Example Results

Jones et al. (2004) presented a sensitivity analysis to assess the benefits of using this EnKF to estimate SOC over time, relative to measurements alone, for different combinations of model parameters, errors, and initial conditions. Here, base case results from this study are summarized, and it is shown how errors of SOC estimation vary over time under different frequencies of measurements (and thus, frequencies of updating estimates of X and R).

To demonstrate numerical values, realistic parameters and error terms were selected for a case study. The variance of SOC measurements (a§) was set at 500,000 (a standard deviation of 707 kg[C] ha-1 or a standard error of measurement of 0.0253% C on a mass basis). The initial value of SOC was assumed to be 16,000 kg[C] ha-1 in the top 20 cm of soil, which is about 0.6% carbon on a mass basis (Yost et al., 2002). Variance of this initial SOC estimate was assumed to be 20,000 (kg[C] ha-1)2, a standard deviation of 141 kg[C] ha-1. It was also assumed that the model error variance (af) was 20,000 (kg[C] ha-1)2. The value of R0 was assumed to be 0.020 (based on a range of values reported by Pieri, 1992, and Bationo et al., 2003); the variance in this parameter (an) was assumed to be 0.0001. The value of Ut was set at a relatively high value of 2000 kg[C] ha-1, constant across all years, and the value of b at 0.20.

Equation 16.3 was used to generate measurements (Zt) for t = 1 through 30 years for a hypothetical field for which SOC is to be estimated, using an initial value of SOC of 16,000 kg[C]/ha and an R value of 0.010. The difference between R0 and R for a particular field conceptually represents the variability among fields that belong to the population of fields

Table 16.5 Values of Parameters, Initial Conditions, and Inputs for Example Ensemble Kalman Filter

Variable Definition

X0 True value of soil organic carbon at tir 0

R True value of mineralization parame'

Ut time

Variance in model estimates of soil organic carbon, each year time step Initial estimate of soil C decomposition parameter Variance of decomposition rate parameter

Units

Value

kg[C]/ha

16,000

1/year

0.010

(kg[C]/ha)2

500,000

(kg[C]/ha)2

20,000

1/year

0.020

(1/year)2

0.0001

kg[C]/ha

2,000

_

b Proportion of annual soil C input tha remains after 1 year

Source: Adapted from Jones, J.W., W.D. Graham, D. Wallach, W.M. Bostick, and J. Koo. 2004. Trans. ASAE, 47(1):331-339.

a that has a mean value of R0. The updated estimates of Rt should converge to the value for the specific field, starting from the initial value of 0.020. The values of parameters and initial conditions used to implement the EnKF are summarized in Table 16.5. The EnKF was used to estimate X and R, and their variances for each year of the 30 years for which measurements were generated. Annual changes in SOC estimated from measurements (Zt - Zt-1) and from EnKF estimates (Updated Xt+1 - Updated Xt), were compared with true values that were generated.

Figure 16.2 shows EnKF estimates of SOC for the inputs used (Table 16.5), as well as annual measurements (generated as discussed above), and "true" values of SOC for the 30-year case study. Estimates made by the EnKF are smooth, and in most years are closer to the "true" values than measured values. Estimates of R evolved from an initial estimate of 0.020 year-1 to values near the "true" value of 0.010 after about 6 years (not shown), and remained near that value for

Years

Figure 16.2 Changes in soil organic carbon (SOC) over time based on measurements (open symbols) and EnKF (heavy line) compared with "true" values of SOC (light line). (Modified from Jones, J.W., W.D. Graham, D. Wallach, W.M. Bostick, and J. Koo. 2004. Trans. ASAE, 47(1):331-339.)

Years

Figure 16.2 Changes in soil organic carbon (SOC) over time based on measurements (open symbols) and EnKF (heavy line) compared with "true" values of SOC (light line). (Modified from Jones, J.W., W.D. Graham, D. Wallach, W.M. Bostick, and J. Koo. 2004. Trans. ASAE, 47(1):331-339.)

the remainder of time in the case study. The effect of the EnKF is clear when one compares annual changes in SOC (Figure 16.3). Over the 30-year study, the EnKF estimates of annual changes were closer to "true" values in all years except one (year 10). EnKF estimates of annual changes in SOC were improved more than estimates of SOC vs. time when compared with measurements because of the smoothing process that occurs when model estimates are combined with measurements.

Three additional runs were made to demonstrate the effect of measurement frequency on standard error of SOC estimates. Errors for measurements made every year, every 2 years, every 3 years, and every 5 years are shown in Figure 16.4. Also shown is the standard error of SOC estimate obtained from using measurements alone (707 kg ha-1). These results indicate that SOC estimation errors using the EnKF and measurements every 3 years would be less than errors based on measurements alone. They also show that errors in SOC estimates decrease over time, after an initial increase,

C1000

c 500

ro 0

c ra

C1000

c 500

ro 0

c ra

■ Measured H EnKF Estimates □ True Values

Figure 16.3 Annual changes in soil organic carbon comparing EnKF estimates with measured and true values. (Modified from Jones, J.W., W.D. Graham, D. Wallach, W.M. Bostick, and J. Koo. 2004. Trans. ASAE, 47(1):331-339.)

■ Measured H EnKF Estimates □ True Values

Figure 16.3 Annual changes in soil organic carbon comparing EnKF estimates with measured and true values. (Modified from Jones, J.W., W.D. Graham, D. Wallach, W.M. Bostick, and J. Koo. 2004. Trans. ASAE, 47(1):331-339.)

1200 1000 800 600 400 200 0

15 20

Years

Figure 16.4 Effect of measurement frequency on errors of soil organic carbon (SOC) estimates. The heavy dashed line is the standard error of SOC estimates based on measurements alone. (Modified from Jones, J.W., W.D. Graham, D. Wallach, W.M. Bostick, and J. Koo. 2004. Trans. ASAE, 47(1):331-339.)

which is the result of more accurate estimates of R and lower model prediction error. Jones et al. (2004) reported on a more comprehensive analysis of the EnKF under different combinations of parameters, initial conditions, and errors in model and measurements. They found that estimates of SOC were better than measurements alone in all combinations when this simple model was used in the EnKF, although estimation errors decreased more under some combinations than others.

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