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Introduction When a forest stand is inventoried by detecting single trees using aerial photographs or laser scanners, a part of the trees remain unobserved. One main reason for this is that trees that are covered by a crown of a larger tree are not visible from the air. This study derived the probability of a tree being observed from the air as a function of crown area and applied it in estimating the distribution of crown areas and the number of stems in two different situations: A. A known proportion of visible trees of a fixed area have been observed B. An unknown proportion of visible trees and the canopy closure have been observed. Assumptions (i) Trees are randomly located in the stand and tree size is not spatially correlated within a stand, (ii) tree height is an increasing function of crown di- ameter, (iii) tree crown forms a circle around tree tip, (iv) a tree is visible from the air if the tree tip is not located within a crown of larger tree, and (v) tree crowns have been observed perpendicularly above the tree. Relationship between canopy closure, stand density and expected crown area The total number of stems in a stand can be estimated by number of stems = - ln(1 - canopy closure) (1) expected crown area Approaches Continuous distribution approach (CDA, situation B) A weighted distribution is fitted to the observed sam- ple of visible crowns. The weight is the probability of a tree being covered by the crown of a larger tree (Fig. 1), which was derived under the stated assumptions. The distribution of crown areas is obtained by writing the obtained parameter estimates to the corresponding unweighted distribution, and the number of stems is es- timated using Eq. 1. Multinomial approach 1 (MNA1, situation A) In this approach, weights are calculated for the observa- tions of the sample and no assumptions about the distri- bution are needed. The weight of each observation is the inverse of its probability for being observed (Fiq 1). If the sampling probability of visible trees is known (e.g., if all visible trees of a fixed area have been observed), dividing the weights with the sampling probability is an estimator for the numbers of stems, which are summed up to estimate the total number of stems in the stand. Multinomial approach 2 (MNA2, situation B) If canopy closure is known in MNA1 the number of stems can be estimated either by the approach of MNA1 or by using Eq. 1. In MNA2 such a value of the sampling probability is searched that makes these two estimates equal. This leads to a multinomial approach that uses the same input information as CDA. Main results and discussion • The RMSE of the number of stems was at a low level with all approaches and the bias was not sta- tistically significant. • The estimator based on crown cover (Eq. 1) gave slightly more accurate estimates of the number of stems than the estimator of MNA1. • In situation B, CDA and MNA2 performed equally well. The former gives always a smooth continuous distribution but a faulty assumption about the distri- bution may violate the estimates. The latter does not need such an assumption but it provides non-zero frequencies only for diameter classes present in the sample. Thus, CDA is suggested for small samples ands MNA2 for large ones. • In real datasets the assumptions are surely invalid to some degree and the results of this study can be regarded as an upper bound for the accuracy. • Since the approaches are based only on the stated assumptions, not on some species or area-specific models, the method is equally applicable in all areas and with all tree species. • The approaches could be further developed in order to work with the distribution of heights. Conclusions General approaches were presented for taking into ac- count the censoring of sample trees when observed from the air. It performed well in a situation where the stated assumptions were valid. The proposed approach can be further developed to take into account other reasons for censoring (e.g. that low density laser beam does not find all small visible trees or that trees smaller than pixel size are not detected from aerial photographs), if one is able to derive the probability of observing a visible tree as a function of crown area. Finnish Forest Research Institute Joensuu Research Centre Tel: +358-10-2111 lauri.mehtatalo@metla.fi ELIMINATING THE EFFECT OF OVERLAPPING CROWNS FROM AERIAL INVENTORY ESTIMATES Lauri Mehtätalo 0.0 0.2 0.4 0.6 0.8 1.0 Crown radius, m p(Crown radius) 0.568 0.769 0.405 0.595 0 1 2 3 4 100 80 60 40 20 0 y coordinate, m 100 80 60 40 20 0 x coordinate, m 2 3 4 5 6 7 8 9 10 1 150 Crown radius, m Number of stems Sample 2 underlying true observed Estimated distributions CDA MNA1 MNA2 1 0 2 3 4 100 50 0 Observed values Area, ha 1.0 Number of stems, 1/ha 973 Number of visible trees, 1/ha 481 Min crown area, m 2 0.021 Max crown area, m 2 44.37 Mean crown area, m 2 11.27 Canopy closure 0.677 Expected values Number of stems, 1/ha 1000 Number of visible trees, 1/ha 479 Crown area, m 2 11.34 Canopy closure 0.678 Sample cc N N visible 1 ˆ N 2 ˆ N 3 ˆ N 4 ˆ N 1 0.733 1140 510 1039 1149 1151 1151 2 0.664 990 530 965 1041 1033 1029 3 0.676 870 440 765 898 843 821 4 0.683 1090 490 1092 919 1099 1219 5 0.631 880 470 786 851 858 861 6 0.690 990 480 1043 899 1056 1148 7 0.568 820 530 885 953 925 910 8 0.630 910 430 836 794 865 909 9 0.769 1030 440 1074 901 1127 1260 10 0.727 1010 490 939 1128 1054 1021 Mean 943 953 1001 1033 95% Lower bound 868 881 928 936 95% Upper bound 1017 1026 1074 1130 Bias (total) -57.5 -46.8 1.1 32.9 RMSE (total) 128 120 112 152 Bias (sampling error eliminated) -30.5 -19.8 28.1 59.9 RMSE (sampling error eliminated) 70.8 102 56.8 104 Table 2. The colums show, respectively, the measured canopy closure, true number of stems, true number of vis- ible trees, and the four estimates of the number of stems obtained by CDA ( 1 ˆ N ), MNA1 ( 2 ˆ N ), MNA1 and Eq. 1 ( 3 ˆ N ), and MNA2 ( 4 ˆ N ) in the ten samples of Fig. 2. Table 1. Characteristics of the simulated stand. The ex- pected values are based on the underlying distribution used in simulation. Figure 3. True and estimated distributions of crown radii in one sample. Figure 2. Crown map of a simulated forest stand and ten sample plots. Red circles show visible trees and blue cir- cles invisible ones. The shadowed area demonstrates the canopy closure. Figure 1. The probability of observing a tree as a func- tion of crown radius in stands where the canopy closure is as shown on the plot and crown radius is distributed according to a unimodal (blue) and a bimodal distribution (orange).
Transcript of ELIMINATING THE EFFECT OF OVERLAPPING CROWNS FROM …cs.uef.fi › ~lamehtat › documents ›...
IntroductionWhen a forest stand is inventoried by detecting single trees using aerial photographs or laser scanners, a part of the trees remain unobserved. One main reason for this is that trees that are covered by a crown of a larger tree are not visible from the air. This study derived the probability of a tree being observed from the air as a function of crown area and applied it in estimating the distribution of crown areas and the number of stems in two different situations:A. A known proportion of visible trees of a fixed area
have been observed B. An unknown proportion of visible trees and the
canopy closure have been observed.
Assumptions(i) Trees are randomly located in the stand and tree size
is not spatially correlated within a stand,(ii) tree height is an increasing function of crown di-
ameter, (iii) tree crown forms a circle around tree tip,(iv) a tree is visible from the air if the tree tip is not
located within a crown of larger tree, and (v) tree crowns have been observed perpendicularly
above the tree.
Relationship between canopy closure, stand density and expected crown area
The total number of stems in a stand can be estimated by
number of stems = - ln(1 - canopy closure) (1) expected crown area
ApproachesContinuous distribution approach
(CDA, situation B)A weighted distribution is fitted to the observed sam-ple of visible crowns. The weight is the probability of a tree being covered by the crown of a larger tree (Fig. 1), which was derived under the stated assumptions. The distribution of crown areas is obtained by writing the obtained parameter estimates to the corresponding unweighted distribution, and the number of stems is es-timated using Eq. 1.
Multinomial approach 1 (MNA1, situation A)
In this approach, weights are calculated for the observa-tions of the sample and no assumptions about the distri-bution are needed. The weight of each observation is the inverse of its probability for being observed (Fiq 1). If the sampling probability of visible trees is known (e.g., if all visible trees of a fixed area have been observed), dividing the weights with the sampling probability is an estimator for the numbers of stems, which are summed up to estimate the total number of stems in the stand.
Multinomial approach 2 (MNA2, situation B)
If canopy closure is known in MNA1 the number of stems can be estimated either by the approach of MNA1 or by using Eq. 1. In MNA2 such a value of the sampling probability is searched that makes these two estimates equal. This leads to a multinomial approach that uses the same input information as CDA.
Main results and discussion• The RMSE of the number of stems was at a low
level with all approaches and the bias was not sta-tistically significant.
• The estimator based on crown cover (Eq. 1) gave slightly more accurate estimates of the number of stems than the estimator of MNA1.
• In situation B, CDA and MNA2 performed equally well. The former gives always a smooth continuous distribution but a faulty assumption about the distri-bution may violate the estimates. The latter does not need such an assumption but it provides non-zero frequencies only for diameter classes present in the sample. Thus, CDA is suggested for small samples ands MNA2 for large ones.
• In real datasets the assumptions are surely invalid to some degree and the results of this study can be regarded as an upper bound for the accuracy.
• Since the approaches are based only on the stated assumptions, not on some species or area-specific models, the method is equally applicable in all areas and with all tree species.
• The approaches could be further developed in order to work with the distribution of heights.
ConclusionsGeneral approaches were presented for taking into ac-count the censoring of sample trees when observed from the air. It performed well in a situation where the stated assumptions were valid. The proposed approach can be further developed to take into account other reasons for censoring (e.g. that low density laser beam does not find all small visible trees or that trees smaller than pixel size are not detected from aerial photographs), if one is able to derive the probability of observing a visible tree as a function of crown area.
Finnish Forest Research Institute Joensuu Research Centre
Tel: +358-10-2111 [email protected]
ELIMINATING THE EFFECT OF OVERLAPPING CROWNS FROM AERIAL INVENTORY ESTIMATES
Lauri Mehtätalo
0.0
0.2
0.4
0.6
0.8
1.0
Crown radius, m
p(C
row
n ra
dius
)
0.568
0.769
0.405
0.595
0 1 2 3 4
100806040200
y co
ordi
nate
, m10
080
6040
200
x coordinate, m
2 3 4 5 6 7 8 9 101
150
Crown radius, m
Num
ber
of s
tem
s
Sample 2 underlyingtrueobserved
Estimated distributionsCDAMNA1MNA2
10 2 3 4
100
50
0
Observed values
Area, ha 1.0Number of stems, 1/ha 973Number of visible trees, 1/ha 481Min crown area, m2 0.021Max crown area, m2 44.37Mean crown area, m2 11.27Canopy closure 0.677
Expected values
Number of stems, 1/ha 1000Number of visible trees, 1/ha 479Crown area, m2 11.34Canopy closure 0.678
Sample cc N Nvisible 1N̂
2N̂ 3N̂ 4N̂
1 0.733 1140 510 1039 1149 1151 11512 0.664 990 530 965 1041 1033 10293 0.676 870 440 765 898 843 8214 0.683 1090 490 1092 919 1099 12195 0.631 880 470 786 851 858 8616 0.690 990 480 1043 899 1056 11487 0.568 820 530 885 953 925 9108 0.630 910 430 836 794 865 9099 0.769 1030 440 1074 901 1127 126010 0.727 1010 490 939 1128 1054 1021
Mean 943 953 1001 103395% Lower bound 868 881 928 93695% Upper bound 1017 1026 1074 1130Bias (total) -57.5 -46.8 1.1 32.9RMSE (total) 128 120 112 152Bias (sampling error eliminated) -30.5 -19.8 28.1 59.9RMSE (sampling error eliminated) 70.8 102 56.8 104
Table 2. The colums show, respectively, the measured canopy closure, true number of stems, true number of vis-ible trees, and the four estimates of the number of stems obtained by CDA ( 1N̂ ), MNA1 ( 2N̂ ), MNA1 and Eq. 1 ( 3N̂ ), and MNA2 ( 4N̂ ) in the ten samples of Fig. 2.
Table 1. Characteristics of the simulated stand. The ex-pected values are based on the underlying distribution used in simulation.
Figure 3. True and estimated distributions of crown radii in one sample.
Figure 2. Crown map of a simulated forest stand and ten sample plots. Red circles show visible trees and blue cir-cles invisible ones. The shadowed area demonstrates the canopy closure.
Figure 1. The probability of observing a tree as a func-tion of crown radius in stands where the canopy closure is as shown on the plot and crown radius is distributed according to a unimodal (blue) and a bimodal distribution (orange).
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