward improved calibration of hydrologic models: Combining the strengths of manual and automatic methods. Water Resources Research, 36(12):3663–3674, 2000. [2] C. C. Holmes and N. M. Adams. A probabilistic nearest neighbour method for statistical pattern recognition. Journal Of The Royal Statistical Society Series B-Statistical Methodology, 64:295– 306, 2002. [3] M. C. Kennedy and A. O’Hagan. Bayesian calibration of computer models. Journal of the Royal Statistical Society Series B-Statistical Methodology, 63:425–450, 2001. [4] M. J. Lees, P. C. Young, S. Ferguson, K. J. Beven, and J. Burns. An adaptive flood warning scheme for the river nith at dumfries. In W. W.R. and J. Watts, editors, 2nd Inter-national Conference on River Flood Hydraulics. Wiley, 1994.
The denominator and numerator of the fraction can be evaluated in a similar fashion using, for example Z P yt,1:k yˆt,1:k = P yt,1:k yˆt,1:k , κ, φt P φt κ, yˆt,1:k , k ∂φt
[5] R. J. Moore. The Probability-Distributed Principle and Runoff Production at Point and Basin Scales. Hydrological Sciences Journal-Journal Des Sciences Hydrologiques, 30(2):273–297, 1985.
(5) so that an informative gamma distribution to assigning P φt κ, yˆt,1:k allows the integral to be evaluated using Gaussian quadrature. The posterior distribution of the hydrological model parameters θ can be computed, given the rainfall-runoff data D from P (θ, κ, k |D, k) ∝ P (D |θ, k, κ) P (θ, κ, k) (6)
[7] H. Moradkhani, S. Sorooshian, H. V. Gupta, and P. R. Houser. Dual state-parameter estimation of hydrological models using ensemble Kalman filter. Advances in Water Resources, 28(2):135–147, 2005.
[6] H. Moradkhani, K. L. Hsu, H. Gupta, and S. Sorooshian. Uncertainty assessment of hydrologic model states and parameters: Sequential data assimilation using the particle filter. Water Resources Research, 41(5):art. no.–W05012, 2005.
[8] R. M. Neal. Slice sampling. Annals Of Statistics, 31(3):705–741, June 2003. [9] J. A. Vrugt, H. V. Gupta, L. A. Bastidas, W. Bouten, and S. Sorooshian. Effective and efficient algorithm for multiobjective optimization of hydrologic models. Water Resources Research, 39(8):art. no.–1214, 2003. [10] T. Wagener, D. P. Boyle, M. J. Lees, H. S. Wheater, H. V. Gupta, and S. Sorooshian. A framework for development and application of hydrological models. Hydrology and Earth System Sciences, 5(1):13–26, 2001. [11] T. Wagener, N. McIntyre, M. J. Lees, H. S. Wheater, and H. V. Gupta. Towards reduced uncertainty in conceptual rainfall-runoff modelling: Dynamic identifiability analysis. Hydrological Processes, 17(2):455–476, 2003.
where P (D |θ, κ, k) =
n Y
P yt yˆt, yt,1:k , yˆt,1:k , κ .
(7)
t=1
The expression for P (D |θ, k, D) takes the form of a crossvalidation.
Uncertainty in Computer Models, Sheffield, July 2010
Figure 2: Summaries of the posterior distributions for the parameters of HyMOD (θ) and κ the precision of the additive noise.
Acknowledgements The first author is funded by the EU as part of IMPRINTS FP7 project.
