Modeling the Catchment Via Mixtures: an Uncertainty Framework for

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1 Modeling the Catchment Via Mixtures: an Uncertainty Framework for Dynamic Hydrologic Systems Lucy Marshall Assistant Professor of Watershed Analysis Department of Land Resources and Environmental Sciences Montana State University Thanks to: Kelsey Jencso, Tyler Smith, Brian McGlynn- MSU Ashish Sharma- University of New South Wales David Nott- National University of Singapore

2 Conceptualizing first order watershed processes E F t 7 nested watersheds Lodgepole pine vegetation Melt driven runoff Freezing temperatures t can occur in every month Tenderfoot Cr. Exp. Forest Flume Well Transect , Meters SNOTEL The Tenderfoot Creek Experimental Forest 555 ha Full range of slope and topographic convergence, divergence Elevation ranges from 184m to 242

3 Snowmelt/Rain mm/day Snow Melt Rain SWE Runoff 24 transect total -binary connectivity- (a) 3.5. ST5W ST2W SWE (mm) Ru noff (mm m/hr) III Winter cumulated Arearea (m 2 ) m 2 umulated Upslope acc Ac 1 TFT2S TFT4N II TFT1S ST1E ST7E TFT1N TFT5S TFT3S ST6W ST7W TFT2N ST4E ST3W ST6E ST2E ST3E Spring Up 1 Stream-Riparian-Hillslope i l Water Table Connection No Connection TFT3N ST5E ST1W TFT5N TFT4S I ummer S ST4W 1/6 12/6 2/7 4/7 6/7 8/7 1/7 Kelsey Jencso

4 Conceptualizing first order watershed processes Unknown Process/Model Implementation Snow melt 5 Soil Moisture -Temp/energy dependent? -Elevation effects? -Rain on snow? Runo off (mm) Accounting/sub surface flow -Thresholds? -Seasonal? observed runoff SWE rainfall 3/1 4/1 5/1 6/1 7/1 8/1 9/1 1/1 Date runoff mm/hr.9.8 Storages/.7.6 Residence times MS runoff Stringer mm/hr Sun Runoff Sun mm/hr -Slope effects? -Seasonal? 4/1/ /1/6 5/31/6 6/3/6 7/3/6 8/29/6 date

5 Conceptual rainfall-runoff modeling in hydrology Watershed is represented as a variable series of storages. Model uses rainfall, S 1 evapotranspiration etc. S 2 time series as inputs to simulate runoff P E S 3 Q r Conceptual distributed models: discretize A 1 A 2 A 3 catchment into individual units, or use hydrologic response units BS Q b Q s

6 An uncertainty framework ways of incorporating hydrologic complexity Standard d Multiple l Hierarchical Bayesian Sources Model Model of Data Ensemble Model Data y 1 θ 1, x y 2 θ 2, x y ya, yb y θ 1, x Process y θ θ, x 1 Parameters θ 1, θ 2, x y A θ, x 1 y B θ, x 1 θ 1 θ 2, x y ~ variable of interest x ~ input data, climatological variables θ ~ parameters Adapted after Clark, Ecology Letters, 25.

7 Base conceptual models Two base structures: asimple bucket model and the probability distributed model (PDM, Moore, 1985). Three semi distribution substructures: based on aspect, elevation and their combination to account for spatial variability in inputs. Three snowmelt accounting routines: temperature index, radiation index and the combination.

8 Difficulties in characterizing hydrologic model uncertainty Hydrological models: often have highly correlated and interdependent parameters 9 Histogram for S1 in AWBM A solution is provided by an adaptive MCMC algorithm using 2 the history of the sampled ld 1 parameter states

9 Inference results (a) Bucket model semi-distributed by aspect and accounting for snowmelt using the temperature- and radiation-index index approach

10 Assessing model uncertainty via Bayesian Model Averaging Model 1 Model 2 m y M ) = f ( y θ, M ) p( θ M ) dθ ( P ( M 1 y) P( M 1) m( y M 1) Probabilistically weight each model Ensemble of models is an increasingly accepted way of representing model structural uncertainty The Bayesian approach accounts for multiple sources of uncertainty

11 The utility of multi model ensembles Models represent competing hypotheses about the first order processes Both models provide information on the processes occurring so that the data is better captured % Conf idence Observed Simple Average of Two Models % Confidence Observed % Confidence Observed

12 Hierarchical Mixtures of Experts. Each conceptual model can be q cast as: Q t = f ( x ; θ ) + ε ( σ 2 i, t t i i, t i The probability of selecting individual models is based on the gating function, using catchment predictors X t : g t, 1 e = 1+ e G( X t G( X, β ) t, β ) ) Logistic gating function q 1 q 2 Model 1 Model 2 x A single-level two-component Hierarchical Mixture of Experts model x Models are sampled via a conditional simulation of independent Bernoulli random variables z t, with probability specified as: p( z 2 p( zt, 1 = 1 Q t, β, θ, σ ) = i 2 p( z t, 1 = i= 1 = 1 β, θ, σ t,i 2 = 1 β, θ, σ )P( Q 2 t )P( Q z t t, 1 z = 1) t,i = 1)

13 Mixture models- alternative models suitable at different times 2.5E-3 Probabilistically split the data according to some catchment 2.E-3 Different indicators models selected for Fit separate models 1.5E-3 parts of the to the data and data data errors. Models may 1.E-3 then specialize Can be likened to 5.E-4 Bayesian Model Averaging, where.e+ the weights vary in time Can fit same model structure with different parameterizations: assumes that model uncertainty does not arise solely out of the assumed model structure

14 Mixture models- alternative parameterizations suitable at different times Flow (mm) Probability Model 1 Modeled Observed Probability Fit two parameterizations of the single best model (combined temperature/radiation index melt, pdm model)

15 Mixture models- alternative parameterizations suitable at different times Flow (mm) Probability Model 1 Modeled.3 Observed Probability Model preference changes according to: Response to event Time of season.4 Comparison of alternate model.2 simulations can indicate which parameters are most sensitive to selected calibration period HME approach gives good fit to data, but has problems with: Identifiability Interpretation Predictions

16 Combining multiple model parameterizations: catchment states Hydrologic model: Topmodel q Logistic gating function q 1 q 2 Model 1 Model 2 x x A single-level two-component Hierarchical Mixture of Experts model Tarrawarra Catchment 91 Two Component HME 81 Contour Map 11 From Hornberger,

17 Combining multiple model parameterizations: catchment states Simulations from individual mixture component models.15 Q1 Q

18 Combining Multiple Model Parameterizations: Model States rge (m) Discha Qobs 5.5 Qmean Probability Hour

19 What about prediction? To use the model for prediction means finding an appropriate catchment descriptor and a function relating this to the probability switching between models Possible predictors Antecedent rainfall Modelled catchment storage Time of the year The best predictors are often related to the most The best predictors are often related to the most dynamic catchment mechanisms

20 Model Aggregation as a Predictive Tool- Comparison of predictors Model Predictor -.5 BIC Topmodel ode N/A 32685

21 Model Aggregation as a Predictive Tool- Comparison of predictors Model Predictor -.5 BIC Topmodel ode N/A Component HME Preceding rainfall Change in storage deficit Change in unsaturated zone storage Unsaturated zone storage 33455

22 Model Aggregation as a Predictive Tool- Comparison of predictors Model Predictor -.5 BIC Topmodel ode N/A Component HME Preceding rainfall Change in storage deficit Change in unsaturated zone storage Unsaturated zone storage 33455

23 Benefits of the HME approach HME provides an improved framework for incorporating multiple sources of model uncertainty in hydrology The HME approach allows combination of multiple models and parameterizations in a single framework

24 Using Mixture Modeling as a Method of Comparing Model Structures, Parameters and Errors HME can highlight problems in the model structure For conceptual models: different responses in wet and dry periods; different ways to model the catchment storage For distributed models: different patterns of soil moisture in wet and dry periods; different assumptions about the recession properties A mixture of error distributions can provide better A mixture of error distributions can provide better prediction limits and better model heteroscedasticity

25 Alternative approach: hierarchical model Flow (mm) Most temporally sensitive parameters are conditioned on observed/modeled exogenous data Easier to interpret in light of the conceptualized hydrologic processes Look at extent to which parametric variability informs model structural uncertainty Modeled Storage Parameter Storage parameter differentiates alternative HME components Condition this on the watershed melt and temperature Observed.97 Model Max log- Storage Recession.9695 likelihood Hierarchical HME 1868

26 Comparison of aggregation and hierarchical approaches How do these approaches compare for: 1. Assessing model structural uncertainty Ensemble methods span the breadth of model space with varying degrees to give a better assessment of model uncertainty. HME and hierarchical formulations can highlight problems in the assumed model structure. 2. Improving model predictions The ensemble approach should give a more consistent performance for the main variable of interest. The hierarchical approach does hold promise in improving model performance. 3. Interpretability Ensemble and HME approaches are less useful beyond the variable of interest. A more complex hierarchical model may give a model structure greater flexibility and a simulation more consistent with internal watershed processes.

27 Can we use multi-model approaches for better model building? For improved conceptual model assessment we should consider that parameter variability and model structural uncertainty are linked. The HME approach and multi model model approaches can be used to determine the utility of alternative models under different watershed conditions These approaches can be used to improve existing models for better interpretability of internal watershed dynamics and their variability

28 Modeling the Catchment Via Mixtures: an Uncertainty Framework for Dynamic Hydrologic Systems Lucy Marshall Assistant Professor of Watershed Analysis Department of Land Resources and Environmental Sciences Montana State University

29 References Rainfall Runoff Models Moore, R. J. (1985). "The probability-distributed principle and runoff production at point and basin scales." Hydrological Sciences 3(2): Beven, K., et al. (1995), Topmodel, in Computer Models of Watershed Hydrology, edited d by V. P. Singh, pp , 668 Water Resources Publications, Highlands Ranch, Colorado. Bayesian Inference and Adaptive MCMC Algorithms Clark JS (25) Why environmental scientists are becoming Bayesians. Ecology Letters 8(1) Haario, H., et al. (21), An adaptive Metropolis algorithm, Bernoulli, 7(2), Haario, H., M. Laine, et al. (26). "DRAM: Efficient adaptive MCMC." Statistics and Computing 16(4): Hierarchical mixtures of Experts Jacobs, R. A., et al. (1997), A Bayesian approach to Model Selection in Hierarchical Mixtures-of-Experts Architectures, Neural Networks, 1(2), Marshall, L., et al. (26), Modeling the catchment via mixtures: Issues of model specification and validation, Water Resour. Res., 42(11), 1-14.

30 Adaptive Bayesian algorithms AM Jump Space Initial DRAM Jump Space Figure 2. A theoretical parameter surface, diagramming AM & DRAM algorithms ability to explore the parameter surface. Rings represent distance from the current location an algorithm can explore. These exploration limits illustrate DRAM s ability to search more space and AM s tendency to falsely converge to local maxima because of its more constricted search area. The Adaptive Metropolis (AM) algorithm (Haario et al., 21): The covariance of the proposal distribution is updated using the information gained from the simulation thus far. Often plagued by initialization problems, causing the algorithm to become trapped in local optima. The Delayed Rejection Adaptive Metropolis (DRAM) algorithm (Haario et al. 26): Reduces the probability that the algorithm will remain at the current state.