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Terrestrial carbon cycle parameter estimation from the ground-up: A case study for Australia.

Parameter estimation of a terrestrial C-Cycle model using multiple datasets of ground based observations.

Model-Data Integration and Network Design for Biogeochemical Research Advanced Study Institute, National Center for Atmospheric Research, May 2002.

Dr Damian Barrett CSIRO Plant Industry, GPO Box 1600 Canberra ACT Australia.Damian.Barrett@CSIRO.au

Thanks: Michael Raupach, Dean Graetz, Ying Ping Wang, Peter Rayner, Ray Leuning, John Finnigan and other ‘Carbon Dreaming’ participants...

• Background: Motivations for developing ‘yet another’ terrestrial BGC model of the C-cycle.

• Forward Model: Conservation equations, parameters, state variables, forcing functions, and driver data.

• Parameter estimation: Recast the forward model as a steady state model, multiple observation datasets, search algorithm (GAs) and parameter covariance matrix

• Output: Parameter estimates (turnover time of C in soil and litter pools, depth profiles of soil C efflux, light use efficiency of NPP).

Topics:

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Science Motivations 1: Reducing uncertainty in carbon cycle

• Large uncertainties in the global C-cycle particularly with terrestrial biogeochemistry, particularly below ground processes.

• Limited capability to observe below-ground dynamics, fluxes and transformations of carbon.

• Depth distribution of turnover time of C in soil? • Depth distribution of soil C flux?

• Limited observations: very patchy & disparate data

• Limited process understanding: • Some processes are well understood (photosynthesis & 13C discrimination, decomposition of litter and soil organic matter).

• Other processes are poorly understood (C-allocation among plant tissues, T sensitivity of humus decomposition, 13C discrimination of decomposition…)

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• An application of the parameter estimation problem • using a forward model of NEE of C (VAST) &• multiple datasets of observations of plant production and pool sizes to constrain parameters in VAST.

• Approach is distinct from Data Assimilation:

• are not estimating initial conditions, updating model state variables in time nor estimating time dependent control parameters

• are estimating steady state model parameters (ie no time- or space-dependency in parameters)

• We use algebraic ‘scaling functions’ in the forward model to introduce time- and space-dependency in parameters

Science Motivations 2: Approach in a nutshell

Scene setting: Australia and North America

• Statistics Australia N. America (Conterminous USA)• Land area (km2): 7.6 x 106 24.2 x 106 (9.2 x 106 km2)• MAR: 479 mm 630 mm • Evaporation: 437 mm 301 mm• Runoff: 50 mm 329 mm• onset of agriculture: 1860 ~1750

Australia is characterized by high year-year climate variability, high vapor pressure deficits, highly weathered soils, high biodiversity and an evergreen vegetation evolved in isolation and adapted to these conditions

• A linear compartmental model of C-dynamics of the terrestrial biosphere

• Linear dependence of qk and Pn on parameters.

• 10 pools

• Plant Production: Light Use Efficiency approach

• Mortality and Decomposition: modeled as first-order kinetics.

• Forced T, P, NDVI, n,s.

• 3 classes of parameters • 12 Partitioning• 10 Timescale• Additional process *,

VAST1.1: Forward Model – schematic diagram

PnqL

qW

qR1

qR2

qR3

qF

qC

qS1

qS2

qS3

F

C

S1

S2

S3

L

W

R1

R2

R3

F

C

S1

S2

S3

Pg

L

W

R

1

2

3

Ra

Rh

Biomass

Littermass

Soil-C

L, W, R, R1, R2, R3, F, C, , S1, S2, S3

L, W, R1, R2, R3, F,

C, S1,

S2, S3

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• VAST1.1 Input C-flux: Light Use Efficiency model

• Mass conservation equations: a system of 10 coupled first-order ODE.

*,

1

1n

n i p ii

P

VAST1.1: Forward Model

LLnLL qPdtdq

WWnWW qPdtdq

jjj RRnRjR qPdtdq

FFCCLLF qqqdtdq

CCWWC qqdtdq

11111 SSCCCFFFRRS qqqqdtdq

jjjjjjjj SSSSSRRS qqqdtdq 111

VAST1.1: Forcing data

Data:• Climate data: BoM 0.25o Monthly max/min

T(oC) (1950 - 2000) & Monthly rainfall(mm) (1890 - 2000)

• NASA PAL 8km-10day NDVI: Noise Filtered (Lovell & Graetz) & re-georef (Barrett) (1981 - 2000)

• NASA Langely SRB: Monthly Shortwave down & net radiation (1983 - 1991)

• Digital Atlas of Australian soils + Interpretation (McKenzie and Hook 1992): depth, ksat.

• Digital Atlas of Australian historic vegetation: “Pre-European” growth form of tallest stratum and FPC.

VAST1.0: Multiple observations dataset

VAST 1.0 Observation Dataset:

183 obs NPP 105 obs above ground biomass 94 obs fine littermass 346 obs soil [C] 55 obs soil bulk density

From 174 published studies.

Available:

http://www-eosdis.ornl.gov

VAST: Multiple observation datasets: interpretation

• Observation sites: vegetation is ‘minimally disturbed’ • where the return period of stand replacing disturbance is longer than the recovery period of vegetation to maximum biomass.

• Assume vegetation is at steady-state• ie when averaged over space and time, the rate of change of C-mass in any pool is zero (where C influx into the biosphere = C efflux from biosphere).

• Criteria to meet steady state assumption • Author’s description of overstorey vegetation was equivalent with AUSLIG 1788 Historical Australian Vegetation Classification

• Age sequence: oldest age vegetation used

• Multiple sites at a single lat/long: data were averaged among sites.

• Only data from published literature was used (Quality control = peer review)

• Only geo-referenced data used (lat/long)

5

10

15

20

25

30

Mean annual rainfall (mm)

0 500 1000 1500 2000

Mea

n an

nual

tem

pera

ture

(o C

)

5

10

15

20

25

0 500 1000 1500 2000 2500

• Open circles depict individual grid cells of continental raster in climate space.

• Colored circles show location of observations in climate space.

• NPP, Biomass and Littermass observations are biased towards higher rainfall/productivity sites

• Bias in the landscape (more productive sites)

• Soil observations are more representatively distributed

VAST: Observation datasets in climate space

NPP qL + qW

qF Soil [C]

AS Savannahs TF

VAST: Parameter estimation – weighted least squares

• Aim: Estimate a by minimising the objective function, O(a), given ŷ, x & y, :

where• y vector of observations for multiple datasets (ie. the VAST 1.0 Obs Dataset)

• ŷ(.) corresponding vector of model predictions (based on steady state equations)

• x vector of forcing variables (climate, radiation, NDVI…)

• a vector of model parameters

• Cy-1 is inverse of the error covariance matrix (a symmetric weighting matrix containing

information on measurement error and correlations among measurement errors).

• where measurement errors are gaussian, uncorrelated and errors constant variance (Cii are equal & Cij = 0): Ordinary least squares• where Cii are not equal & Cij = 0: Weighted least squares. • where Cii are not equal & Cij ≠ 0: Generalised least squares.

• Multiple constraints case: need to deal with observations of unequal magnitude & consequently have unequal variances.

• VAST1.1: we assume that measurement errors are independent and gaussian and that the error variances are equal to the sample variances for each dataset.

1ˆ ˆ( ; ) ( ; )O T

ya y y x a C y y x a

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• Since observations are from ‘minimally’ disturbed sites (ie. are assumed to represent steady state conditions) we need to express the conservation equations in steady state form.

• Recalling that at steady state:

• Re-arrange conservation equations steady-state form:

Where fk is the fraction of NPP which has passed through pools upstream of qk.

k n k kq P f

0, 0dq q

Idt

VAST: Specification of steady state model

• fk in VAST1.1 are :

• Subject to

VAST: Specification of inverse model

WCWLFRSRSRS

WCWLFRSRS

WCWLFRS

WC

WLF

RjjR

WW

LL

f

f

f

f

f

f

f

f

112233

1122

11

3

1

1and,1j

jRWL

VAST: Uncertainty in estimated parameters• The uncertainty in parameters is given by the parameter covariance matrix

Cb = sy [JT J]-1

where J is the Jacobian; the matrix of model derivatives with respect to parameters

The Jacobian is of dimensions n rows x p columns (n = Total Number of observations and p = No. of parameters)

Each element of the Jacobian is

sy is the residual sum of squares

sy = [y – ŷ(x; a)]T [y – ŷ(x; a)] / (n – p)

,

ˆ; 1,..., ; 1,..., ; 1,...,

n m

ijijk

k

yJ i n j m k p

a

x

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Parameter estimation using multiple datasets

• Equations in each ŷ must share at least some parameters in common

• otherwise there is no mutual constraint imposed by the multiple datasets (off diagonal elements of [JT J] = 0)

• This is equivalent to independent parameter estimates

• Shared parameters between equations must be on an equivalent SCALE

• eg. Photosynthesis models at leaf and canopy scales.

• leaf scale Jmax: e- transport processes in chloroplast

• canopy scale Jmax: a statistical average over a pop.

• observations used to constrain the canopy model cannot constrain estimates of the leaf scale parameter (unless we have a way of disaggregating the large scale information among the population leaves).

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Parameter estimation using multiple datasets (continued)…

•Highly correlated datasets add little information to constrain parameter estimates

• eg. Do N concentration datasets provide a further constraint on C fluxes?

• Due to conserved C:N ratios in terrestrial pools, C & N data are highly correlated

• Therefore including N data does not necessarily add much new information to more tightly constrain parameters.

• In practice, Cy may be very difficult to specify

• particularly for multiple datasets where information on error correlation between datasets is unavailable.

Search method: Genetic algorithms

• a type of stochastic search technique that is particularly useful in

optimisation where...the region of the global minimum of O occupies a small fraction

of parameter spaceparameter space is rough (numerous local minima)parameter space is discontinuous (jacobian is unavailable)

• Example: shows the evolution of a solution to the global minimum

of a particularly difficult function

• Start with a random selection of population members which are

solutions to the problem and evolves the population towards the

global minimum within 90 trials

• Note: local minima are not retained in the population if other “fitter”

members are found

even though the global minimum is found in < 90 trials,

mutation maintains diversity in the search of parameter space

1

60

90

global minimum

Genetic algorithms: a Primer

• Population is made up of a set of “Chromosomes” = (a

set of parameters) comprising “Genes” (1 per parameter).

• Each parameter value (gene) is encoded into a binary

string.

• Crossover operator: generates offspring from mating

parent chromosomes.

• Mutation operator: creates new genes stochastically.

• Selection Operator: selects chromosomes based on a

‘Fitness’ function.

• GAs generate solutions to problems by evolving the

population over time and selecting for fitter solutions.

They increase the average “fitness” of a population of

model solutions by exploiting prior knowledge of

parameter values retained in the population.

• For difficult objective functions: Can use monte carlo

approaches to obtain estimates of parameter

uncertainties. (not done here)

...10010111 | 100111010 | 100010111...

...1100000 | 1110110 | 111111101...

314

3.14

{...pi-1, pi, pi+1...}

...10010111 | 1001110 | 101111101...

...1100000 | 111011010 | 100010111...

0.784.74

0.78

P2

P1

O2

O1

O2O1

chromosomegene

encoding

decoding

mutationcrossover

selection

Turnover Time (years)

0 50 100 150 200 250

Dep

th (

cm)

-100

-80

-60

-40

-20

0

VAST: Turnover time of soil C pools

• Estimated turnover time of C as a function of soil depth (+/- 1) corrected for temperature and moisture effects on decomposition

•In situ turnover time at any time and place is modified by climate, soil moisture content of the soil and vegetation type.

• Faster turnover of carbon in surface soil.

• Turnover time of C not significantly different between 20 – 50 cm and 50 – 100 cm depths.

• Increasing turnover time with depth reflects decreasing decomposition rate, due to less labile C, lower nutrient or oxygen availability, increasing physical protection of C by higher soil clay contents,…

• Plots show realizations of the fraction of soil C-flux originating from fine and coarse litter pools and from different soil horizons for each of the 3 vegetation types.

• More than 89% of total soil C-flux originates from < 20cm depth (>98% < 50cm)

• Largest source of C flux from soil is fine litter

• Model is relatively insensitive to uncertainty in the estimated turnover time (predicted soil respiration in 50 - 100 cm horizon has low uncertainty).

VAST: Depth profiles of soil carbon flux

‘Tall’ productive forests Open Woodlands Arid shrublands

0.00 0.20 0.40 0.60

-20

0

20

40

60

80

100

0.00 0.20 0.40 0.60

-20

0

20

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100

0.00 0.20 0.40 0.60

-20

0

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100

Soi

l Dep

th (

cm)

FineCWD

Summary points:

To integrate inventory data, remote sensing, flux station and atmospheric [CO2] data for parameter estimation we need to consider the following:

• We need a comprehensive set of forward models to predict system state

• predict fluxes = f(NPP, stores, …)• predict fluxes = f(near surface [CO2], u, …)• predict radiance measures = f(LAI, n, …)• predict atmospheric [CO2] = f(fluxes, atmospheric transport, …)

• We need the forward models to share common parameters

• otherwise no benefit obtained using multiple constraints approach

• Need to address issues of scale in order to relate data obtained on different time and space scales

• eg the need to relate near surface [CO2] to atmospheric [CO2] in order to combine eddy flux and atmospheric CO2 datasets

Summary points: continued…

• We need an objective means for specifying the network design (ie. a quantitative means of identifying the types and locations of measurements)

• How do you decide who’s network is better?• network design is an optimization problem! • so its possible to include in the objective function a cost term for new observations• “Is it better to generate extensive datasets of cheap and uncertain observations over the scale of interest, than few expensive accurate observations?”

• We need analysis of the information content of different types of datasets

• because adding new datasets may not lead to further constraint on model parameters if:

• new data are highly correlated with existing data

• if by adding new data we also add new model equations having new unknown parameters (just shifts the problem of insufficient information to one of estimation of new parameters).