Martyn Clark (NCAR/RAL)
Decisions when building a model: Spatial approximations, process parameterizations, and
time stepping schemes
SWGR Snow Modeling School: Snowpack modeling for practitioners and modelers 23 June 2014, NCAR, Boulder, Colorado, USA
Outline • The necessary ingredients of a model (modeling in general) ▫ State variables, process parameterizations, model parameters, model
forcing data, and the numerical solution ▫ Two examples:
� Temperature-index snow model � Conceptual hydrologic model
• Physically-motivated snow modeling ▫ Major model development decisions
• Impact of key model development decisions ▫ General philosophy underlying SUMMA ▫ Case studies: Reynolds Creek and Umpqua
• Summary and research needs
The art of modeling: A realistic portrayal of dominant processes
Need to define: 1) State variables (storage of
water and energy); and 2) Fluxes that affect the
evoluAon of state variables
The necessary ingredients of a model: Model forcing data, model state variables, flux parameterizations, model parameters, and the numerical solution
• Example 1: A temperature-index snow model ▫ The state equation
▫ Flux parameterizations and model parameters
▫ Numerical solution � Simple in this case, since fluxes do not depend on state variables
dS a mdt
= − State variable (also known as prognostic variable)
Fluxes
State variable: S = Snow storage (mm)
Fluxes: a = Snow accumulation (mm/day) m = Snow melt (mm/day)
0a f
a f
p T Ta
T T<⎧
= ⎨ ≥⎩
( )0 a f
a f a f
T Tm
T T T Tκ
<⎧⎪= ⎨
− ≥⎪⎩
Forcing data
Forcing data Model parameter Physical constant (can also be treated as a model parameter)
Model forcing: p = Precipitation rate (mm/day) Ta = Air temperature (K)
Parameters: κ = Melt factor (mm/day/K)
Physical constants: Tf = Freezing point (K)
The necessary ingredients of a model: Model forcing data, model state variables, flux parameterizations, model parameters, and the numerical solution
• Example 2: A conceptual hydrology model
• State equation
Figure from Hornberger et al. (1998) “Elements of Physical Hydrology” The Johns Hopkins University Press, 302pp.
t sdS p e rdt
= − −
The necessary ingredients of a model: Model forcing data, model state variables, flux parameterizations, model parameters, and the numerical solution
• Example 2: A conceptual hydrology model ▫ The state equation
▫ Flux parameterizations
▫ Numerical solution � Care must be taken: model fluxes depend on state variables (numerical daemons)
t bdS p e qdt
= − − State variable
Fluxes
State variable: S = Soil storage (mm)
Model forcing: p = Precipitation rate (mm/day)
Model fluxes: et = Evapotranspiration (mm/day) qb = Baseflow (mm/day
p pspst
p ps
Se S SSe
e S S
⎧ ⎛ ⎞<⎪ ⎜ ⎟⎜ ⎟= ⎨ ⎝ ⎠
⎪ ≥⎩
maxb s
cSq kS
⎛ ⎞= ⎜ ⎟
⎝ ⎠
Forcing data
Forcing data
Model forcing: ep = Potential ET rate (mm/day)
Parameters: Sps = Plant stress storage (mm) Smax = Maximum storage (mm) ks = Hydraulic conductivity (mm/day) c = Baseflow exponent (-)
Model parameter
Model parameter Model parameter
Model parameter
State variable
State variable
Pulling it all together: The general modeling problem
Propositions: 1. Most hydrologic modelers share a common understanding
of how the dominant fluxes of water and energy affect the time evolution of thermodynamic and hydrologic states
▫ The collective understanding of the connectivity of state variables and fluxes allows us to formulate general governing model equations in different sub-domains
▫ The governing equations are scale-invariant
2. Key modeling decisions relate to a) the spatial discretization of the model domain; b) the approaches used to parameterize individual
fluxes (including model parameter values); and c) the methods used to solve the governing model
equations. General schematic of the terrestrial water cycle, showing dominant fluxes of water and energy
Given these propositions, it is possible to develop a unifying model framework The SUMMA approach defines a single set of governing equations, with the capability to use different spatial discretizations (e.g., multi-scale grids, HRUs; connected or disconnected), different flux parameterizations and model parameters, and different time stepping schemes
Outline • The necessary ingredients of a model (modeling in general) ▫ State variables, process parameterizations, model parameters, model
forcing data, and the numerical solution ▫ Two examples:
� Temperature-index snow model � Conceptual hydrologic model
• Physically-motivated snow modeling ▫ Major model development decisions
• Understanding the impact of key model development decisions ▫ General philosophy underlying SUMMA ▫ Case studies: Reynolds Creek and Umpqua
• Summary and research needs
Snow modeling • How should we simulate the
dominant snow processes in this environment? ▫ What are the dominant processes
from a hydrologic perspective? � Snow accumulation: drifting
and avalanching; non-homogenous precipitation; rain-snow transition
� Snow melt: Net energy flux for the snowpack; meltwater flow
� Changes in snow properties: grain growth; snow compaction
▫ What information do we need to simulate the dominant processes? � Model forcing data: Precip;
temperature; wind; humidity; sw and lw radiation; (air pressure)
� Model parameters: Drifting; snow albedo; turbulent heat fluxes; storage and transmission of liquid water in the snowpack
Starting point • Governing equations that describe temporal evolution
of thermodynamic and hydrologic states ▫ Thermodynamics
▫ Hydrology � Volumetric liquid water content
� Volumetric ice content
ssssss icep ice fus
mf
T FC Lt t z
θρ
⎡ ⎤∂∂ ∂− = −⎢ ⎥∂ ∂ ∂⎣ ⎦
change in temperature melt/freeze fluxes at the
boundaries
,snow snow snowsnowliq liq z evapice ice
liq liqmf
q Et t z
θ ρ θρ ρ
∂ ∂⎡ ⎤∂+ = − +⎢ ⎥∂ ∂ ∂⎣ ⎦
change in liquid water melt/freeze fluxes at the
boundaries evaporation
sink
snow snow snow snow snowice ice ice ice ice ice ice sub
liq liq liq liqmf cs
q Et t t z
ρ θ ρ θ ρ θρ ρ ρ ρ
⎡ ⎤ ⎡ ⎤∂ ∂ ∂ ∂− − = − +⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦
change in ice content melt/freeze compaction fluxes at the
boundaries sublimation
sink
Notes:
1) Fluxes are only defined in the vertical dimension, meaning that there is no lateral exchange of water and energy among elements (isolated vertical columns)
2) Spatial variability can be represented through spatial variability in model forcing (e.g., non-homogenous precipitation represented as drift factors; spatial variability in solar radiation), and spatial variability in model parameters (e.g., dust loading).
3) Most snow models follow these governing equations
Model decisions • 1) Spatial discretization of the
model domain � The size and shape of the model
elements � Vertical discretization of each
model element
Model decisions • 2) Parameterization of the
model fluxes (and properties) � Spatially distributed forcing data � Vertical flux parameterizations
sfc sfc sfc sfcswnet lwnet h l pF Q Q Q Q Q= + + + +
ssTFz
λ∂
= −∂
How do we represent snow albedo?
How do we represent atmospheric stability?
How do we represent thermal conductivity?
Model decisions • 3) Specifying the model
parameters
sfc sfc sfc sfcswnet lwnet h l pF Q Q Q Q Q= + + + +
( ) ( )max,d min,d min,d
snowliq sf snow snow snow snowd
dref
qddt S α
ραα α κ α α= − − −
How much snow is necessary to refresh albedo?
What is the albedo decay rate?
What is the minimum albedo?
Model decisions • 4) Time stepping schemes
� Operator splitting: It can be very difficult to solve equations simultaneously; most models follow a solution sequence
� Iterative solution procedure: Many fluxes are a non-linear function of the model states; iterative methods typically used to estimate the state at the end of the time step (iSNOBAL exception)
� Numerical error monitoring and adaptive sub-stepping: Dynamically adjust the length of the model time step to improve efficiency and reduce temporal truncation errors
• The necessary ingredients of a model (modeling in general) ▫ State variables, process parameterizations, model parameters, model
forcing data, and the numerical solution ▫ Two examples:
� Temperature-index snow model � Conceptual hydrologic model
• Physically-motivated snow modeling ▫ Major model development decisions
• Understanding the impact of key model development decisions ▫ General philosophy underlying SUMMA ▫ Case studies: Reynolds Creek and Umpqua
• Summary and research needs
Outline
Objectives • Advance capabilities in hydrologic prediction through a unified
approach to hydrological modeling ▫ Improve model fidelity ▫ Better characterize model uncertainty
Motivation • Develop a Unified approach to modeling to understand model
weaknesses and accelerate model development
• Address limitations of current modeling approaches ▫ Poor understanding of differences among models
� Model inter-comparison experiments flawed because too many differences among participating models to meaningfully attribute differences in model behavior to differences in model equations
▫ Poor understanding of model limitations � Most models not constructed to enable a controlled and systematic approach to
model development and improvement
▫ Disparate (disciplinary) modeling efforts � Poor representation of biophysical processes in hydrologic models � Community cannot effectively work together, learn from each other, and
accelerate model development
The method of multiple working hypotheses
• Scientists often develop “parental affection” for their theories
T.C. Chamberlain
• Chamberlin’s method of multiple working hypotheses
• “…the effort is to bring up into view every rational explanation of new phenomena… the investigator then becomes parent of a family of hypotheses: and, by his parental relation to all, he is forbidden to fasten his affections unduly upon any one”
• Chamberlin (1890)
Numerical modeling as a (subjective) decision-making process
• Some modeling decisions can be based on relatively well understood physical principles ▫ Explicitly simulate snow surface energy exchanges rather than
simulating melt “just” as an empirical function of temperature?
• Other modeling decisions more ambiguous ▫ How should saturation-excess runoff be represented? ▫ What about macropore flow – is it significant or even dominant, and,
if so, how should it be represented? ▫ What is the best way to quantify (unknown) bedrock topography/
permeability on sub-surface water retention?
• Other modeling decisions are more pragmatic, based on the computer budget and other considerations ▫ What is the best way to represent the spatial variability of snow depth
across a hierarchy of scales? ▫ Is the application of Beer’s Law to a single canopy layer sufficient to
simulate the transmission of shortwave radiation through the forest canopy, or are more sophisticated methods required?
21
(1) Model architecture
soil soil
aquifer
(e.g., Noah) (e.g., VIC)
aquifer
soil soil
(e.g., PRMS) (e.g., DHSVM)
aquifer
soil
- spatial variability and hydrologic connectivity
(2) Process parameterizations
SUMMA: The unified approach to hydrologic modeling
Governing equaAons
Hydrology
Thermodynamics
Physical processes
XXX Model opAons
Evapo-‐transpiraAon
InfiltraAon
Surface runoff
Solver Canopy storage
Aquifer storage
Snow temperature
Snow Unloading
Canopy intercepAon
Canopy evaporaAon
Water table (TOPMODEL)
Xinanjiang (VIC)
Roo<ng profile
Green-‐Ampt Darcy
Frozen ground
Richards’ Gravity drainage
Mul<-‐domain
Boussinesq
Conceptual aquifer
Instant ouPlow
Gravity drainage
Capacity limited
WeRed area
Soil water characteris<cs
Explicit overland flow
Atmospheric stability
Canopy radiaAon
Net energy fluxes
Beer’s Law
2-‐stream vis+nir
2-‐stream broadband
Kinema<c
Liquid drainage
Linear above threshold
Soil Stress func<on Ball-‐Berry
Snow driRing
Louis Obukhov
Melt drip Linear reservoir
Topographic driZ factors
Blowing snow model
Snow storage
Soil water content
Canopy temperature
Soil temperature
Phase change
Horizontal redistribuAon
Water flow through snow
Canopy turbulence
Supercooled liquid water
K-‐theory L-‐theory
VerAcal redistribuAon
Differences among models are defined by the methods used to represent spatial heterogeneity • Implicit representation of spatial heterogeneity ▫ Statistical-dynamical models
� TOPMODEL, VIC � Sub-grid probability distributions of SWE, or frozen ground
▫ Explicit representation of within-grid differences for a subset of processes � PFTs; separate energy balance calculations for snow covered / snow free areas � Separate stomatal conductance calculations for sunlit and shaded leaves
▫ Empirical flux parameterizations � “New” equations based on relating area-average small-scale fluxes to area-average model variables � The “smoother” equations common in bucket-style hydrologic models (empirical guesses?)
▫ “Effective” parameter values � Richards equation, etc., some based on upscaling operators to match fluxes across scales
• Explicit representation of spatial heterogeneity ▫ Different spatial units
� e.g., grids, HRUs, TINs, etc, ▫ Different degrees of hydrologic connectivity
� e.g., lateral subsurface flow among soil columns
The behavior of different flux parameterizations depends on the model parameter values, especially methods to relate geophysical attributes to model parameters
Example Application: Simulation of snow in open clearings
• Different model parameterizations (top plots) do not account for local site characteristics, that is dust-on-snow in Senator Beck
• Model fidelity and characterization of uncertainty can be improved through parameter perturbations (bottom plots)
Reynolds Creek
Senator Beck
Example application: Interception of snow on the vegetation canopy
• Again, model fidelity and characterization of uncertainty can be improved through parameter perturbations
Different interception formulations Simulations of canopy interception (Umpqua)
Example Application: Transpiration
Biogeophysical representations of transpiration necessary to represent diurnal variability
Interplay between model parameters and model parameterizations
Rooting depth
Hydrologic connectivity
Soil stress function
Example Application: Importance of model architecture (spatial variability and hydrologic connectivity)
! 1-D Richards’ equation somewhat erratic ! Lumped baseflow parameterization produces ephemeral behavior ! Distributed (connected) baseflow provides a better representation of runoff
• The necessary ingredients of a model (modeling in general) ▫ State variables, process parameterizations, model parameters, model
forcing data, and the numerical solution ▫ Two examples:
� Temperature-index snow model � Conceptual hydrologic model
• Physically-motivated snow modeling ▫ Major model development decisions
• Understanding the impact of key model development decisions ▫ General philosophy underlying SUMMA ▫ Case studies: Reynolds Creek and Umpqua
• Summary and research needs
Outline
Summary of SUMMA
• Objectives ▫ Better representation of observed processes (model fidelity) ▫ More precise representation of model uncertainty
• Approach: Detailed evaluation of competing modeling approaches ▫ Recognize that different models based on the same set of governing equations ▫ Defines a “master modeling template” to reconstruct existing modeling
approaches and derive new modeling methodologies ▫ Provides a systematic and controlled approach to model and evaluation
• Outcomes ▫ Provided guidance for future model development ▫ Improved understanding of the impact of different types of model
development decisions ▫ Improved operational applicability of process-based models
Research needs
• Model fidelity ▫ Comprehensive review/analysis of different modelling approaches has helped
identify a preferable set of modeling methods � Some obvious: biophysical representation of transpiration, two-stream canopy radiation, dust
deposition on snow, etc. ▫ Need to place much more emphasis on parameter estimation
� A science problem rather than a curve-fitting exercise � Focus on relating geophysical attributes to model parameters � Use multiple datasets at different scales to reduce compensatory errors
• Model uncertainty ▫ Improved understanding of suitable methods to characterize uncertainty in
different parts of the model � Distinguish between decisions on process representation versus decisions on choice of
constitutive functions ▫ Recognize shortcomings of using multi-physics and multi-model approaches
to characterize uncertainty � Competing models can provide the wrong results for the same reasons (albedo example)
▫ Need to approach uncertainty quantification from a physical perspective � Inverse methods are plagued by compensatory interactions among different sources of
uncertainty… difficult to infer meaningful uncertainty estimates � Progress possible through a more refined analysis of individual model development decisions
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There are known knowns. These are things we know that we know. There are known unknowns. That is to say, there are things that we know we don't know. But there are also unknown unknowns. There are things we don't know we don't know.
Donald Rumsfeld.
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