spatial approximations, process parameterizations, and ... · (1998) “elements of physical...

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)


•  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

= − −


•  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




•  Understanding the impact of key model development decisions ▫  General philosophy underlying SUMMA ▫  Case studies: Reynolds Creek and Umpqua


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






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






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

heli drop

volunteers

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.

QUESTIONS ???

spatial approximations, process parameterizations, and ... · (1998) “elements of physical...

Documents