Ecological Archives M072-004-A2

H. Jochen Schenk and Robert B. Jackson. 2002. The global biogeography of roots. Ecological Monographs 72:311-328.

Appendix B. Methods used for interpolation and extrapolation of root profiles. Tests of the accuracy of inter- and extrapolations of root profiles were conducted with subsets of the global database of 475 root profiles.  These subsets included the following:

1.  The test profiles:  The 76 profiles sampled to at least 0.8 m depth and to depths at which no further roots were found or profiles which had been sampled to 2 m depth or more.  For tests of extrapolation accuracy, each of these test profiles was divided into an upper and a lower part.  The cutoff depth for the division into these two parts depended on the number and depth of sample intervals in each individual profile:  Each of the 76 upper test profiles contained at least four sample intervals and reached to depths of >0.8 to 1.2 m, or where the upper meter contained less than four sample intervals, at most to 1.6 m.  The lower part reached to the maximum sample depth.

2.  The comparison profiles:  The 181 profiles from the whole database that had been sampled to depths ranging from 0.8 to 1.6 m.  These profiles were used as a comparison to the upper test profiles.  This comparison was necessary to test whether there was a systematic difference between the root distributions in the deeply sampled test profiles and other, less deeply sampled profiles in the database.

Root distributions in all subsets of profiles were characterized by two values:  The depths above which 50% and 95%, respectively, of all roots (i.e. either biomass, number, or length) were located in the profile (hereafter termed the 50% and 95% rooting depths).

Testing for a systematic difference between test and comparison profiles

The test profiles differed from all other profiles in the database in that they were completely or deeply sampled (see above), and it may be that this was occasionally done because observations on root distributions in the upper test profiles suggested to the researchers the necessity of sampling to greater depths.  To test whether root distributions within the upper test profiles were representative for those in the rest of the database, we calculated non-extrapolated 50% and 95% rooting depths within these profiles and compared them to non-extrapolated 50% and 95% rooting depths determined for the 181 comparison profiles (see above).  The null hypothesis was that the 76 upper test profiles had different mean 50% and 95% rooting depths than the 181 comparison profiles.  The comparison was done by a bootstrap analysis.  A thousand random subsamples of size n = 76 were taken from the total sample of 181 comparison profiles, and the mean 50% and 95% rooting depth was calculated for each of these subsamples. The mean rooting depths in the upper test profiles were not different from those observed in the comparison profiles (Fig. B1).  This suggests that there was no systematic difference between the root distributions in the test profiles and root distributions in the other profiles in the database.


 

Fig. B1.  Comparisons of vertical root distributions (characterized by their 50% and 95% rooting depths) within the upper 0.8 to 1.6 m of the 76 test profiles to root distributions in the comparison profiles sampled incompletely to between 0.8 and 1.6 m depth.  The rooting depths for the comparison profiles represent means of 1000 random sub-samples of sample size n = 76 from a total sample of 181 profiles, and the error bars represent 95% confidence intervals.


Interpolation of root profiles

Root profiles differed greatly in the number and depth of intervals sampled, which made it necessary to standardize them in order to enable statistical analyses to weigh each individual profile equally.  To achieve this, profiles were interpolated by fitting a nonlinear smoothing function to the data.  The goal of interpolations was to calculate 50% and 95% rooting depths for each profile, hereafter denoted as D50 and D95.  The accuracies of these nonlinearly interpolated rooting depths were tested for the 76 test profiles against rooting depths determined by linear interpolations between individual sampled depth intervals.  These linearly interpolated rooting depths will be denoted as Di50 and Di95, respectively.

Six nonlinear mathematical functions were evaluated as smoothing functions.  These included four log-log and log-linear functions used previously for inter- and extrapolation of soil carbon profiles (Jobbágy and Jackson 2000) and two other functions for use with cumulative data:  a modified version of the Gale and Grigal (1987) function used by Jackson et al. (1996) and a logistic equation known as the logistic dose-response curve (LDR).  Only the latter two are discussed here, because they provided far better fits than the other four functions.

Mathematical functions were fitted to cumulative root profile data using the software TableCurve for Windows, version 1.11 (Jandel Scientific, San Rafael, California, USA).  The Gale and Grigal (1987) model (hereafter termed the -model) in its original version sets the total amount of roots equal to one.  Because the total amount of roots is not a known quantity in most profiles, we modified this equation by adding an additional parameter, Rmax, for the total amounts of roots (i.e. total biomass, length, number etc.) in the profile.  The equation is:

r (D) = Rmax (1 - D
(B.1)

where r (D) is the cumulative amount of roots above profile depth D (in cm, including the organic layer), and is the depth coefficient of the Gale and Grigal model.  The second function was the logistic dose-response curve (hereafter termed the LDR model):

(B.2)

where D50 is the depth (cm) at which r (D) = 0.5 Rmax, and c is a dimensionless shape-parameter.  These functions were fitted to each of the 76 test profiles, allowing Rmax to vary to obtain the best fit.

Both nonlinear interpolation methods discussed here used cumulative data, which meant that conventional regression techniques could not be used to evaluate the goodness of fit for individual profiles, because individual data points in a cumulative curve are not statistically independent.  Because of this, we evaluated overall interpolation accuracies by regressing D50 and D95 determined by nonlinear interpolation against the corresponding rooting depths determined by linear interpolation between sample intervals (Di50 and Di95).


 

Fig. B2.  Comparison of rooting depths calculated by two nonlinear 
interpolation-methods (beta: Eq. .B1, and LDR: Eq. B.2) 
to rooting depths calculated by linear interpolation between individual 
sampling intervals.


Extrapolation of root profiles

The goal of profile extrapolation was to estimate the distributions of roots to their maximum depth or to 3 m depth at most.  Extrapolations were done using LDR-models (Eq. B.2) that were fitted to the measured parts of the profiles.  To avoid excessive extrapolation errors, profiles were extrapolated to no more than twice their sampled depth or to 3 m depth at most.  Profiles sampled to the maximum rooting depth or to at least 3 m were not extrapolated.  The cumulative amounts of roots at the bottom of the extrapolated profiles were set to Rmax = 100%.  Extrapolated rooting depths are denoted as Dx50 and Dx95 for the extrapolated 50% and 95% rooting depths, respectively.

For tests of extrapolation accuracy, each of the 76 upper test profiles was extrapolated.  All parameters in the LDR model (Eq. B.2) were allowed to vary to obtain the best fit, but the D50 parameter was constrained to be >= 80 cm, because none of the completely and deeply sampled test profiles, including those measured to 3 m or more, had D50 values of > 80 cm.  Extrapolated rooting depths (Dx50 and Dx95) were evaluated against the corresponding, interpolated D50 and D95 for the whole test profiles by linear regression.  Regression lines were forced through the origin, because extrapolated rooting depths theoretically could only equal zero when interpolated rooting depths equaled zero.

A bootstrap analysis was conducted to develop estimates of total errors for mean rooting depths, including inter- and extrapolation errors, as a function of the number of profiles used to estimate the means.  Mean Dx50 and Dx95 were calculated for 1000 random subsets of 10 to 70 test profiles and were compared to the corresponding mean interpolated Di50 and Di95 for the whole data set of test profiles.  Errors were expressed as the deviation (in %) of the mean Dx50 and mean Dx95 from the mean Di50 and mean Di95, respectively.  Error ranges were estimated as the 95 percentiles of these deviations.

Linear regressions of extrapolated rooting depths against rooting depths interpolated for the whole profiles were highly significant (P < 0.0001) and had coefficients of determination of 0.905 (D50) and 0.786 (D95).  Slopes were slightly, but significantly, lower than one (D50: 0.94 ± 0.04 CI95%, D95: 0.95 ± 0.05 CI95%), and this was largely caused by underestimation of a few large rooting depths (Fig. 2A in paper).  Total errors, incl. inter- and extrapolation errors, of estimated mean rooting depths decreased with the number of profiles used to derive the estimate from up to ±40% of the mean for samples of 10 profiles to less than ±10% of the mean for samples of 60 profiles or more (Fig. 2B in paper).  There was a slight tendency towards underestimating mean rooting depths by about 1 to 3%.  Based on these tests, the procedure described above for extrapolating the upper test profiles was used for extrapolating all other profiles in the database.  The 95% error ranges depicted in Fig. 2B (in the paper) were used to estimate 95% confidence intervals for all mean rooting depths calculated in this study, as a function of the number of profiles used to calculate the means.


Literature Cited

Gale, M. R., and D. F. Grigal. 1987. Vertical root distribution of northern tree species in relation to successional status. Canadian Journal of Forest Research 17:829-834.

Jackson, R. B., J. Canadell, J. R. Ehleringer, H. A. Mooney, O. E. Sala, and E. D. Schulze. 1996. A global analysis of root distributions for terrestrial biomes. Oecologia 108:389-411.

Jobbágy, E. G., and R. B. Jackson. 2000. The vertical distribution of soil organic carbon and its relation to climate and vegetation. Ecological Applications 10:423-436.
 
 
 


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