66
William K. Lauenroth
TABLE 4.4. Example of using Monte Carlo uncertainty analysis to propagate variance through calculations of belowground net primary production. The example uses data from Publicover and Vogt (1993).
The information necessary for Monte Carlo uncertainty analysis is the mean, the variance, and a reasonable range for all of the
input variables and an assumption about the distribution for each variable. If the distribution type is not known, a uniform
distribution can be used (O'Neill et al. 1982). The uncertainty of the final result is determined by making a large number of
calculations (;;:' 10,000) each with a sample of the input variables drawn from their respective distributions. A mean and
variance of the result is then calculated from the results of the Monte Carlo simulations. This example uses data from months
3 and 4 from Table 1 on page 1180 of Publicover and Vogt (1993) and calculates net fine root production for that single
interval.
Month
o
2
3
4
LFR
554.3
671.1
DFR
369.8
391.2
P
200
MR
0.1
0.15
DR
0.15
0.15
M
45
83
D
61
62
I assumed that live fine roots (LFR, g m- 2 ) and dead fine roots (DFR; g m- 2 ) had coefficients of variation of 10% (Publicover
and Vogt 1993), and that mortality rate (MR, g g-l) and disappearance rate (DR, g g-l) also had coefficients of variation of
10%. All, but especially those for MR and DR, are conservative estimates (Publicover and Vogt 1993, Burke and Raynal
1994). I also assumed normal distributions for all of the variables and made 100,000 calculations of net fine root production
(P" g m -2) using the following formulas from Publicover and Vogt (1993):
P, = (LFR, - LFR'_I) + (DRF, - DFR'_I) + D,
(1)
Equation 1 was suggested by McClaugherty et al. (1982) as appropriate for estimating net fine root production during periods
when live and dead fine root biomass both increase. D, was calculated as proposed by Publicover and Vogt (1993):
(
LFR'_IMR)
D, = DFR'_1 +
2
DR,
(2)
The following table contains a sample of 50 values for each of the variables and for P,. These 50 were selected from the 100,000
values generated for the analysis. The mean and coefficient of variation of Pt for these 50 values are 221 g m -2 and 48%. The
results for the entire analysis indicate that a coefficient of variation of 10% for all of the input data produces a mean Pt of
199.7 g m -2 and a coefficient of variation of 50%. See Figure 4.2 for the results of an analysis for a range of coefficients of
variation of the input variables.
LFR(t)
LFR(t - 1)
DFR(t)
DFR(t - 1)
MR(t)
DR(t)
pet)
739.8
567.2
426.9
418.5
0.16
0.15
251
659.7
575.0
388.6
406.5
0.16
0.15
135
591.9
486.0
457.0
285.3
0.15
0.15
327
686.1
512.3
413.4
387.2
0.14
0.17
271
626.1
576.8
333.7
355.8
0.14
0.16
92
657.8
482.1
376.5
403.9
0.13
0.15
214
745.5
586.1
319.8
385.6
0.18
0.18
170
726.2
450.5
396.5
376.7
0.12
0.15
357
687.0
480.5
296.0
344.5
0.14
0.15
213
606.4
529.6
391.1
337.4
0.16
0.16
192
786.8
528.5
331.8
375.0
0.16
0.14
275
569.4
550.3
449.1
373.3
0.13
0.17
164
697.2
562.7
392.1
355.3
0.16
0.16
234
694.6
619.4
349.4
351.1
0.14
0.12
123
739.2
485.7
400.9
453.9
0.15
0.15
272
741.4
508.7
399.2
389.3
0.15
0.16
311
637.5
596.0
395.7
349.6
0.15
0.18
157
(continued)
William K. Lauenroth
TABLE 4.4. Example of using Monte Carlo uncertainty analysis to propagate variance through calculations of belowground net primary production. The example uses data from Publicover and Vogt (1993).
The information necessary for Monte Carlo uncertainty analysis is the mean, the variance, and a reasonable range for all of the
input variables and an assumption about the distribution for each variable. If the distribution type is not known, a uniform
distribution can be used (O'Neill et al. 1982). The uncertainty of the final result is determined by making a large number of
calculations (;;:' 10,000) each with a sample of the input variables drawn from their respective distributions. A mean and
variance of the result is then calculated from the results of the Monte Carlo simulations. This example uses data from months
3 and 4 from Table 1 on page 1180 of Publicover and Vogt (1993) and calculates net fine root production for that single
interval.
Month
o
2
3
4
LFR
554.3
671.1
DFR
369.8
391.2
P
200
MR
0.1
0.15
DR
0.15
0.15
M
45
83
D
61
62
I assumed that live fine roots (LFR, g m- 2 ) and dead fine roots (DFR; g m- 2 ) had coefficients of variation of 10% (Publicover
and Vogt 1993), and that mortality rate (MR, g g-l) and disappearance rate (DR, g g-l) also had coefficients of variation of
10%. All, but especially those for MR and DR, are conservative estimates (Publicover and Vogt 1993, Burke and Raynal
1994). I also assumed normal distributions for all of the variables and made 100,000 calculations of net fine root production
(P" g m -2) using the following formulas from Publicover and Vogt (1993):
P, = (LFR, - LFR'_I) + (DRF, - DFR'_I) + D,
(1)
Equation 1 was suggested by McClaugherty et al. (1982) as appropriate for estimating net fine root production during periods
when live and dead fine root biomass both increase. D, was calculated as proposed by Publicover and Vogt (1993):
(
LFR'_IMR)
D, = DFR'_1 +
2
DR,
(2)
The following table contains a sample of 50 values for each of the variables and for P,. These 50 were selected from the 100,000
values generated for the analysis. The mean and coefficient of variation of Pt for these 50 values are 221 g m -2 and 48%. The
results for the entire analysis indicate that a coefficient of variation of 10% for all of the input data produces a mean Pt of
199.7 g m -2 and a coefficient of variation of 50%. See Figure 4.2 for the results of an analysis for a range of coefficients of
variation of the input variables.
LFR(t)
LFR(t - 1)
DFR(t)
DFR(t - 1)
MR(t)
DR(t)
pet)
739.8
567.2
426.9
418.5
0.16
0.15
251
659.7
575.0
388.6
406.5
0.16
0.15
135
591.9
486.0
457.0
285.3
0.15
0.15
327
686.1
512.3
413.4
387.2
0.14
0.17
271
626.1
576.8
333.7
355.8
0.14
0.16
92
657.8
482.1
376.5
403.9
0.13
0.15
214
745.5
586.1
319.8
385.6
0.18
0.18
170
726.2
450.5
396.5
376.7
0.12
0.15
357
687.0
480.5
296.0
344.5
0.14
0.15
213
606.4
529.6
391.1
337.4
0.16
0.16
192
786.8
528.5
331.8
375.0
0.16
0.14
275
569.4
550.3
449.1
373.3
0.13
0.17
164
697.2
562.7
392.1
355.3
0.16
0.16
234
694.6
619.4
349.4
351.1
0.14
0.12
123
739.2
485.7
400.9
453.9
0.15
0.15
272
741.4
508.7
399.2
389.3
0.15
0.16
311
637.5
596.0
395.7
349.6
0.15
0.18
157
(continued)