Ecological Archives A023032A1
Peter Schall, Christian Ammer. 2013. Can land use intensity be reliably quantified by using a single selfthinning relationship?. Ecological Applications 23:675–677. http://dx.doi.org/10.1890/120847.1
Appendix A. Conceptual shortcomings and inaccuracies in the inventory data used by L11.
In this appendix we use the definition of the terms N, N_{obs}, N_{maxx}, N_{pr}, c_{dbh}, dbh, dbh_{obs}, dbh_{max}, dbh_{maxx}, dbh_{maxxobs}, LUDI_{o}, LUDI_{p}, LUDI_{ppr} given in L11. Readers may refer to this paper for definitions.
Conceptual shortcomings
LUDI is supposed to measure “the difference between potential and actual biomass storage” at “long and short timescales associated with management and disturbance” (by components LUDI_{p}, the “planning intensity”) and LUDI_{o}, the “operational intensity”. However, the components of LUDI (Fig. A1) definitely fail do not measure the difference between potential and actual biomass storage:
(i) The mismatch is obvious for LUDI_{o}, which is defined in L11 as the relative deviation of the observed quadratic mean diameter at breast height (dbh_{obs}) from the corresponding maximum dbh indicated by the selfthinning function (dbh_{max}) for the observed stand density N_{obs}. As LUDI_{o} relates linearly to diameter  not to biomass  a value of, e.g., 0.5 (i.e., dbh_{obs} is half of dbh_{max} for N_{obs}) actually means that observed biomass is 0.19 (= 0.52.4) of the potential biomass of a fully stocked stand for a given tree number, since tree biomass scales to dbh to the power of ~2.4 (Djomo et al. 2010, Zianis and Mencuccini 2004).
FIG. A1. Conceptual figure of land use and disturbance intensity LUDI of L11 with components LUDI_{p} “planning intensity” and LUDI_{o} “operational intensity” showing reference, minimum and maximum stand density N and the corresponding diameters dbh under selfthinning. Note that dbh = 750 cm for N = 10 trees/ha and not 520 cm, as mistakenly given in Fig. 1 of L11.
(ii) For LUDI_{p} the mismatch is masked in L11 by a definition that lacks a physical meaning, since it is based on the “length” of segments of the selfthinning function in the diameterstand density space. In order to clarify the meaning of LUDI_{p} we present here the mathematical definition for the selfthinning function dbh = c_{dbh}·N^{3/4}, which was not specified in L11:
Function f(N) describes the length of line segments along the selfthinning function for infinitesimal changes in N (Fig. A1). The value of the proportionality constant c_{dbh} is subject to units of measure of N and dbh: c_{dbh} = 4220 for N (trees/ha), dbh (cm); c_{dbh} = 42.2 for N (trees/ha), dbh (m); c_{dbh} = 0.0422 for N (trees/m²), dbh (m). The sensitivity of the LUDI_{p} equation to units of measure is evident. This property of LUDI_{p}, which is generally inappropriate for a scientific measure, is due to the fact that the “length” of line segments of the selfthinning function in the diameterdensity space has no physical meaning. Instead, LUDI_{p} is controlled by N and dbh, dependent on the scaling of the variables, i.e., the actual numerical contribution of a variable to the euclidean distance (Fig A2). Only if dbh and N are specified in centimeters and trees per hectare, LUDI_{p} behaves as described in Appendix B of L11. Then LUDI_{p} is almost exclusively controlled by stand density if N > 100 trees/ha. However, this setting is arbitrary. If for example basic units are used to describe dbh (m) and N (trees/m²) the picture changes completely (Fig. A2); then LUDI_{p} is controlled by the deviation in diameter from reference. Thus, the densest stands considered in L11 (3600 trees/ha) are characterized by LUDI_{p} = 7.1% if using basic units compared to 76.7% if using centimeters and hectares (Fig. A2).
FIG. A2. LUDI_{p} as defined in L11 is sensitive to the units used to characterize stand density N and mean diameter dbh. Values for LUDI_{p} are exemplarily given for three combinations of units of measure of N (trees/ha vs. trees/m²) and dbh (cm vs. m).
In summary, LUDI_{p} as defined by L11 is scientifically inaccurate and, when calculated with the combination of units arbitrarily used in L11, almost exclusively measures the deviation of actual stand density N_{obs} from the reference stand density N_{ref} irrespective of diameter (Fig. A2).
(iii) The index LUDI, defined by L11 as the sum of LUDI_{o} and LUDI_{p}, thus combines a relative deviation in dbh for a given N and the relative deviation of N from a reference stand density irrespective of dbh. Unlike suggested by L11, neither LUDI nor its components relate to biomass storage or specific timescales. In our opinion the authors ignored the only interpretation of the selfthinning relationship that makes sense in terms of biomass storage: namely the difference between actual N and maximum N for a given dbh, which is linearly related to the deficit in standing biomass. According to the approach of L11, LUDI is lowest  signaling minimum intensity  for a forest that comprises 415 trees/ha (LUDI_{p} = 0) with mean diameter 45.9 cm (LUDI_{o} = 0), irrespective of tree species identity and site quality (Fig. A1). Any thinning or disturbance that does not reduce stand density below 415 trees/ha decreases “planning intensity” LUDI_{p} while stand age related density changes increase “planning intensity” afterwards. The thinning depicted in Fig. A1, affecting 50% of trees (and biomass), is measured by LUDI as intensity change from 62.3% (N = 3000 trees/ha, dbh = 10.4 cm) to 66.7% (N = 1500 trees/ha, dbh = 10.4 cm). The biomass accumulation with aging, depicted in Fig. A1, is measured by LUDI as intensity increase from 4.1% (N = 250 trees/ha, dbh = 67.1 cm) to 5.3% (N = 200 trees/ha, dbh = 79.3 cm).
In summary, LUDI fails to quantify the intensity of a thinning or disturbance (Fig. A1 and Fig. A2), but at least the response direction is correct. If stands accumulate biomass through ageing, even the direction of response is questionable given that stand density falls below 415 trees/ha (Fig. A2).
Inconsistency between results and methods
With respect to the summary characteristics of the plots of pristine forests in Slovakia (L11Appendix E), we found:
The values provided for total plots are often mathematically inaccurate. This inaccuracy affects the results as the average stem number of pristine forests N_{pr} serves as reference for LUDI_{p} of managed forests (Eq. 4 and Fig. 1b, L11). We assume that the incorrect value (i.e., 415 trees/ha) was used in calculations because it is provided in L11Appendix B (Procedure for the calculation of the distance along the selfthinning line).
With respect to values for individual pristine forest plots in Slovakia (L11Appendix E) we found that:
The findings suggest that the stateoftheart study by Drößler (2006), in which pristine forests were systematically sampled using replicates and an adequate plot size (cf. Drößler, 2006 and references therein), was attributed less value than data from Korpel’ (1995).
Table A1. Data from "Ecological Archives A021146A5" (Appendix E, Table E1) for pristine forests with LUDI_{o} computed using equations 3 and 5 (cdbh = 4220, page 3277) and LUDI_{ppr}, which is LUDI_{p} for pristine forests, computed using equations 4, 6 and 7 in Luyssaert et al. (2011).
original data  calculated  
using: dbh_{max} = N_{obs}^(3/4) * 4220  
A = (N_{maxx}: 108 stems/ha, dbh_{maxx}: 126 cm) 

Species  Location  Plot  N 
basal area 
dbh 
wood volume 
dbh_{max} (cm) 
LUDI_{o}  d_{AB}  LUDI_{ppr} (d_{max} = 4152) 

Fagus/ 
Sitno 
DVF1 
658 
41.80 
27.30 
605 
32.5 
16.0 
565.3 
13.6 

Hardwood/ 
DVF2 
276 
42.90 
42.70 
661 
62.3 
31.5 
181.9 
4.4 

Abies 
DVF3 
1026 
41.20 
21.70 
530 
23.3 
6.8 
933.4 
22.5 

DVF1 
524 
36.70 
28.66 
547 
38.5 
25.6 
431.1 
10.4 

DVF2 
294 
37.30 
38.58 
607 
59.4 
35.1 
200.1 
4.8 

DVF3 
850 
41.30 
23.87 
615 
26.8 
11.0 
757.4 
18.2 

Fagus 
Kyjov 
Drößler 
252 
35.35 
40.56 
500 
66.7 
39.2 
157.5 
3.8 

Fagus 
Kyjov 
DVF1 
384 
29.04 
29.78 
405 
48.6 
38.8 
290.8 
7.0 

DVF2 
644 
35.40 
25.39 
447 
33.0 
23.1 
551.3 
13.3 

DVF3 
398 
32.34 
30.87 
448 
47.4 
34.8 
304.8 
7.3 

DVF1 
360 
34.01 
33.29 
455 
51.1 
34.8 
266.6 
6.4 

DVF2 
382 
33.72 
32.18 
456 
48.8 
34.1 
288.7 
7.0 

DVF3 
350 
34.84 
34.17 
512 
52.2 
34.5 
256.6 
6.2 

Fagus 
Rozok 
DVF1 
298 
37.75 
38.55 
776 
58.8 
34.5 
204.1 
4.9 

DVF2 
292 
44.32 
42.20 
794 
59.7 
29.4 
198.1 
4.8 

DVF3 
310 
32.61 
35.13 
577 
57.1 
38.5 
216.3 
5.2 

DVF1 
197 
33.03 
44.35 
709 
80.3 
44.7 
100.8 
2.4 

DVF2 
254 
46.42 
46.30 
914 
66.3 
30.2 
159.5 
3.8 

DVF3 
332 
34.63 
34.98 
709 
54.3 
35.5 
238.5 
5.7 

Fagus 
Hovesova 
Drößler 
193 
40.00 
49.31 
870 
81.5 
39.5 
96.6 
2.3 

Fagus 
Hovesova 
DVF1 
222 
32.40 
41.38 
655 
73.4 
43.6 
126.7 
3.1 

DVF2 
236 
32.50 
40.19 
687 
70.1 
42.7 
141.1 
3.4 

DVF3 
214 
35.60 
44.17 
773 
75.4 
41.4 
118.5 
2.9 

Fagus/Abies 
Badin 
DVF1 
436 
40.80 
33.13 
745 
44.2 
25.1 
342.9 
8.3 

Fagus/Abies 
Dobrov 
DVF3 
358 
71.00 
48.23 
1066 
51.3 
5.9 
264.6 
6.4 

Fagus/Picea 
Kotlov 
DVF1 
548 
52.41 
33.49 
475 
37.3 
10.1 
455.2 
11.0 

DVF2 
868 
71.36 
31.05 
697 
26.4 
17.7 
775.4 
18.7 

DVF3 
806 
50.71 
27.17 
500 
27.9 
2.6 
713.3 
17.2 

DVF4 
466 
68.30 
41.46 
838 
42.1 
1.5 
373.0 
9.0 

DVF1 
386 
44.52 
36.78 
449 
48.5 
24.1 
292.8 
7.1 

DVF2 
698 
65.54 
33.19 
742 
31.1 
6.8 
605.3 
14.6 

DVF3 
944 
49.63 
24.83 
495 
24.8 
0.2 
851.4 
20.5 

DVF4 
368 
53.77 
41.40 
737 
50.2 
17.6 
274.7 
6.6 

Picea 
Kosodrevina 
DVF3 
376 
65.00 
45.03 
805 
49.4 
8.9 
282.7 
6.8 

Picea 
Polana 
DVF2 
270 
56.00 
49.32 
670 

63.4 
22.2 

175.8 
4.2 

Result of recalculation of Table 2 from Luyssaert et al. (2011, page 3278) ) for pristine forests based on the values for plots shown above (confirmed values are printed in bold) 

Species 
# plots 

N 
basal area (m²/ ha) 
dbh 
wood 
dbh_{max} 
LUDI_{o} 

LUDI_{ppr} 

total 
35 
mean 
442.0 
43.8 
36.3 
642.0 
50.4 
24.0 
8.4 

SD 
228.6 
12.2 
7.6 
158.0 
16.3 
16.1 
5.5 



FagusPicea 
33 
mean 
449.2 
42.8 
35.6 
636.2 
50.0 
24.5 
8.6 

SD 
233.3 
11.7 
7.3 
160.1 
16.6 
16.4 
5.6 



Picea 
2 
mean 
323.0 
60.5 
47.2 
737.5 
56.4 
15.5 
5.5 



SD 
75.0 
6.4 
3.0 
95.5 

9.9 
9.4 


1.8 
Literature Cited
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Drössler, L. 2006. Struktur und Dynamik von zwei Buchenurwäldern in der Slovakei. University of Göttingen, Göttingen, Germany.
Korpel’, S. 1995. Die Urwälder der Westkarpaten. G. Fischer: Stuttgart, Jena, New York. 310 pp.
Luyssaert, S., D. Hessenmöller, N. von Lüpke, S. Kaiser, and E. D. Schulze. 2011. Quantifying land use and disturbance intensity in forestry, based on the selfthinning relationship. Ecological Applications 21:3271–3283.
Zianis, D., and M. Mencuccini. 2004. On simplifying allometric analyses of forest biomass. Forest Ecology and Management 187:311–332.