Ecological Archives A023-032-A1

Peter Schall, Christian Ammer. 2013. Can land use intensity be reliably quantified by using a single self-thinning relationship?. Ecological Applications 23:675–677. http://dx.doi.org/10.1890/12-0847.1

Appendix A. Conceptual shortcomings and inaccuracies in the inventory data used by L11.

In this appendix we use the definition of the terms N, Nobs, Nmaxx, Npr, cdbh, dbh, dbhobs, dbhmax, dbhmaxx, dbhmaxxobs, LUDIo, LUDIp, LUDIppr 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 LUDIp, the “planning intensity”) and LUDIo, 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 LUDIo, which is defined in L11 as the relative deviation of the observed quadratic mean diameter at breast height (dbhobs) from the corresponding maximum dbh indicated by the self-thinning function (dbhmax) for the observed stand density Nobs. As LUDIo relates linearly to diameter - not to biomass - a value of, e.g., 0.5 (i.e., dbhobs is half of dbhmax for Nobs) 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).

FigA1

FIG. A1. Conceptual figure of land use and disturbance intensity LUDI of L11 with components LUDIp “planning intensity” and LUDIo “operational intensity” showing reference, minimum and maximum stand density N and the corresponding diameters dbh under self-thinning. Note that dbh = 750 cm for N = 10 trees/ha and not 520 cm, as mistakenly given in Fig. 1 of L11.

(ii) For LUDIp 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 self-thinning function in the diameter-stand density space. In order to clarify the meaning of LUDIp we present here the mathematical definition for the self-thinning function dbh = cdbh·N-3/4, which was not specified in L11:

Eq1

Function f(N) describes the length of line segments along the self-thinning function for infinitesimal changes in N (Fig. A1). The value of the proportionality constant cdbh is subject to units of measure of N and dbh: cdbh = 4220 for N (trees/ha), dbh (cm); cdbh = 42.2 for N (trees/ha), dbh (m); cdbh = 0.0422 for N (trees/m²), dbh (m). The sensitivity of the LUDIp equation to units of measure is evident. This property of LUDIp, which is generally inappropriate for a scientific measure, is due to the fact that the “length” of line segments of the self-thinning function in the diameter-density space has no physical meaning. Instead, LUDIp 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, LUDIp behaves as described in Appendix B of L11. Then LUDIp 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 LUDIp is controlled by the deviation in diameter from reference. Thus, the densest stands considered in L11 (3600 trees/ha) are characterized by LUDIp = 7.1% if using basic units compared to 76.7% if using centimeters and hectares (Fig. A2).

FigA2

FIG. A2. LUDIp as defined in L11 is sensitive to the units used to characterize stand density N and mean diameter dbh. Values for LUDIp are exemplarily given for three combinations of units of measure of N (trees/ha vs. trees/m²) and dbh (cm vs. m).

In summary, LUDIp 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 Nobs from the reference stand density Nref irrespective of diameter (Fig. A2).

(iii) The index LUDI, defined by L11 as the sum of LUDIo and LUDIp, 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 self-thinning 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 (LUDIp = 0) with mean diameter 45.9 cm (LUDIo = 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” LUDIp 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 (L11-Appendix 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 Npr serves as reference for LUDIp 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 L11-Appendix B (Procedure for the calculation of the distance along the self-thinning line).

With respect to values for individual pristine forest plots in Slovakia (L11-Appendix E) we found that:

The findings suggest that the state-of-the-art 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 A021-146-A5" (Appendix E, Table E1) for pristine forests with LUDIo computed using equations 3 and 5 (cdbh = 4220, page 3277) and LUDIppr, which is LUDIp for pristine forests, computed using equations 4, 6 and 7 in Luyssaert et al. (2011).

original data calculated
              using: dbhmax = Nobs^(-3/4) * 4220
                  A = (Nmaxx: 108 stems/ha,
dbhmaxx: 126 cm)
Species Location Plot

N
= Nobs
(trees/ha)

basal area
(m²/ha)

dbh
=dbhobs
(cm)

wood volume
(m3/ha)

dbhmax
(cm)
LUDIo dAB LUDIppr
(dmax = 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
(trees/ha)

basal area (m²/ ha)

dbh
(cm)

wood
(m3/ha)

dbhmax
(cm)

LUDIo

 

LUDIppr

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

 

Fagus-Picea

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

Djomo, A. N., A. Ibrahima, J. Saborowski, und G. Gravenhorst. 2010. Allometric equations for biomass estimations in Cameroon and pan moist tropical equations including biomass data from Africa. Forest Ecology and Management 260:1873–1885.

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 self-thinning 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.


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