Classical and Bayesian nonlinear regression applied to hydraulic rating curve inference.

1. Classical and Bayesian nonlinear regression applied to hydraulic rating curve inference. Construction and uncertainty analysis of stage-discharge rating curves

2. Motivation and key definition • River hydrology => How much water is flowing through the rivers? • Key definition; discharge, Q Amount of water passing through a cross-section of the river each time unit

3. Key problem • Discharge is expensive. How to get a discharge time series? • Solution: Find a relationship between discharge and something that is inexpensive to measure. • Usually, that something is water level. • This job must be done over and over again: Need solid tools for finding such relationships. • Discharge measurements are uncertain => need statistical tools • Program must be easy for hydrologists to use => User friendliness in statistics?

4. Water level definitions • Stage: the height of the water level at a river site h Q h0 Datum, height=0

5. Stage-discharge relationship h Q=C(h-h0)b Q h0 Datum, height=0 Discharge,Q

6. Stage-discharge relationship • Simple physical attributes: • Q=0 for hh0 • Q(h2)>Q(h1) for h2>h1>h0 • Parametric form suggested by hydraulics (Lambie (1978) and ISO 1100/2 (1998)): Q=C(h-h0)b • Parameters may be fixed only in stage intervals - segmentation h h width Q

7. Calibration data and statistical model • n stage-discharge measurements. • Discharge is error-prone. • Statistical inference on C,b,h0 nonlinear regression • Qi=C (hi-h0)b Ei, where Ei~logN(0,2) i.i.d. noise and i{1,…,n} • qi=a+b log(hi-h0) + i, where i~N(0,2) i.i.d. • Problem: Enable hydrological engineers to estimate Q(h)=C(h-h0)b and evaluate the calibration uncertainty.

8. Problems concerning classic one segment curve fitting • Sometimes exhibits heteroscedasticity. • Sometimes there's no finite solution! • Found a set of requirements that ensures finite estimates. • In practice, broken requirements means no finite estimates. • The model can produce broken requirements for any set of stage measurements! • Parameter estimators have infinite expectancy -> Uncertainty inference becomes difficult! • Explored in paper 1, Reitan and Petersen-Øverleir (2006) and in the appendix, Reitan and Petersen-Øverleir (2005).

9. Bayesian one segment fitting • Based in the same data model, but with a prior distribution to the set of parameters. The Bayesian study of this resulted in paper 2, Reitan and Petersen-Øverleir (2008a). • Bayesian analysis of other models done by Moyeeda and Clark (2005) and Árnason (2005).

10. Pros and cons • Upsides: • Encodes hydraulic knowledge. • Can put softer ‘restrictions’ on the parameters. • Finite estimates. • Natural uncertainty measures. • Downsides: • Requires heavier numerical methods (MCMC). • Coming up with a prior distribution can be hard. • Also sometimes exhibits heteroscedasticity.

11. Input - prior distribution • Prior distribution form as simple as possible. • Prior knowledge either local or regional. • Regional knowledge can be extracted once and for all. • At-site prior knowledge can be set through asking for credibility intervals.

12. Output – estimates and uncertainties • Estimates– expectancies or medians from the posterior distribution. • Uncertainty– credibility intervals of parameters and the curve itself, Q(h)=C (h-h0)b.

13. Segmentation • Original idea: Divide the stage-discharge measurement into sets and fit Q(h)=C (h-h0)bseparately for each segment. This can fit a wider range of measurement sets. Segment 2 h Segment 1 Intersection Q

14. Problems with manual segmentation h • Uncertainty analysis of manual decisions not statistically available. • Curves fitted to two neighbouring sets may not intersect. • Two such curves may have intersections only inside the sets. Jump in the curve Q h Q

15. Statistical model for segmentation – the interpolation model • Idea: Make a model with segments and let the data be attached to that model. • Model: for k segments, introducek-1 segment limits parameters, hs,1, …, hs,k-1.For a measurement hs,j-1<hi< hs,jassumeqi=aj+bj log(hi-h0,j) +i. • Make sure there’s continuity by sacrificing one of the parameters in the upper segments (aj for j>1). • Goal: Make inference on all parameters in this model. Also, make inference on k.

16. Frequentist inference on the interpolation model • Segmented model first formulated and treated by using the maximum likelihood method in paper 3, Petersen-Øverleir and Reitan (2005). • Problems: • Possibility of infinite parameter estimates inherited from one segment model. (Much more likely than usual for upper segments.) • Multi-modality and discontinuous derivative of the marginal likelihood of changepoint parameters, {hs,j}. • Inference of k through AIC or BIC?

17. Bayesian inference on the interpolation model • Need prior distribution of changepoints, {hs,j}, and number of segments, k. • MCMC for each sub-model characterized by k. • Importance sampling for posterior sub-model probability, Pr(k|D). • Input: Data, prior probability of each sub-model and prior for the parameter set of each sub-model. (Can be regional or partially regional. Set by asking for credibility intervals.) • Output: Pr(k|D) and posterior dist. of all parameters for each k. (Summarised by estimates and credibility intervals.)

18. Output example for interpolation model inference Q

19. Problems with Bayesian treatment of segmented models • Difficult to make efficient inference algorithms (but a semi-efficient one has been made). • Changepoints only inside the dataset (thus ”the interpolation model”). Extrapolation uncertainty underestimated because there can be changepoints outside the dataset. • Solution(?): The process model, a new model where the segments appear through a process. • Problems with the process model: Very inefficient algorithms. Difficult to implement all sorts of relevant prior knowledge. • Middle ground? Use changepoints of most sub-model from the interpolation model as data in inference about the changepoint process. Process model used for extrapolation of the curve.

20. References • Árnason S (2005), Estimating nonlinear hydrological rating curves and discharge using the Bayesian approach. Masters Degree, Faculty of Engineering, University of Iceland • Clarke, RT (1999), Uncertainty in the estimation of mean annual flood due to rating curve indefinition. J Hydrol, 222: 185-190 • ISO 1100/2. (1998), Stage-discharge Relation, Geneva • Lambie JC (1978), Measurement of flow - velocity-area methods. Hydrometry: Principles and Practices, first edition, edited by R.W. Herschy, John Wiley & Sons, UK. • Moyeeda RA, Clarke RT (2005), The use of Bayesian methods for fitting rating curves, with case studies. Adv Water Res, 28:8:807-818 • Petersen-Øverleir A, Reitan T (2005), Objective segmentation in compound rating curves. J Hydrol, 311: 188-201 • Reitan T, Petersen-Øverleir A (2005), Estimating the discharge rating curve by nonlinear regression - The frequentist approach. Statistical Research Report, University of Oslo, Preprint 2, 2005 Available at: http://www.math.uio.no/eprint/stat report/2005/02-05.html • Reitan T, Petersen-Øverleir A (2006), Existence of the frequentistic regression estimate of a power-law with a location parameter, with applications for making discharge rating curves. Stoc Env Res Risk Asses, 20:6: 445-453 • Reitan T, Petersen-Øverleir A (2008a), Bayesian power-law regression with a location parameter, with applications for construction of discharge rating curves. Stoc Env Res Risk Asses, 22: 351-365 • Venetis C (1970), A note on the estimation of the parameters in logarithmic stage-discharge relationships with estimation of their error, Bull Inter Assoc Sci Hydrol, 15: 105-111