Modeling the Catchment Via Mixtures: an Uncertainty Framework for - PowerPoint PPT Presentation

Modeling the Catchment Via Mixtures: an Uncertainty Framework for Dynamic Hydrologic Systems Dynamic Hydrologic Systems Lucy Marshall Assistant Professor of Watershed Analysis A i P f f W h d A l i Department of Land Resources and Environmental Sciences Montana State University Email: lmarshall@montana.edu Thanks to: Kelsey Jencso, Tyler Smith, Brian McGlynn- MSU Ashish Sharma- University of New South Wales David Nott- National University of Singapore

Conceptualizing first order watershed processes processes Tenderfoot Cr. Tenderfoot Cr. • 7 nested watersheds 7 nested watersheds E Exp. Forest Exp. Forest E F F t t • Lodgepole pine vegetation • Melt driven runoff • F Freezing temperatures can occur i t t in every month ¯ • 555 ha • Full range of slope and topographic convergence, convergence 0 0 0 0 0 0 250 250 250 250 250 250 500 500 500 500 500 500 1,000 1,000 1,000 1,000 1,000 1,000 Meters Meters Meters Meters Meters Meters divergence • Elevation ranges Flume Flume Flume Flume Flume Flume Well Transect Well Transect Well Transect Well Transect Well Transect Well Transect SNOTEL SNOTEL SNOTEL SNOTEL SNOTEL SNOTEL from 1840m to 2420 2420 The Tenderfoot Creek Experimental Forest

Snowmelt/Rain SWE (mm) 0 0 mm/day 20 Snow Melt 300 Rain 40 SWE m/hr) noff 0.5 Runoff (a) ( ) (mm Ru Winter 100000 0.0 24 transect total III -binary connectivity- ST5W ST2W ST2W area m 2 TFT2S ea (m 2 ) TFT4N II cumulated Are umulated Spring TFT1S ST1E 10000 ST7E TFT1N TFT5S TFT3S TFT3S Upslope Ac slope acc ST6W ST7W TFT2N ST4E ST3W ST6E ST3E ST2E Up I TFT3N ummer ST5E ST1W TFT5N TFT4S Stream-Riparian-Hillslope St Ri i Hill l S 1000 No Connection Water Table Connection ST4W Kelsey Jencso 10/06 12/06 2/07 4/07 6/07 8/07 10/07

Conceptualizing first order watershed processes processes Unknown Process/Model Implementation -Temp/energy dependent? Snow melt 1 500 0 -Elevation 1 0.9 450 2 effects? effects? 0.8 400 3 -Rain on snow? 4 0.7 350 Soil Moisture 5 0.6 off (mm) 300 -Thresholds? Accounting/sub 0.5 250 -Seasonal? Seasonal? surface flow surface flow Runo 0.4 200 0.9 0.3 observed runoff 150 MS runoff mm/hr Stringer Sun Sun Runoff mm/hr SWE 0.8 0.2 100 rainfall Storages/ 0.7 0.1 50 -Slope effects? Residence times 0.6 0.5 0 -Seasonal? 0 runoff mm/hr 03/01 04/01 05/01 06/01 07/01 08/01 09/01 10/01 0.4 Date 0.3 0.2 0.2 0.1 0 4/1/06 5/1/06 5/31/06 6/30/06 7/30/06 8/29/06 -0.1 date

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

An uncertainty framework – ways of incorporating hydrologic complexity i ti h d l i l it Standard St d d Multiple M lti l Hi Hierarchical hi l Ensemble Bayesian Sources Model Model Model of Data Data y | yA, yB y 1 | θ 1 , x y 2 | θ 2 , x y | θ 1 , x y | θ , x 1 y | θ x Process θ 1 , θ 2 , x y A | θ , x 1 y B | θ , x 1 Parameters θ 1 | θ 2 , x y ~ variable of interest Adapted after Clark, Adapted after Clark, x ~ input data, climatological variables Ecology Letters, 2005. θ ~ parameters

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

Difficulties in characterizing hydrologic model uncertainty model uncertainty • Hydrological models: often have highly correlated and interdependent parameters interdependent parameters Histogram for S1 in AWBM Histogram for S1 in AWBM 900 800 700 600 • A solution is provided by an 500 400 adaptive MCMC algorithm using 300 th hi t the history of the sampled f th l d 200 100 parameter states 0 128 129 130 131 132 133 134 135 136

Inference results B Bucket model semi-distributed by k d l d b d b aspect and accounting for snowmelt using the temperature- and (a) radiation index approach radiation-index approach

Assessing model uncertainty via Bayesian Model Averaging Bayesian Model Averaging • Probabilistically ∑ weight each model • Ensemble of models E bl f d l is an increasingly accepted way of representing model representing model Model 1 Model 2 ‘structural’ uncertainty • The Bayesian = ∫ θ θ θ approach accounts m ( y | M ) f ( y | , M ) p ( | M ) d 1 1 1 for multiple sources of uncertainty ∝ • P ( M | y ) P ( M ) m ( y | M ) 1 1 1

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 0.45 0.4 Simple Average of Two Models 90% Confidence 0.35 Observed 0.3 0.25 0.45 0.2 0.4 0.15 90% Confidence 90% Confidence 0.35 0.1 Observed 0.05 0.3 0 1 501 1001 1501 2001 2501 3001 0.25 0.2 0.45 0.15 0.4 90% Confidence 0.1 0.35 Observed 0.3 0.05 0.25 0 0.2 1 501 1001 1501 2001 2501 3001 0 15 0.15 0.1 0.05 0 1 501 1001 1501 2001 2501 3001

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

Mixture models- alternative models suitable at different times diff i • • Probabilistically split Probabilistically split 2.5E-03 the data according to some catchment Different 2.0E-03 indicators indicators models selected for • Fit separate models 1.5E-03 parts of the to the data and data data data errors. Models may errors Models may 1.0E-03 then ‘specialize’ • Can be likened to 5.0E-04 Bayesian Model B i M d l Averaging, where the weights vary in 0.0E+00 1000 1500 2000 time time Can fit same model structure with different parameterizations: assumes that model uncertainty does not arise solely out of the assumed model structure

Mixture models- alternative parameterizations suitable at different times 0.45 1.2 0.4 Fit two parameterizations of the 1 Probability Model 1 0.35 single best model (combined g ( Modeled 0.3 0 3 0 8 0.8 Observed temperature/radiation index Flow (mm) Probability 0.25 0.6 melt, pdm model) 0.2 0.15 0.4 0 1 0.1 0.2 0.05 0 0 1 1001 2001 3001 4001 5001

Mixture models- alternative parameterizations suitable at different times Model preference changes 0.45 1.2 according to: 0.4 1 Probability Model 1 0.35 •Response to event Modeled 0.3 0.8 Observed Observed Probability Flow (mm) 0.25 •Time of season 0.6 0.2 Comparison of alternate model 0.15 0.4 simulations can indicate which simulations can indicate which 0.1 0.2 parameters are most sensitive to 0.05 0 0 selected calibration period 1 1001 2001 3001 4001 5001 HME approach gives good fit to data, but has problems with: •Identifiability •Interpretation •Interpretation •Predictions

Combining multiple model parameterizations: catchment states catchment “states” q Hydrologic model: Topmodel Logistic gating function q 2 q 1 Model 2 Model 1 x x A single-level two-component Hierarchical Mixture of Experts model 121 111 Tarrawarra Catchment 101 Two Component HME Two Component HME 91 Contour Map 81 71 61 51 41 31 31 21 11 From Hornberger, 1998 1 1 11 21 31 41 51 61 71 81 91 101 111 121

Combining multiple model parameterizations: catchment states catchment “states” 0.003 Simulations from individual mixture 0.0025 0 0025 component models 0.002 Q1 Q2 0.0015 0 001 0.001 0.0005 0

Combining Multiple Model Parameterizations: Model States Parameterizations Model “States” 0.0025 1 0.9 0.002 0.8 0.7 rge (m) 0.0015 0.6 Qobs 0.5 0 5 Qmean Qmean Discha Probability 0.001 0.4 0.3 0.0005 0.2 0.1 0 0 Hour