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Modeling the Catchment Via Mixtures: an Uncertainty Framework for Dynamic Hydrologic Systems Dynamic Hydrologic Systems

Lucy Marshall

A i P f f W h d A l i Assistant Professor of Watershed Analysis 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. E F t Tenderfoot Cr. E F t

• 7 nested watersheds
• Exp. Forest
• Exp. Forest

7 nested watersheds

• Lodgepole pine vegetation
• Melt driven runoff

F i t t

¯

• Freezing temperatures can occur

in every month

• 555 ha
• Full range of slope

and topographic convergence

500 1,000 250 Meters

Well Transect Flume SNOTEL

500 1,000 250 Meters 500 1,000 250 Meters

Well Transect Well Transect Flume Flume SNOTEL SNOTEL

500 1,000 250 Meters

Well Transect Flume SNOTEL

500 1,000 250 Meters 500 1,000 250 Meters

Well Transect Well Transect Flume Flume SNOTEL SNOTEL

convergence, divergence

• Elevation ranges

from 1840m to 2420

The Tenderfoot Creek Experimental Forest

2420

noff m/hr) 0.5

Runoff

Snowmelt/Rain mm/day 20 40 SWE (mm) 300

Snow Melt Rain SWE

(a) 100000

ST2W ST5W

Ru (mm 0.0

( )

Winter 24 transect total

• binary connectivity-

III

ea (m2)

TFT2S ST2W TFT4N

area m2

cumulated Are

10000

TFT1N TFT5S TFT3S ST7E ST1E TFT1S

Spring umulated

II

Upslope Ac

ST2E ST6E ST3W ST3E ST4E TFT2N ST6W ST7W TFT3S

slope acc

TFT4S TFT5N ST1W ST5E TFT3N

St Ri i Hill l

ummer Up

I

10/06 12/06 2/07 4/07 6/07 8/07 10/07 1000

ST4W

Stream-Riparian-Hillslope Water Table Connection No Connection

S

Kelsey Jencso

Conceptualizing first order watershed processes processes

Unknown Process/Model Implementation

0.9 1

450 500 1 2

Snow melt

• Temp/energy

dependent?

• Elevation

effects?

0.5 0.6 0.7 0.8

• ff (mm)

250 300 350 400 3 4 5

Soil Moisture Accounting/sub surface flow

effects?

• Rain on snow?
• Thresholds?
• Seasonal?

0.2 0.3 0.4 Runo

100 150 200

• bserved runoff

SWE rainfall

Storages/ surface flow

Seasonal?

0.7 0.8 0.9 MS runoff mm/hr Sun Runoff mm/hr Stringer Sun

03/01 04/01 05/01 06/01 07/01 08/01 09/01 10/01 0.1 Date

50

Residence times

• Slope effects?
• Seasonal?

0.2 0.3 0.4 0.5 0.6

runoff mm/hr

• 0.1

0.1 0.2 4/1/06 5/1/06 5/31/06 6/30/06 7/30/06 8/29/06

date

Conceptual rainfall-runoff modeling in hydrology hydrology

Watershed is represented

P E Qs

p as a variable series of storages.

Model uses rainfall

S1

S2 S3 Qr

Model uses rainfall,

evapotranspiration etc. time series as inputs to l ff

A1 A2 A3

simulate runoff

Conceptual distributed

models: discretize

BS Qb

models: discretize catchment into individual units, or use hydrologic response units

BS Qb

response units

An uncertainty framework – ways of i ti h d l i l it incorporating hydrologic complexity

St d d M lti l Hi hi l Standard Bayesian Model Multiple Sources

• f Data

Ensemble Model Hierarchical Model Data

y | θ x y | θ1, x y1 | θ1, x y | yA, yB y2 | θ2, x

Process

y | θ, x1 yA | θ, x1 θ1, θ2, x yB | θ, x1

Parameters

y ~ variable of interest Adapted after Clark, θ1| θ2, x x ~ input data, climatological variables θ ~ parameters Adapted after Clark, Ecology Letters, 2005.

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 and their elevation and 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

Histogram for S1 in AWBM

interdependent parameters

600 700 800 900

Histogram for S1 in AWBM

300 400 500

• A solution is provided by an

adaptive MCMC algorithm using th hi t f th l d

128 129 130 131 132 133 134 135 136 100 200

the history of the sampled parameter states

Inference results

B k d l d b d b

(a)

Bucket model semi-distributed by aspect and accounting for snowmelt using the temperature- and radiation index approach radiation-index approach

Assessing model uncertainty via Bayesian Model Averaging Bayesian Model Averaging

• Probabilistically

weight each model E bl f d l

∑

• Ensemble of models

is an increasingly accepted way of representing model representing model ‘structural’ uncertainty

• The Bayesian

Model 1 Model 2

θ θ θ d M p M y f M y m ) | ( ) , | ( ) | (

1 1 1

∫

=

approach accounts for multiple sources

• f uncertainty

) | ( ) ( ) | (

1 1 1

M y m M P y M P

• ∝

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 0.15 0.2 0.25 0.3 0.35 0.4 90% Confidence Observed

Simple Average of Two Models

0.4 0.45 90% Confidence

0.05 0.1 1 501 1001 1501 2001 2501 3001 0.45

0.2 0.25 0.3 0.35 90% Confidence Observed

0 15 0.2 0.25 0.3 0.35 0.4 90% Confidence Observed

0.05 0.1 0.15 1 501 1001 1501 2001 2501 3001

0.05 0.1 0.15 1 501 1001 1501 2001 2501 3001

Hierarchical Mixtures of Experts Hierarchical Mixtures of Experts

E h t l d l b

q Logistic gating

Each conceptual model can be cast as:

) ( ) ; (

2 , , i t i i t t i t

x f Q σ ε θ + =

q

1

q 2 Model 1 Model 2 function

The probability of selecting individual models is based on the gating function, using catchment

x x

A single-level two-component Hierarchical Mixture of Experts model

.

predictors Xt:

) , X ( G ) , X ( G , t

t t

e e g

β β

+ = 1

1

Mixture of Experts model

Models are sampled via a conditional simulation of independent Bernoulli ) z | Q ( P ) , , | z ( p ) , , , Q | z ( p

i , t t , t t , t

∑

=

= σ θ β = = σ θ β =

2 2 1 2 1 2 1

1 1 1

e + 1

independent Bernoulli random variables zt, with probability specified as: ) z | Q ( P ) , , | z ( p

i i , t t i , t

∑

=

= σ θ β =

1 2

1 1

Mixture models- alternative models suitable at diff i different times

• Probabilistically split

2.0E-03 2.5E-03

• Probabilistically split

the data according to some catchment indicators

Different

1.5E-03

indicators

• Fit separate models

to the data and data errors Models may

models selected for parts of the data

5.0E-04 1.0E-03

• errors. Models may

then ‘specialize’

• Can be likened to

B i M d l

data

0.0E+00 1000 1500 2000

Bayesian Model Averaging, where the weights vary in 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 3 0.35 0.4 0.45 0 8 1 1.2 Probability Model 1 Modeled

Fit two parameterizations of the single best model (combined

0 1 0.15 0.2 0.25 0.3

Flow (mm)

0.4 0.6 0.8

Probability

Observed

g ( temperature/radiation index melt, pdm model)

0.05 0.1 1 1001 2001 3001 4001 5001 0.2

Mixture models- alternative parameterizations suitable at different times

0.3 0.35 0.4 0.45 0.8 1 1.2 Probability Model 1 Modeled Observed

Model preference changes according to:

• Response to event

0.1 0.15 0.2 0.25

Flow (mm)

0.4 0.6

Probability

Observed

• Time of season

Comparison of alternate model simulations can indicate which

0.05 1 1001 2001 3001 4001 5001 0.2

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
• Interpretation
• Predictions

Combining multiple model parameterizations: catchment “states” catchment states

Hydrologic model: Topmodel

q Logistic gating function q

1

q 2 x x Model 1 Model 2

Two Component HME

A single-level two-component Hierarchical Mixture of Experts model

91 101 111 121

Tarrawarra Catchment Two Component HME

31 41 51 61 71 81

Contour Map From Hornberger, 1998

1 11 21 31 41 51 61 71 81 91 101 111 121 1 11 21 31

Combining multiple model parameterizations: catchment “states” catchment states

0 0025 0.003

Simulations from individual mixture

0.002 0.0025

component models

0 001 0.0015 Q1 Q2 0.0005 0.001

Combining Multiple Model Parameterizations Model “States” Parameterizations: Model States

0.002 0.0025 0.8 0.9 1 0.0015 rge (m) 0 5 0.6 0.7 Qobs Qmean 0.001 Discha 0.3 0.4 0.5 Qmean Probability 0.0005 0.1 0.2 Hour

What about prediction? p

To use the model for prediction means finding an To use the model for prediction means finding an

appropriate catchment descriptor and a function relating this to the probability switching between relating this to the probability switching between models

Possible predictors Possible predictors Antecedent rainfall Modelled catchment storage 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

Model Aggregation as a Predictive Tool- Comparison of predictors Comparison of predictors

Model Predictor

• 0.5 BIC

Topmodel N/A 32685

• p
• de

/ 3 685

Model Aggregation as a Predictive Tool- Comparison of predictors Comparison of predictors

Model Predictor

• 0.5 BIC

Topmodel N/A 32685

• p
• de

/ 3 685 2 Component HME Preceding rainfall 33445 Change in storage deficit 33487 Change in unsaturated Change in unsaturated zone storage 33428 Unsaturated zone storage 33455

Model Aggregation as a Predictive Tool- Comparison of predictors Comparison of predictors

Model Predictor

• 0.5 BIC

Topmodel N/A 32685

• p
• de

/ 3 685 2 Component HME Preceding rainfall 33445 Change in storage deficit 33487 Change in unsaturated Change in unsaturated zone storage 33428 Unsaturated zone storage 33455

Benefits of the HME approach pp

HME provides an improved framework for

incorporating multiple sources of model uncertainty in p g p y hydrology

The HME approach allows combination of multiple

models and parameterizations in a single framework

Using Mixture Modeling as a Method of Comparing Model Structures Parameters and Errors 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 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

Alternative approach: hierarchical model

• Most temporally sensitive parameters are conditioned on
• bserved/modeled exogenous data
• Easier to interpret in light of the conceptualized hydrologic processes
• Look at extent to which parametric variability informs model

Look at extent to which parametric variability informs model structural uncertainty

0 4 0.45 0 973 0.9735

• Storage parameter differentiates

alternative HME components

0 2 0.25 0.3 0.35 0.4 w (mm) 0 9705 0.971 0.9715 0.972 0.9725 0.973 e Parameter

p

• Condition this on the watershed

melt and temperature

0.05 0.1 0.15 0.2 Flow 0 968 0.9685 0.969 0.9695 0.97 0.9705 Storage Modeled Observed Storage Recession

Model Max log- likelihood Hierarchical 15791

1 1001 2001 3001 4001 5001 0.968

HME 18068

Comparison of aggregation and hi hi l h hierarchical approaches

• How do these approaches compare for:

pp p

1.

Assessing model structural uncertainty

• Ensemble methods span the breadth of model space with varying

degrees to give a better assessment of model uncertainty 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

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. 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

• interest. A more complex hierarchical model may give a model structure

greater flexibility and a simulation more consistent with internal watershed processes.

Can we use multi-model approaches for better model building? 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 approaches can be used The HME approach and multi 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 These approaches can be used to improve existing models for

better interpretability of internal watershed dynamics and their variability

Modeling the Catchment Via Mixtures: an Uncertainty Framework for Dynamic Hydrologic Systems Dynamic Hydrologic Systems

Lucy Marshall

Assistant Professor of Watershed Analysis Department of Land Resources and Environmental Sciences Department of Land Resources and Environmental Sciences Montana State University

Email: lmarshall@montana.edu

References

Rainfall Runoff Models

Moore, R. J. (1985). "The probability-distributed principle and runoff

production at point and basin scales." Hydrological Sciences 30(2): 273-297.

Beven, K., et al. (1995), Topmodel, in Computer Models of Watershed Hydrology,

dit d b V P Si h 627 668 W t R P bli ti Hi hl d R h edited by V. P. Singh, pp. 627-668, Water Resources Publications, Highlands Ranch, Colorado.

Bayesian Inference and Adaptive MCMC Algorithms

Clark JS (2005) Why environmental scientists are becoming Bayesians. Ecology Letters

( ) y g y gy 8(1)

Haario, H., et al. (2001), An adaptive Metropolis algorithm, Bernoulli, 7(2), 223-242.

Haario, H., M. Laine, et al. (2006). "DRAM: Efficient adaptive MCMC."

S d C (4) 339 3 4 Statistics and Computing 16(4): 339-354.

Hierarchical mixtures of Experts

Jacobs, R. A., et al. (1997), A Bayesian approach to Model Selection in Hierarchical

Mixtures-of-Experts Architectures Neural Networks 10(2) 231-241 Mixtures-of-Experts Architectures, Neural Networks, 10(2), 231-241.

Marshall, L., et al. (2006), Modeling the catchment via mixtures: Issues of model

specification and validation, Water Resour. Res., 42(11), 1-14.

Adaptive Bayesian algorithms p y g

• The

Adaptive Metropolis (AM) algorithm (Haario et al (AM) algorithm (Haario et al., 2001):

• The covariance of the proposal

distribution is updated using the

AM Jump Space Initial DRAM Jump Space

distribution is updated using the information gained from the simulation thus far.

• Often

plagued by initialization bl i h l i h

p p

problems, causing the algorithm to become trapped in local optima.

• The

Delayed Rejection Adaptive Metropolis (DRAM) Adaptive Metropolis (DRAM) algorithm (Haario et al. 2006):

• Reduces the probability that the

algorithm will remain at the current

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

algorithm will remain at the current state.

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.