Data assimilation in terrestrial biosphere models: exploiting mutual constraints among the carbon, water and energy cycles Michael Raupach, Cathy Trudinger, Peter Briggs, Luigi Renzullo, Damian Barrett, Peter Rayner Thanks: colleagues in the Australian Water Availability Project (CSIRO, BRS, BoM); participants in OptIC project CarbonFusion (Edinburgh, 9-11 May 2006)
29
Embed
Michael Raupach, Cathy Trudinger, Peter Briggs, Luigi Renzullo, Damian Barrett, Peter Rayner
Data assimilation in terrestrial biosphere models: exploiting mutual constraints among the carbon, water and energy cycles. Michael Raupach, Cathy Trudinger, Peter Briggs, Luigi Renzullo, Damian Barrett, Peter Rayner - PowerPoint PPT Presentation
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Data assimilation in terrestrial biosphere models: exploiting mutual constraints among the
carbon, water and energy cycles
Michael Raupach, Cathy Trudinger, Peter Briggs, Luigi Renzullo, Damian Barrett, Peter Rayner
Thanks: colleagues in the Australian Water Availability Project (CSIRO, BRS, BoM); participants in OptIC project
CarbonFusion (Edinburgh, 9-11 May 2006)
Outline
Data assimilation challenges for carbon and water
Multiple-constraint data assimilation
Using water fluxes (especially streamflow) to constrain carbon fluxes
Observation models for streamflow (with more general thoughts on scale)
Example: Murrumbidgee basin
Model-data fusion: comparison of two methods
Carbon DA
Challenges for carbon cycle science (including data assimilation)
• Science: finding state, evolution, vulnerabilities in C cycle and CCH system
• Policy: supporting role: IPCC-SBSTA-UNFCCC, national policy
• If d[water store]/dt can be neglected (small store or long averaging time):
• [streamflow] = [runoff] + [drainage]
• [water store] includes groundwater within catchment, rivers, ponds ...
Requirements for unimpaired catchment
• All runoff and drainage finds its way to the river (no farm dams)
• No other water fluxes from the river (eg irrigation, urban water use)
• No major dams (otherwise d[store]/dt dominates streamflow)
• Groundwater does not leak horizontally through catchment boundaries
Snow
• needs a separate balance
Streamflow (and other) data issues
Requirements on catchments
• Unimpaired, gauged at outlet
• Catchment boundary must be known
Requirements on measurement record
• Well maintained gauge
• The water agency must be prepared to give you the data
Requirements on other data
• Need spatial distribution of met forcing (precip, radiation, temperature, humidity)
• Need spatial distribution of soil properties (depth, water holding capacity ...)
• Catchments are hilly:
• Downside: everything varies
• Upside: exploit covariation of met and soil properties with elevation
(eg: d(Precipitation)/d(elevation) ~ 1 to 2 mm/y per metre
• ANUSplin package (Mike Hutchinson, ANU)
Modelling at multiple scales
We often have to predict large-scale behaviour from given small-scale laws:
Small-scale dynamics Large-scale dynamics
Four generic ways of approaching this problem:
1. Full solution: Forget about F, integrate dx/dt = f(x,u) directly
2. Bulk model: Forget about f, find F directly from data or theory
3. Upscaling: Find a probabilistic relationship between small scales (f) and large scales (F), for example by:
4. Stochastic-dynamic modelling: Solve a stochastic differential equation for PDF of x (small scale), and thence find large-scale F:
,state vectorexternal forcing
d dttt
x f x uxu
, , withd dt
X xX F X U
U u
, , , d d xuF X U f x u x u x u
Raupach, Barrett, Briggs, Kirby (2006)
Steady-state water balance: bulk approach
Steady state water balance:
Dependent variables: E = total evaporation, R = runoff
Independent variables: P = precipitation, EP = potential evaporation
Similarity assumptions (Fu 1981, Zhang et al 2004)
Solution finds E and R (with parameter a)(Fu 1981, Zhang et al 2004)
1
1
,
,
aa aP P P
aa aP P P
E P E P E P E
R P E P E E
1 1
2 2
, with 0 0 (wet limit)
, with 0 0 (dry limit)
P
P P
E P f E E P f
E E f P E E f
0 P T S R DdW dt Q Q Q Q Q
P E R
Fu (1981)Zhang et al (2004)
Normalise with potential evap EP:plot E/EP against P/EP
Normalise with precipitation P:plot E/EP against EP/P
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2 2.5 3
P/Q
E/Q
NECoastSECoastTasAgricNtropicsArid2345
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2 2.5 3
Q/P
E/P
NECoastSECoastTasAgricNtropicsArid2345
Steady water balance: bulk approach
dry wet
wet dry
a=2,3,4,5
a=2,3,4,5
Fu (1981)Zhang et al (2004)
P/EP
E/E
P
EP/P
Stochastic-dynamic modelling
Forcing (u) on small-scale process (x(t)) is often random, and dynamics (dx/dt = f(x,u)) often has strong nonlinearities such as thresholds
Examples: soil moisture, dust uplift, fire, many other BGC processes
If we can find x(x), the PDF of x, we can find any average (large-scale) property
Equation for state (x) Equation for PDF of state [px(x)]
Deterministic system
Deterministic dynamic equation Liouville equation
d dt x f x,u xx
d
dt
x f
Stochastic-dynamic modelling
Forcing (u) on small-scale process (x(t)) is often random, and dynamics (dx/dt = f(x,u)) often has strong nonlinearities such as thresholds
Examples: soil moisture, dust uplift, fire, many other BGC processes
If we can find x(x), the PDF of x, we can find any average (large-scale) property
Equation for state (x) Equation for PDF of state [x(x)]
Deterministic system
Deterministic dynamic equation Liouville equation
Stochastic system
(deterministic system with
random perturbations)
Stochastic dynamic equation
u(t) is a Markov process, with transition prob obeying CK eq
Stochastic Liouville equation
-------------- and then --------------
d dt x f x,u
u u uT t L T
xx
d
dt
x f
xuxu u xu
dL
dt
x f
, ,d dt t tx f x u
d d xuF f f x u
Steady-state water balance: stochastic-dynamic approach
Dynamic water balance for a single water store w(t):
Then:
• Let precipitation p(t) be a random forcing variable with known statistical properties (Poisson process in time, exponential distribution for p in a storm)
• Find and solve the stochastic Liouville equation for w(w), the PDF of w
• Thence find time-averages: <w>, E = <e(w)>, R = <r(w)>
dw dt p t e w t r w t
Rodriguez-Iturbe et al (1999)Porporato et al (2004)
0.2 0.4 0.6 0.8 1w
0.5
1
1.5
2
2.5
3
3.5
PDF PDF of w
w(w)
w = relative soil water
increasing precipitation
event frequency
<w>
1 2 3 4PQ
0.2
0.4
0.6
0.8
w parameterbb: zzrrQ
dry wet
P/EP
increasing precipitation
event frequency
Water and carbon balances: dynamic model
Dynamic model is of general form dx/dt = f(x, u, p)
All fluxes (fi) are functions fi(state vector, met forcing, params)
Governing equations for state vector x = (W, Ci):
Soil water W:
Carbon pools Ci:
Simple (and conventional) phenomenological equations specify all f(x, u, p)
Carbon allocation (ai) specified by an analytic solution to optimisation of NPP
precipitation transpiration runoff to drainage tosoilChange of soil
Basic components• Model: containing adjustable "target variables" (y)• Data: observations (z) and/or prior constraints on the model• Cost function: to quantify the model-data mismatch z – h(y)• Search strategy: to minimise cost function and find "best" target variables
Quadratic cost function:
1 1T TJ z yy z h y C z h y y y C y y
Cost function
MeasurementsPrior information
about target variables
Target variables
Model prediction of observations
Covariance matrix of observation error
Covariance matrix of prior information error
Observations Prior information
Estimates the time-evolving hidden state of a system governed by known but noisy dynamical laws, using data with a known but noisy relationship with the state.
Dynamic model:
• Evolves hidden system state (x) from one step to the next
• Dynamics depend also on forcing (u) and parameters (p)
Observation model:
• Relates observations (z) to state (x)
Target variables (y): might be any of state (x), parameters (p) or forcings (u)
Kalman filter steps through time, using prediction followed by analysis
• Prediction: obtain prior estimates at step n from posterior estimates at step n-1
• Analysis: Correct prior estimates, using model-data mismatch z – h(y)
Kalman Filter
1 , , noise with covariancen n n Q x φ x u p
, noise with covariance R z h x u
Parameter estimation with the Kalman Filter
Dynamic model includes parameters p = pk (k=1,…K) which may be poorly known:
Include parameters in the state vector, to produce an "augmented state vector"
The dynamic model for the augmented state vector is
1
1
, state variables: 1,...,
parameters: 1,...,
n n nj
n nj j
X j N
X X j N N K
φ X u
11, , , with ,...n n n n n n
Mx x x φ x u p x
1 1,... , ,... lengthn n n n nM Kx x p p M K X
Parameter estimation from runoff data
Compare 2 estimation methods
• EnKF with augmented state vector (sequential: estimates of p and Cov(p) are functions of time)
• Levenberg-Marquardt (PEST)(non-sequntial: yields just one estimate of p and Cov(p))
Model runoff predictions with parameter estimates from EnKF
Final thoughts
Applications of "Multiple constraints"
• Data sense: atmospheric CO2, remote sensing, flux towers, C inventories ...
• Process sense: measuring one cycle (eg water) to learn about another (eg C)
Requirement for multiple constraints (in process sense)
• "Confluence of cycles"
• Fluxes: cycles share a process pathway controlled by similar parameters
• Pools: cycles have constrained ratios among pools (eg C:N:P)
Streamflow as a constraint on water cycle, thence carbon cycle
• Strength: Independent constraint on water-carbon (and energy-water) cycles (strongest in temperate environments with P/EP ~ 1)
• Limitation 1: obs model = full hydrological model (sometimes can be simplified)
• Limitation 2: streamflow data (availability, quality, access)
Model-data fusion
• Several methods work (focus on EnKF in parameter estimation mode)
• OptIC (Optimisation InterComparison) project: see poster by Trudinger et al.