If soil C sequestration is to become an accepted mechanism for reducing atmospheric CO2 levels, a soil carbon accounting system needs to be developed (Antle and Uehara, 2002). Mass of carbon accumulation in soils is of interest, so measurements will include field sampling, laboratory determination of carbon, and its conversion to mass basis using soil bulk density. Thus, errors in such measurements would include errors associated with each step. As a result, yearly changes in soil C are small relative to errors associated with the measurement process, and such measurements are expensive. Biophysical models can be used to estimate SOC and its changes under different weather, soil, and management practices (Parton et al., 1988, 1994; Jones et al., 2002b). However, although these models may produce precise estimates, they are imperfect and parameters for specific field situations are uncertain. Thus, errors exist in estimates of SOC from field measurements and from model predictions.

Techniques exist to combine models and measurements to obtain better estimates of system states and model parameters. The Kalman filter (Maybeck, 1979; Welch and Bishop, 2002) approach first uses a model to predict the state of a system, and then uses measurements to update the estimates in an optimal way, taking into account errors in measurements and predictions. Variations of the Kalman filter, originally developed for linear models, have been developed for non linear models (e.g., Albiol et al., 1993; Graham, 2002). One variation, the ensemble Kalman filter (Burgers et al., 1998; Eknes and Evensen, 2002; Margulis et al., 2002), was used by Jones et al. (2004) to evaluate its use for estimating SOC and a decomposition rate parameter over time for a single field using a nonlinear model.

Here the use of ensemble Kalman filter (EnKF) methodology to combine measurements with models to estimate SOC is demonstrated. Analyses in this chapter focus on estimation of SOC over time (years) for a single field following the work of Jones et al. (2004); they do not address spatial variability or aggregation of estimates over space.

A simple discrete-time model is used to simulate SOC (Xt, kg ha-1) as it changes over time, using a time step of 1 year. It is assumed that only one pool of C exists in the soil, and that biomass organic carbon (Ut, kg ha-1) may increase this pool, while during the same annual time step, microbial activity decomposes both Xt and Ut. The model also has one parameter (SOC decomposition rate constant, R, year-1) that is constant over time, but is not known with certainty. The resulting model has one state variable (Xt) and one parameter (R), which are estimated using the EnKF. Uncertainties in model predictions of SOC and R are assumed. State equations for the nonlinear, stochastic model follow:

where

X = soil organic carbon in year t (kg[C] ha-1) R = rate of decomposition of existing SOC (year-1) R0 = initial estimate of SOC decomposition rate (year-1) b = fraction of fresh organic C that is added to the soil in year t that remains after 1 year Ut = amount of C in crop biomass added to the soil in year t (kg[C] ha-1 year-1) £t = model error for SOC (kg[C] ha-1) 0 = error in estimate of decomposition rate R (year-1)

Model error (£t) includes uncertainties in U and b, as well as uncertainties due to the fact that the model is a simplification of reality. It is also assumed that model and parameter errors are normally distributed and uncorrelated. Thus,

where a2 = variance of model error for Xt = variance of model error for R

The model error (£t) is a random process that changes over time but is uncorrelated with time (i.e., white noise), whereas decomposition rate parameter error (n) is a random variable that does not change with time.

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