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Ph.D. Program of Agriculture Science
National Chiayi University
Ph. D. Dissertation
L an ds a t TM
Modeling forests stand structures, and aboveground biomass
estimation using Landsat TM imagery in Mongolia
Advisor: Chinsu Lin, Ph. D.
Graduate Student: Khongor Tsogt
:
January, 2013
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Acknowledgements
This thesis would not have been possible unless with support of Taiwanese government and
National Chiayi University scholarships and financial support of my supervisor, Chinsu Lin. I am
heartily thankful to my supervisor whose encouragement, guidance and support from the initial to
the final level enabled me to develop an understanding of the subject. Besides I would like to thank
my father, Tsogt Zandraabal, and my colleagues for providing me great information resources. It is
an honor for me to study in Taiwan with support of my mother Dagiisuren Tserennadmid and
family members. Lastly, I am indebted to my many of my friends, Liang-Ting and Marco Lee to
morally support me and lab mates, Jia Wei and A-Zhi boosted me in any respect during the
completion of the study.
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Table of Contents
Table of Contents ........................................................................................................................... ivList of Figures ................................................................................................................................ ixList of Tables ................................................................................................................................. xiPart I. Modeling forest stand structures for volume estimation ........................................................ 1 Part I. Background .......................................................................................................................... 1Chapter 1 ........................................................................................................................................ 41 Diameter structure analysis of a larch forest stand and selection of suitable model................... 4
1.1 Abstract ........................................................................................................................... 41.2 Introduction ..................................................................................................................... 41.3 Materials and Methods..................................................................................................... 5
1.3.1 Field measurement ....................................................................................................... 51.3.2 Data analysis................................................................................................................ 5
1.4 Results ............................................................................................................................. 6Table 1.1 Parameter estimates of the three distribution models for the study plot ........................ .... 6Fig. 1.1Model comparison for the study plot .................................................................................. 7
1.5 Discussion and Conclusion .............................................................................................. 7Chapter 2 ........................................................................................................................................ 92
Diameter and height distributions of natural even-aged Pine forests............................ ......... .... 9
2.1 Abstract ........................................................................................................................... 92.2 Introduction ..................................................................................................................... 92.3 Materials and Methods................................................................................................... 10
2.3.1 Field measurement ..................................................................................................... 10Table 2.1 Inventory information of the study site .......................................................................... 11Table 2.2 Descriptive diameter and height statistics of the study site ........... ..................... ........... .. 11
2.3.2 Data analysis.............................................................................................................. 112.4 Results ........................................................................................................................... 12
Table 2.3 Parameter estimates of the dbh and h distribution models for pine forests, P1-P3 ........... 122.5 Discussion ..................................................................................................................... 132.6 Conclusion .................................................................................................................... 16
Chapter 3 ...................................................................................................................................... 173 Modeling diameter and height distributions of a spruce-larch mixed forest ............. ............... 17
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3.1 Abstract ......................................................................................................................... 173.2 Introduction ................................................................................................................... 173.3 Material and Methods .................................................................................................... 19
3.3.1 Field measurement ..................................................................................................... 19Table 3.1 D and H characteristics of the spruce-larch forest in the study site ................................. 19
3.3.2 Data analysis.............................................................................................................. 193.3.3 A flexible mixture model for dbh structure modeling ................................................. 203.3.4 Model comparison ..................................................................................................... 21
3.4 Results ........................................................................................................................... 213.5 Discussion ..................................................................................................................... 263.6 Conclusion .................................................................................................................... 28
Chapter 4 ...................................................................................................................................... 294 A flexible modeling of irregular diameter structure for the volume estimation of larch foreststands ............................................................................................................................................ 29
4.1 Abstract ......................................................................................................................... 294.2 Introduction ................................................................................................................... 294.3 Material and Methods .................................................................................................... 31
4.3.1 Field measurement ..................................................................................................... 314.3.2 Data analysis.............................................................................................................. 324.3.3 dbh-h relationship modeling and assumption tests ..................... ........... ........... ........... 324.3.4 Volume estimation and accuracy assessment .............................................................. 33
4.4 Results ........................................................................................................................... 344.4.1 Simple dbh distribution models of larch forest ........................................................... 344.4.2 Mixture dbh distribution models of larch forest ........... ......................................... ...... 354.4.3 dbh-h relationship of larch forest................................................................................ 374.4.4 Accuracy assessment of the volume estimation for the models derived by unimodalapproach and multimodal approach ....................................................................................... 40
4.5
Discussion and Conclusion ............................................................................................ 40
Chapter 5 ...................................................................................................................................... 425 Forests aboveground biomass and carbon storage estimation in Eastern Khentii, Mongolia ... 42
5.1 Abstract ......................................................................................................................... 425.2 Introduction ................................................................................................................... 425.3 Material and Methods .................................................................................................... 43
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5.3.1 Study area .................................................................................................................. 43Fig. 5.1 Tree species of study area, Mongon morit soum, Tov aimag, Mongolia. ............ ............... 44
5.3.2 Field measurement ..................................................................................................... 45Table 5.1Generalcharacteristics of studied plots .......................................................................... 45
5.3.3 Biomass calculation ................................................................................................... 465.3.4 Carbon and other elemental composition estimation ............. ...................................... 47
Table 5.2 Four main elemental compositions of larch and birch tree species in Mongolia .............. 475.4 Results and Discussion .................................................................................................. 475.5 Conclusion .................................................................................................................... 49
Part I. Summary ............................................................................................................................ 50Part II. Aboveground biomass estimation using Landsat TM imagery in Mongolia ..................... .. 52Part II. Background ....................................................................................................................... 526 Regression models for forest aboveground biomass and leaf area index in North-EasternMongolia ...................................................................................................................................... 53
6.1 Abstract ......................................................................................................................... 536.2 Introduction ................................................................................................................... 53
6.2.1 Background ............................................................................................................... 536.2.2 Spectral characteristics of vegetation ......................................................................... 556.2.3 Objective of the research ............................................................................................ 57
6.3 Material and Methods .................................................................................................... 586.3.1 Study area and field measurement .............................................................................. 586.3.2 Abovegound biomass estimation ................................................................................ 586.3.3 LAI-2000 Plant canopy analyzer (Li-COR) ................................................................ 596.3.4 Landsat TM data ........................................................................................................ 596.3.5 Vegetation indices ..................................................................................................... 60
Table 6.1 Vegetation indices (VIs) used in this study .................................................................... 60 6.3.6 Data analysis.............................................................................................................. 61
6.4
Results ........................................................................................................................... 63
6.4.1 Correlations between biomass of tree components and/or LAI and Landsat TM bands636.4.2 Correlation between AGB and/or LAI, and band ratios .............................................. 646.4.3 Correlation between AGB and/or LAI, and VIs ........... ......................................... ...... 656.4.4 Models for AGB and LAI .......................................................................................... 666.4.5 Index validation ......................................................................................................... 69
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6.5 Discussion ..................................................................................................................... 696.5.1 Relationship between AGB, LAI, Landsat TM bands and band ratios ...................... .. 696.5.2 Relationship between AGB, LAI and VIs................................................................... 716.5.3 AGB and LAI estimation using regression equation ............. ...................................... 726.5.4 Effect of the undergrowth vegetation and background reflectance .......... .............. ...... 726.5.5 Model verification ..................................................................................................... 736.5.6 Suggestions................................................................................................................ 73
6.6 Conclusion .................................................................................................................... 74Chapter 7 ...................................................................................................................................... 767 Mapping land cover and forest types using multi-temporal Landsat imageries with biophysicalinformation of dominant forest types in mountainous area ....................... ......... .............. ............... 76
7.1 Abstract ......................................................................................................................... 767.2 Introduction ................................................................................................................... 767.3 Materials and Methods................................................................................................... 78
7.3.1 Study area .................................................................................................................. 787.3.2 Landsat TM data ........................................................................................................ 78
Fig. 7.1 Framework of image processing and forest type map........................................................ 79 7.3.3 Image preprocessing .................................................................................................. 807.3.4 Classifiers .................................................................................................................. 80
7.3.4.1 Support vector machines (SVM) ........................................................................ 807.3.4.2 Maximum Likelihood ......................................................................................... 81
7.3.5 Region of interest (ROI) for classification .................................................................. 81 Table 7.1 Training pixels and reference pixels used for image classification and validation ........... 82
7.3.6 Classification accuracy assessment ............................................................................ 837.3.7 Ancillary biophysical information for post-classification ........................................... 83
Table 7.2 Forests altitudinal belt zones in Eastern Khentey province, Mongolia ...................... ...... 84Table 7.3 Conditional distributions of forest types by elevation .......... .............. ......... .................... 86
7.4
Results ........................................................................................................................... 87
7.4.1 Land cover and forest type classification result and accuracy assessment for multi-temporal imagery .................................................................................................................. 87
Table 7.4 Confusion matrix of multi-temporal image classification result by SVM ................. ...... 88Table 7.5 Confusion matrix of multi-temporal image classification result by Maximum likelihood 89Table 7.6 Classification accuracy assessment of Maximum likelihood and SVM................ ........... 89
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7.4.2 Land cover and biophysically adjusted forest type image classification result andaccuracy assessment .............................................................................................................. 90
Table 7.9 Post-classification accuracy assessment of Maximum likelihood and SVM.................. .. 917.4.3 Dominant forest type, larch and cedar mapping accuracy assessment .......... ............. .. 92
7.5 Discussion ..................................................................................................................... 947.5.1 Classification accuracy improvement in multi-temporal imagery ............................... 947.5.2 Land cover mapping and accuracy ............................................................................. 957.5.3 Dominant forest type mapping and accuracy .............................................................. 95
7.6 Conclusion .................................................................................................................... 96Part II. Summary ........................................................................................................................... 97References .................................................................................................................................... 99Appendixes ................................................................................................................................. 110
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List of Figures
Fig. 0.1 Framework of Part I study .................................................................................................. 3Fig. 1.1Model comparison for the study plot .................................................................................. 7Fig. 2.1dbh (left) and h (right) model comparisons for pine forests, study plots P1-P3. The
histogram represents the observed distribution and the short dashed line (Burr), long dashed
line (Dagum), and solid line (Johnson SB) show the estimated distributions. ............. ........... 14Fig. 3.1 Model comparison for the Spruce-Larch mixed forest. The histogram represents the
observed diameter at breast height (dbh) and height (h) distributions with simple distribution
model (solid line), mixture distribution model (dashed line). a) dbh of spruce-larch mixed
trees fit by simple distribution, Burr (4P) and fit by mixture distribution, Burr (4P) and
Johnson SB.b) dbh of spruce trees fit by simple distribution, Burr (4P) and fit by mixture
distribution Burr (4P) and Johnson SB. c) dbh of larch trees fit by simple distribution, Dagum
(3P) and fit by mixture distribution, double Johnson SB. d) h of spruce-larch mixed trees fit by
simple distribution Johnson SB and fit by mixture distribution double Johnson SB. e) h of
spruce trees fit by simple distribution, Dagum (3P) and fit by mixture distribution Dagum (4P)
and Johnson SB. f) h of larch trees fit by simple distribution, Dagum (3P) and fit by mixture
distribution Dagum (3P) and Johnson SB. ............................................................................. 23Fig. 3.2 Spruce and larch trees a. D and b. H structures. a. Spruce trees are dominant in D 2-10 cm
while larch trees are dominant in mid D classes around 12-16 cm and then equally distributed
around 18-24 cm and around 26-30 cm only spruce trees inventoried. b. Spruce trees are
dominant 3-11 m H while larch trees are dominant in 12-19 m. ........... ..................... .......... .. 27Fig. 4.1 The diameter distribution of the study plots in larch forests. The histogram represents the
observed dbh distribution, the dashed line represents the derived simple distribution model
and the solid line represents the derived mixture distribution model. ........... .............. ......... .. 36Fig. 4.2 Curvilinear relationships of tree height to dbh of larch forests. The relationship is
exponential for plots 1-2 and sigmoid for plots 3-6. .............................................................. 39Fig. 5.1 Tree species of study area, Mongon morit soum, Tov aimag, Mongolia. ............ ............... 44Fig. 6.1 Spectral reflectance of healthy Red cypress (
Chamaecyparis formosensis), green vegetation
for the wavelength interval 350-2500 nm. The dominant factors controlling leaf reflectance
are the various leaf pigments in the palisade mesophyll (e.g., chlorophyll a andb, and-
carotene), the scattering of near-infrared energy in the spongy mesophyll, and the amount of
water in the plant. The chlorophyll absorption regions are at 430-450 nm and 650-660 nm and
the water absorption regions occur at 970, 1190, 1450, and 1940 nm. ............................ ...... 56
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Fig. 6.2Additive reflectance ofCinnamomum camphora leaves. 1, 3 and 9 layers. Near-infrared
reflectance increases as the leaf layers in the canopy increases. Direct relationship exists
between response in the near-infrared region and biomass measurements. ............. ............... 56Fig. 6.3Yearly pattern of Red cypress (Chamaecyparis formosensis) trees absorption change in the
blue spectral region, 400-550 nm (Tsogt et al. 2008). ............ ............. .......... ...................... .. 57Fig. 6.4 Framework of aboveground biomass (AGB) and leaf area index (LAI) study relationship
with vegetation indices (VIs). ............................................................................................... 58Fig. 6.5 Relationship between Landsat TM bands 3, 5 and 7 and aboveground biomass (AGB) (left),
and relationship between Landsat TM bands 2, 3, 4 and 5 and leaf area index (LAI) (right).
Correlation coefficients (r) are given for bands and AGB and/or LAI. .............. .................... 64Fig. 6.6 Relationship between VIs and AGB for forest stands. Log transformed Brightness index
and aboveground biomass (AGB) (top left), log transformed Surface albedo and AGB (top
right), visible atmospherically resistant index and AGB (middle left), NDVIc and AGB
(middle right), and SWIR corrected NDGVI and AGB (lower left). The SWIR correction
substantially reduced NDVI values over low AGB. .............................................................. 67Fig. 6.7 Relationship between VIs and LAI. Logarithm transformed (log) Brightness and
exponential transformed (exp) leaf area index (LAI) (top left), log-Surface albedo and exp-
LAI (top right), VARI and log-LAI (middle left), exp-NDVIc and log-LAI (middle right), and
exp-NDGVIc and log-LAI (bottom left). .............................................................................. 68Fig. 7.1 Framework of image processing and forest type map........................................................ 79 Fig. 7.2 Biophysical information rules of forests distributions along elevation for dominant forest
types post classification. ....................................................................................................... 85Fig. 7.3 Alternative, accuracy assessment for image post-classification. ........... ............................. 86Fig. 7.4 Multi-temporal image composite differences between SVM and Maximum likelihood
classification results before and after applying ancillary biophysical information in dominant
fores type, larch and cedar classes. a and c - SVM and Maximum likelihood classification
results; b and d - SVM and Maximum likelihood classification results adjusted by forests
biophysical distribution characteristics along altitudinal-belt zones ....................... ......... ...... 94
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List of Tables
Table 1.1 Parameter estimates of the three distribution models for the study plot ........................ .... 6Table 1.2 Summary of empirical diameter distribution for larch forest (=0.05) .............................. 6Table 2.1 Inventory information of the study site .......................................................................... 11Table 2.2 Descriptive diameter and height statistics of the study site ........... ..................... ........... .. 11Table 2.3 Parameter estimates of the dbh and h distribution models for pine forests, P1-P3 ........... 12Table 2.4Goodness-of-fit and ranking (rank in parentheses) of Burr, Dagum and Johnson SB
distributions for the empirical dbh and h distributions as measured by MLE criterion ( = 0.05)
............................................................................................................................................ 15Table 3.1 D and H characteristics of the spruce-larch forest in the study site ................................. 19Table 3.2 Parameter estimates of diameter at breast height distribution models for the spruce-larch
mixed, spruce and larch trees (=0.05) ................................................................................. 24Table 3.3 Parameter estimates of tree height distribution models for the spruce-larch mixed, spruce
and larch trees (=0.05) ....................................................................................................... 25Table 3.4 RMSE and
2 test of the simple and the mixture models for the spruce-larch mixed,
spruce and larch trees (N/ha) ................................................................................................ 26Table 4.1 Descriptive statistics of the study plots. ......................................................................... 32 Table 4.2 Coefficients of tree volume equation (with bark) by diameter at breast height and height
............................................................................................................................................ 33Table 4.3 Measures of the goodness of fit tests for the unimodal approach derived dbh distribution
models. ................................................................................................................................ 35Table 4.4 Measures of the goodness of fit tests for the multimodal approach derived sub-group dbh
distribution models............................................................................................................... 37Table 4.5 Measures of the goodness of fit tests for the empirical dbh-h models. ............. ............... 38Table 4.6 A comparison of the accuracy of volume estimation using the simple dbh models and the
mixture dbh models*. ........................................................................................................... 40Table 5.1Generalcharacteristics of studied plots .......................................................................... 45Table 5.2 Four main elemental compositions of larch and birch tree species in Mongolia .............. 47Table 5.3Aboveground biomass and carbon of larch, birch and mixed forest stands in Eastern
Khentii, Mongolia (AGC Aboveground carbon) ................................................................ 47Table 6.1 Vegetation indices (VIs) used in this study .................................................................... 60
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Table 6.2Biomass of tree components and LAI of study plots (0.09 ha, regards to Landsat TM pixel
size 30x30 m)....................................................................................................................... 62Table 6.3 Correlations of the biomass components of forest stand and leaf area index (LAI) against
the Landsat TM band reflectance ......................................................................................... 64Table 6.4 Correlations of the aboveground biomass of forest stand and leaf area index against the
band ratios. ........... ................................ ..................... ........... ......................................... ...... 65Table 6.5 Correlations of the aboveground biomass of forest stand and leaf area index against the
VIs. ...................................................................................................................................... 66Table 6.6 Regression models for aboveground biomass (AGB) and leaf area index (LAI) ............. 69Table 6.7 Correlations of the AGB of forest stand and LAI against the VIs. ................................ .. 69Table 7.1 Training pixels and reference pixels used for image classification and validation ........... 82Table 7.2 Forests altitudinal belt zones in Eastern Khentey province, Mongolia ...................... ...... 84Table 7.3 Conditional distributions of forest types by elevation .......... .............. ......... .................... 86Table 7.4 Confusion matrix of multi-temporal image classification result by SVM ................. ...... 88Table 7.5 Confusion matrix of multi-temporal image classification result by Maximum likelihood 89Table 7.6 Classification accuracy assessment of Maximum likelihood and SVM................ ........... 89Table 7.7 Confusion matrix of post-classification result by SVM. .................... ............ ........... ...... 90Table 7.8 Confusion matrix post-classification result by Maximum likelihood. .............. ............... 91Table 7.9 Post-classification accuracy assessment of Maximum likelihood and SVM.................. .. 91Table 7.10 Accuracy assessment for the larch and cedar cover type map of the study area with
comparison of inventory map-data ....................................................................................... 92Table 7.11 Accuracy assessment for the larch and cedar cover type (that was adjusted based on
forests distribution characteristics along elevation) map of the study area with comparison of
inventory map-data .............................................................................................................. 93
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Part I. Modeling forest stand structures for volume estimation
Part I. Background
Volume estimation of forest stands is an important task in the accounting of forest stocks as
well as carbon storage and sequestration. Modeling diameter at breast height (dbh) structure anddbh-height (h) relationship is essential in order to carry out the volume and carbon accounting task.
In past decades, the mainstream approach to forest volume estimation generally operates at the
stand level (Wiant et al. 1989, Hamilton and Brack 1999) and therefore is also named stand-level
approach. It uses the stand diameter and height to obtain the information needed for forest
management such as the calculation of forest productivity, annual or periodical yield, and economic
benefits. The stand-level approach lacks accuracy because generalized stand parameters may not be
able to represent the natural variation of a larger area of forest stands and hence might cause a high
uncertainty of prediction. Recently, a large variety of research has been conducted based on using
forest structures, such as dbh, height and age to overcome the inaccuracy of the stand-level
approach. These kinds of methods generate more suitable models based on the probability density
functions (pdf) of the inventoried forest samples to present more accurately the dbh and height
structure of a forest stand.
In a forest, there are always variations in tree size. These variations are caused by biological
reasons and by spatial locations across stands, and by microclimatic conditions. These variations
easily can be seen by tree diameter and height.
In forestry, we often use probability density functions to reveal forest stands structures. Based
on forest stand structure, we can understand forests development stages, phases and dynamics. But,
depending on development stages, forests size structural shapes are different. Then it is necessary to
use various theoretical distributions to fit stands trees dbh distributions to explain their development
cycle or phase. It is also possible to build growth models. to predict stand changes and dynamics.
For instance, dbh distribution models can be used to complete missing dbh class of life table
information.
Modeling dbh distributions of different development stages of forests is important to define
for theoretical and practical forestry. Depend on development stages, forest structural distributions
possible to be explained by different distributions. Then it is able to define theoretical forests by
distribution shape. Especially, if the stand trees are highly skewed or irregular shaped, diameter
distribution modeling is very important in order to choose proper strategy management plan.
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Forest volume measures expressed by per hectare area, but researchers often do not develop a
hectare area for study and we divide volume stock by study area to transfer for a hectare. But, each
sample plots has got unique structural characteristics. This could bring huge bias to forest planning
and an end product. One of the advantages of using pdf to predict forest structure is that they could
predict forest structure by their general tendency.
pdf s can manipulate to produce detailed diameter distribution of forest stand with statistically
promised probability. Traditional dbh distribution models falls between one sided an exponential
distribution and a symmetrical normal distribution (or Weibull distribution). There are more
sensitive and flexible distribution functions are available to model dbh distributions. Modeling dbh
distribution becomes easy task with the availability of specified statistical software packages.
Forest is a complex dynamic system which may consist of several individual patches with
different variations. If the variations are similar among the patches, the forest size distribution is
regular shaped (unimodal) and could be modeled by simple distribution function. If the variations
are different among the patches, the forest structure is irregular shaped. Then simple distribution
functions are no longer sufficient for modeling irregular shaped diameter and height distributions.
Therefore, mixture distribution models are known better for structurally heterogeneous forests.
In Fig. 0.1 shows general framework of Part I study. Chapters 1 and 2 represent simple
distribution modeling while Chapters 3 and 4 compares the results of simple and mixture
distribution modeling. Chapter 1 shows suitable distribution model selection for forest diameter
structure modeling in case of a pure larch (Larix sibirica) forest stand. Chapter 2 shows also
distribution model selection, but for both diameter and height structures in case of pure pine (Pinus
sylvestris) forest stands. Rather than distribution model selection, Chapter 3 shows a comparison of
simple and mixture distribution modeling in structural parameters, both diameter and height in case
of spruce-larch (Picea obovata andLarix sibirica) mixed forest. Chapter 4 shows flexible mixture
distribution modeling in case of larch (Larix sibirica) forest stands and further diameter-height
relationships discovered for volume estimation. In chapter 5, we are concentrated on one of the
important tasks of quantifying forests AGB.
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Fig. 0.1 Framework of Part I study
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Chapter 1
1 Diameter structure analysis of a larch forest stand and selection of suitable model
1.1 AbstractEcologically and economically it is important to understand how many tree stems are in each
diameter at breast height (dbh) class. The purpose of this study was to find larch forest (Larix
sibirica) dbh distribution model among Weibull, Burr and Johnson S B distributions. Inventory was
conducted near Gachuurt village, Ulaanbaatar, Mongolia. The goodness of fit test were
accompanied with Kolmogorov-Smirnov (K-S), Anderson-Darling (A-D) and Chi-Square (2) tests
for distribution models. Study result shows Johnson SB distribution gave the best performance in
terms of quality of fit to the diameter distribution of larch forest.
1.2 IntroductionDetailed information of forest stand is crucial for forest research and planning. This
information used for input of ecosystem modeling and/or forest growth and yield models. In the
analysis of stand dynamics, detailed data for all trees on a plot is often lacking. In such case, we
may generate missing data using various theoretical at breast height (dbh) distributions. For many
years there were various activity and interest in describing the frequency distribution of dbh
measurements in forest stands using probability density functions. First study of dbh distribution
mathematical description was negative exponential (DeLiocourt 1898), and since then, researchers
used various distributions.
All distribution models have their advantage and sensitive in specific shape. Weibull
distribution able to describe Exponential, Normal and Lognormal distribution shapes (Bailey & Dell,
1973; Lin et al., 2007), while Burr distribution cover much larger area of skewness and kurtosis
plane than the Weibull distribution (Lindsay et al., 1996). Moreover, it is closely approximate with
above mentioned distributions plus Gamma, Logistic and several Pearson type distributions.
Johnson SB distribution cover different region of skewness and kurtosis plane than the Burr
(Johnson, 1949; Hafley & Schreuder, 1977), and it is closely approximate Beta and generalizedWeibull distributions.
In case of Mongolian forests, Khongor et al. (2011a) published the birch forest dbh study
using Weibull and Lognormal distributions and compared the accurateness of these models. For
larch forest dbh distribution, Khongor et al., (2011b) used Exponential, Lognormal and Gaussian
(or Normal) distributions, but they did not used Weibull, Burr and Johnson SBbefore.
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The purpose of this study is to investigate the suitability of the Weibull, Burr and Johnson SB
distributions for modeling dbh distribution of larch forest (Larix sibirica).
1.3 Materials and Methods1.3.1 Field measurement
Study plot was selected near the Gachuurt village in the vicinity of Ulaanbaatar city,
Mongolia, located at 4800'18.9''N and 10713'23.1''E with altitudinal elevation 1607-1627 m above
sea level. The forest consisted of natural stands and any management activity had taken previously.
Inventory was conducted in summer of 2009. Composition of the stands is pure larch. Plot size was
0.2 ha, i.e. 40 x 50 m in area. D were measured for all trees >1.3 m, and totally 275 stems were
counted. Average dbh of tree stands in plot was 14.6 cm with standard error mean 0.497 cm. Dbh of
tree stems ranges from 2 to 32 cm. Dbh distribution skewness value was 0.38 indicating that the tail
on the right side of the probability density function is longer than the left side and kurtosis value -
0.97 indicating statistically flattered peak.
1.3.2 Data analysis
The goodness of fit of empirical D distribution was tested using three theoretical distributions:
Johnson SB, Weibull and Burr (Appx 1). A suitable model was chosen for further application based
on the results of the goodness of fit test for each plot. EasyFit 5.5 Professional software was used
to estimate the parameters and to examine the goodness of fit of the distribution models mentioned
above. Empirical models of the derived stand dbh structure were examined by the Kolmogorov-
Smirnov (K-S), Anderson-Darling (A-D) and chi square (2) tests. The K-S test is a nonparametric
test for the equality of a continuous, one-dimensional probability distribution that can be used to
compare a sample with a reference probability distribution. It is based on the largest vertical
difference between the theoretical and the empirical cumulative distribution function. The K-S
critical values used in this study are based on the table published in statistical literature (DAgostino
1986). The A-D test is also a nonparametric test of whether there is evidence that a given sample of
data did not arise from a given probability distribution. It is more sensitive to the tails of adistribution than the K-S test. The 2testis used for binned data and checks if the sample data came
from a specific distribution. So the value of the test statistic depends on how the data is binned. For
this study, the dbh data grouped into intervals of equal probability. Hypothesis tests of dbh structure
model were carried out by examining the p-value that is associated with a goodness of fit statistic.
When the p-value is less than the pre-defined critical value or the significant probability level, the
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null hypothesis is rejected and it is concluded that the data did not come from the specified
distribution.
1.4 ResultsThe parameter estimates of the three models are given in Table 1.1 Parameter estimates of the
three distribution models for the study plot. The predictions from each model were compared with
observed frequencies. The K-S, A-D and 2 tests and P value for K-S and 2 tests were computed for
each model (Table 1.2 Summary of empirical diameter distribution for larch forest (=0.05)). All
tested distribution models were statistically fitted with observed dbh distribution and among them
Johnson SB distribution was more flexible than Weibull and Burr distributions.
Table 1.1 Parameter estimates of the three distribution models for the study plot
Johnson SB Weibull Burr
k
0.38167 0.68757 31.863 1.81 1.8296 16.335 1474.1 1.8545 841.49
1By the definition, the area under the pdf graph must equal 1, so the theoretical pdf values have to be multiplied by the
total number of stems to match the 1histogram and the D coverage of bin width to calculate the number of stems in each
dbh class.
Table 1.2 Summary of empirical diameter distribution for larch forest (=0.05)
Distribution
Kolmogorov-Smirnov
(critical value 0.08189)
Anderson-Darling
(critical value 2.5018)
Chi-Squared
(critical value 15.507)
statistic P-value statistic statistic P-value
Johnson SB 0.03106 0.94589 0.18795 7.6044 0.47303
Weibull 0.06891 0.14004 1.8136 10.535 0.22946
Burr 0.07506 0.08567 1.909 13.562 0.09392
Tree stems are smoothly distributed in dbh classes and it is statistically unimodal. It is easy to
fit such distribution, but here flattered peak is problem that causes under/over prediction. Though all
models passed on goodness of fit test, Weibull and Burr models over-predict dbh classes around 10-
16 cm and lower-predict 4-6 cm and 24-30 cm classes. It is evident that the Johnson S B model was
more flexible in fitting flattered dbh distribution of larch forest stand (Fig. 1.1Model comparison
for the study plot).
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Fig. 1.1Model comparison for the study plot
1.5 Discussion and ConclusionWeibull and Burr theoretical distributions fit the best for right tailed dbh distributions whilst
Johnson SB distribution has ability to represent equally well right and left tailed distributions. With
this reason we have chosen the Weibull, Burr and Johnson SB distributions to test their suitability
and flexibility for larch forest dbh distribution. Then, our study result suggest Johnson SB
distribution for larch forest and that is would not be necessary to believe about the Johnson SB
distribution is the best for all over larch forest in Mongolia.
Forests stand dbh distributions are different depending on the site quality, climatic condition
and history of natural or human disturbances. Supposedly, empirical distribution models would
accurately work in big scale if the geographic and climate conditions are same. But, random
disturbances, such as forest fire, insect invasion or selective logging are change the forest structure
and shape in different forms. Specially, every forest ever influenced with forest fire in Mongolia
and near urbanized areas all forests under danger of illegal timber logging.
However, it is still important that stand specific forest structure information for model
development and research or management planning in small scale forest area. If we needed bigger
scale as regional forest dbh structure, we have to collect more stand dbh data to fit general dbh
distribution. The required amount of stem numbers or sample plots for regional dbh distribution
study would be defined by stability of a chosen model. If the one fails we need to collect more stand
samples and do it again until it become statistically stable. Westphal (2006) suggested that for the
0
5
10
15
20
25
30
35
40
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
Numbero
fstems
Diameter (cm)
Observed diameter distribution
Johnson SB distribution
Weibull distribution
Burr distribution
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regional scale dbh distribution is reverse J shaped because of many small stems and relatively fewer
big stems.
Strong intensity disturbances or high intensity regeneration may change dbh structure as
bimodal. If disturbance happened repeatedly in same forest, then the dbh distribution would forms
multimodal shape. In this study, we used unimodal dbh distribution. However, it may not be
sufficient when a frequency distribution is reverse J with hump, bimodal or multimodal, and
therefore, irregular shaped dbh distributions should have tested by mixture distribution (Zhang &
Liu, 2006).
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Chapter 2
2 Diameter and height distributions of natural even-aged Pine forests
2.1 AbstractThe purpose of this study is to find a suitable probability density function (pdf) to model diameter
breast height (dbh) and height (h) distributions of even-aged pine (Pinus sylvestris L.) forests. For
the study, three different age-class (AG) pine forests were used. Burr, Dagum, and Johnson SB
distributions were applied due to their flexible properties. Result showed that dbh distributions of
the 10~15 yr (AG1) and the 40~45 yr (AG2) stands are left tailed and the 60~65 yr (AG3) stand is
normally skewed. h distribution of AG1 and AG3 are left tailed while AG2 show no obvious
distribution shape, due to its discrete h distribution. Distribution study reveals that in the left tailed
forests, dbh and h distribution shapes were the best approximated by the Dagum distribution. In
case of normal distribution shape, Johnson SB is better than the Burr and Dagum. Based on the
results, it is concluded that dbh distributions of even-aged AG1 and AG2 forests are heavy left
tailed and forest structure tend to normal at the AG3. The h distribution is left tailed (AG1 and AG3)
if a forest h growth is not constrained for space, while it will become discrete in a high density stand
(AG2).
2.2 IntroductionForest managers often require information concerning the size-class distribution of a forest
stand, usually in the form of a tabulation of numbers of trees by dbh and h class. This dbh and h
distribution information is often used to predict volume production, the primary variable that forest
managers are interested in. dbh and h distribution models are also important in forecasting the range
of products that might be expected from a stand. The forest volume estimates are usually based on
the dbh distributions and for improved forest volume estimates, tree hs are used. Unmanaged forests
are used as a standard for comparison of different types of managed stands. Thus, detailed modeling
of the variation of dbh and h classes is required.
A wide range of pdfs have been used in forestry to model tree dbh and h structuraldistributions (Bailey and Dell 1973, Hafleyand Schreuder 1977, Gove et al. 2008, Wang et al. 2010)
as well as age distribution (Lin et al. 2007).
Distribution having unimodal shape (single high point) is easy to simulate by traditional pdfs.
However, regular shaped distributions may take different shapes therefore one needs to choose
appropriate theoretical distribution function that can express the empirical distribution shape. In
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forestry, more often flexible pdfs are used for forest size-structure modeling. For instance: Beta,
Burr, Dagum (inverse Burr), Gamma, Johnson SB and Weibull.
Weibull is generally the most favored distribution used in forestry as it is easy to use and able
to describe both positive and negative skewness. Comparison study of lognormal and Weibull
distribution on a regular shaped birch forest dbh distribution structure show that Weibull was
superior (Tsogt et al. 2011a). However these studies did not include any other distribution
functions. According to Lindsay et al. (1996) study Burr distribution outperforms Weibull
distribution in its ability to drive dbh distribution and Dagum has the ability to fit rotated sigmoid
dbh distribution (Gove et al. 2008). Both Weibull and Burr distributions are family of Dagum
distribution and the Dagum distribution has a wider region of applicability then either of two
(Lindsay et al, 1996).
Hafley and Schreuders (1977) study showed that Johnson SB distribution gave the best
performance, while normal, lognormal and gamma distributions were inferior to the Weibull and
beta distributions in terms of their general performance over variety of even-aged stands. Generally
beta was the second best fitting distribution and Weibull was third best. From the viewpoint of
practical application, they believed Johnson SB distribution has important advantage over the beta
distribution. The reason being that it spans a slightly broader range of the skewness-kurtosis space
than the beta distribution (ie., it covers, in addition, the region between the lognormal and the
gamma).
Based on pdfs, few studies have paid attention to different species (birch and larch) of
Mongolian forests dbh and h (Tsogt and Lin 2012, Tsogt et al. 2011a, b, c) distribution structures.
But, no one has studied dbh and h distribution of pine forests. The purpose of this study is to find
the appropriate distribution for modeling dbh and h distributions of even-aged pine forests using
Burr, Dagum and Johnson SB distributions.
2.3 Materials and Methods2.3.1 Field measurement
The distribution models investigated in this paper were fitted to dbh and h measurements of 3pine forest plots: P1 and P2 in Khuder soum Selenge aimag and P3 in Shariin gol soum Darkhan-
Uul aimag. The species found in 3 forest-plots were pure pine (Pinus sylvestris L.). dbh and h were
measured for all trees higher than 1.3 m. General characteristics of the study plots are given in
Table 2.1 and 2.2
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Table 2.1 Inventory information of the study site
Plot #Year of
Plot sizeNumber of trees Average age
inventory by plot area in a hectare of plot trees
P1 2005 50 x 40 m 239 1195 10~15P2 2005 40 x 40 m 250 1563 40~45
P3 2001 50 x 20 m 96 960 60~65
Table 2.2 Descriptive diameter and height statistics of the study site
Plot ID Variables Mean Standard deviation Min-Max Skewness Kurtosis
P1 dbh (cm) 16.10 6.03 2.50-30.00 -0.23 -0.75
h (m) 14.06 3.58 3.30-19.66 -0.94 0.21P2 dbh (cm) 15.80 6.18 2.50-30.00 -0.17 -0.82
h (m) 11.88 3.25 6.19-15.02 -0.82 -0.91
P3 dbh (cm) 22.10 5.58 8.30-36.80 -0.02 -0.1
h (m) 15.89 2.58 6.11-21.62 -1.36 3.67
P1 and P2 forests result from the expansion of the forest area caused by favorable site
conditions and abundant parent material (seed sources and advance regeneration). The P1 stand
does contain old trees, however they are not inside our plot area. P2 is the neighbor of an olderforest stand. P3 has no obvious seed source. Perhaps it had already died and/or been removed from
the stand.
All 3 plots, including dbh and h, were negatively skewed but P3 h is very close to zero. Which
means P3 h would be a symmetric distribution and the rest of the distributions are left tailed. In
addition, P1 and P2 dbhs are relatively short tailed and the hs for P1, P2, and P3 are long tailed.
Four of 6 kurtosis values are negative. P1, and P2 dbh, and P2 h are more flat than the normal
distribution according to their kurtosis value. The P3 dbh is less flat and similar to normal
distribution The P1 and P3 h are peaked distribution. In fact, P3 h is high peaked distribution.
2.3.2 Data analysis
This study intended to model dbh distribution of the forest plots with 3 functions (Appx 1),
including Burr, Dagum, and Johnson SB. The goodness-of-fit test descriptions are given in heading
1.3.2, chapter 1.
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2.4 ResultsThe parameters, estimated by likelihood with the measured data are shown in Table 2.3. All
parameters were successfully computed. The observed and estimated frequency distributions of
each plot are shown in Fig. 2.1 and Table 2.4. The dbh distributions are plotted by widths of 2 cm
classes and the h distributions by 1 m classes. The histogram represents the observed distribution
and the curves represent the estimated distributions. Clearly, the Dagum is superior to the other two
distributions in terms of their general performance over the left tailed distributions. The Burr is
second best and it was not failed at any test statistics of goodness-of-fit but in the case of P2 h all 3
models failed. Because of discrete distribution characteristic, none of them could model the P2 h
distribution among the 3 pdfs. The fact that Burr and Dagum lines fall rather close to each other in
comparison with the Johnson SB distribution helps to explain why sets of data can often be fitted
equally well (or equally poorly), by either of these distributions. For the P3 dbh, Johnson SB
distribution showed the best result. In this study, either Burr and Dagum or Johnson SB were the
best fitted models. In each case the Weibull model did not show the best or the second best
performance of goodness-of-fit test (results not shown) and generally was similar to the Burr
distribution results. Therefore we decided to not include the Weibull distribution in this study.
Table 2.3 Parameter estimates of the dbh and h distribution models for pine forests, P1-P3
Plot # dbh h
Burr Dagum JohnsonSB
Burr Dagum JohnsonSB
P1 k = 96.872 k = 0.12207 = -0.33446 k = 3422.4 k = 0.10296 = -1.0667
= 4.3718 = 16.329 = 1.0467 = 489.29 = 32.122 = 0.83265
= 70.738 = 24.025 = 30.387 = 1328.3 = 18.134 = 18.118
=-6.4707 = 0 = -1.0791 = -1290.7 = 0 = 0.79728
P2 k = 108.79 k = 0.09438 = -0.23789 k = 1045.9 k = 0.00778 = -0.55124
= 3.8956 = 15.688 = 0.99499 = 4.7765 = 442.93 = 0.27967
= 76.963 = 22.283 = 29.794 = 56.121 = 15.101 = 9.1613
= -5.0039 = 2.3031 = -0.52346 = 0 = 0 = 5.5655
P3 k = 8.253 k = 0.43323 = -0.30612 k = 3.0013 k = 0.398 = 0.51806
= 4.7694 = 9.8894 = 4.3887 = 1.4860E+6 = 147.07 = 3.5092
= 36.962 = 25.513 = 99.144 = 2.4671E+6 = 123.03 = 84.116
= 0 = 0 = -29.188 = -2.4671E+6 = -105.52 =-16.76
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2.5 DiscussionIn our study, young pine forests dbh distributions are heavy left tailed for P1 and P2, and
normal for P3. h distribution of P1 and P3 are left tailed while P2 show discrete distinct h groups.
P1 and P2 dbh distributions are identical even though their geographical locations and ages are
different. That means, their regeneration and growth patterns are same, according to their dbh
distribution. In P1 and P2, most stems were established shortly after the disturbance, and this still
left enough growing space for stems to become established later, during 1990-1995 (P1) and 1960-
1965 (P2). During the 5 yr many seedlings successfully grew and survived after regeneration; those
are 10 yr old in P1 and 40 yr old in P2. They are recognizable on the dbh distribution P1 and P2 of
Fig. 2.1, where plateaus are on the left side of the peak. The left tailed dbh distributions indicate that
growth space is still sufficient (in free growth situations) for the major trees which occupy the main
canopy of the stand and the dbh distribution structure will not change until the forest reaches the
maximum capacity of stem number and mean size ratio. Once, a forest reaches maximum tree
density-size ratio, individual tree growth can continue only if the number of individuals is reduced.
Thus, the forest dbh distribution structure may change with different distribution shapes depending
on the size of tree stems removed from the stand.
P3 dbh distribution indicates that some trees dominate the major population while some are
suppressed. In theory, dominant trees are located in more favorable microclimate conditions than
suppressed trees or they may have just inherited good genetic materials. Either way, dominant trees
grow bigger, faster and stronger while other trees will grow more slowly and they will be
suppressed. However, still the dbh structure of main population is well normal in P3 forest.
h distributions of P1 and P2 are very different. The plateau is clearer on h distribution of P1
but P2 is not showing any distribution shape. P1 is just in crown closure stage, so the explanation is
same as for the dbh distribution of P1. Because P2 stem density is higher than P1 and P3, the crown
development is strained. When a tree respiration and foliage and root growth consume all
photosynthates, normal h growth is not maintained. Tree h repression occurs first in trees with small
crowns. In our case, those are 6 and 7 m high trees in P2 h distribution. It can occur in individual
trees as they become suppressed or in whole stands as they approach stagnation (Eversol 1955).This explains why the P2 average h is shorter than P1. h distribution structure of P3 indicates that
the major trees h growth has not yet stopped, while a few trees h growth is constantly suspended.
That results in long left tailed distribution structure.
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Fig. 2.1 dbh (left) and h (right) model comparisons for pine forests, study plots P1-P3. Thehistogram represents the observed distribution and the short dashed line (Burr), long dashed line
(Dagum), and solid line (Johnson SB) show the estimated distributions.
0
50
100
150
200
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
NUM
BEROFSTEMS,ha
DBH, cm
P1 dbh
0
50
100
150
200
250
3 5 7 9 11 13 15 17 19
NUM
BEROFSTEMS,ha
H, m
P1 h
0
50
100
150
200
250
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
NUMBEROFSTEMS,ha
DBH, cm
P2 dbh
0
90
180
270
360
450
6 7 8 9 10 11 12 13 14 15
NUMBEROFSTEMS,ha
H, m
P2 h
0
40
80
120
160
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
NUMBEROFSTEMS,ha
DBH, cm
P3 dbh
0
50
100
150
200
250
6 8 10 12 14 16 18 20 22
NUMBEROFSTEMS,ha
H, m
P3 h
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Table 2.4Goodness-of-fit and ranking (rank in parentheses) of Burr, Dagum and Johnson SB
distributions for the empirical dbh and h distributions as measured by MLE criterion ( = 0.05)
Plot #Distribution
Kolmogorov-Smirnov Anderson-Darling Chi-Squared
Structure statistic statistic statistic
P1 dbh
critical value 0.08784 2.5018 14.067
Burr (4P) 0.06589 (3) 1.1113 (2) 7.9501 (2)
Dagum (3P) 0.04047 (1) 0.40807 (1) 7.7654 (1)
Johnson SB 0.04462 (2) 4.3858 (*) -
P1 h
critical value 0.08784 2.5018 14.067
Burr (4P) 0.07234 (2) 1.4021 (2) 13.7770 (2)
Dagum (3P) 0.06231 (1) 0.95645 (1) 8.4542 (1)
Johnson SB 0.07286 (3) 38.763 (*) -
P2 dbh
critical value 0.08589 2.5018 14.067
Burr (4P) 0.06888 (3) 1.3299 (2) 9.136 (2)
Dagum (4P) 0.03904 (1) 0.26976 (1) 4.6169 (1)
Johnson SB 0.04521 (2) 4.4621 (*) -
P2 h
critical value 0.08589 2.5018 7.8147
Burr (3P) 0.25849 (*) 22.676 (*) 30.646 (*)
Dagum (3P) 0.26938 (*) 25.977 (*) 14.728 (*)
Johnson SB 0.16567 (*) 142.13 (*) -
P3 dbh
critical value 0.13675 2.5018 12.592
Burr (3P) 0.04506 (2) 0.20612 (2) 3.8621 (3)
Dagum (3P) 0.05222 (3) 0.24601 (3) 2.7432 (2)
Johnson SB 0.04361 (1) 0.18449 (1) 0.9937 (1)
P3 h
critical value 0.13675 2.5018 12.592
Burr (4P) 0.04699 (2) 0.3492 (2) 1.8253 (2)
Dagum (4P) 0.04166 (1) 0.22303 (1) 1.6842 (1)Johnson SB No fit
* assumption rejected at = 0.05
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2.6 ConclusionNaturally generated, even-aged young pine forest dbh distribution is left tailed in Western
Khentii. Few distribution have the ability to simulate left skewed distribution. All the 3 pdfs, which
applied in this study, can simulate both left and right skewness. However, the Dagum distribution
was superior according to test statistics. The Burr, generally, was good enough to simulate the
distributions. The Johnson SB was only the best in case of normal distribution. In fact, Johnson SB
is good at simulating heavy tailed shapes which is not given in this study. In addition, forest h
distribution is more sensitive than dbh distribution to reveal forest structural relationships within
forest stands.
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Chapter 3
3 Modeling diameter and height distributions of a spruce-larch mixed forest
3.1 AbstractWe derived diameter breast height (D) and height (H) structural distribution model for
Siberian spruce and Siberian larch (Picea obovata andLarix sibirica) mixed and uneven aged stand.
The aim of this study is to fit distribution models for irregular shaped uneven aged mixed stands D
and H distributions. For this reason we apply simple unimodal and mixture multi modal distribution
models. More than dozen probability density functions are used to describe the D and H
distributions of the species groups and entire forest stand. Moreover, we tried to explain their
distributional structures from ecological perspective. Inventory was conducted in river valley of
near Gachuurt village, Ulaanbaatar, Mongolia. The goodness of fit test were accompanied with
Kolmogorov-Smirnov, Anderson-Darling and Chi-Squared (2) tests for distribution models. Root
mean square error, and chi-squared tests were used for comparing how closely predict given D and
H distributions can predict by simple and mixture distribution models. The study result shows that
all distribution models goodness of fit test accepted. Mixture distributions are more similar to
observed distributions than simple distribution models according to the root mean square error
(RMSE) and 2
test. Both RMSE and 2 tests values are bigger in simple distribution case than in
mixture distribution case. The overall study result indicates that the mixture models were suitable
for modeling irregular and bimodal structural distributions. Mixture models have a potential to
characterize tree structural diversity.
3.2 IntroductionSiberian spruce grows on marshy soils around rivers and bogs (Dilis, 1981), and is able to
withstand strong shading (Kellomaki, 1987). It can regenerate under canopies of all forest types
(Kujala, 1924). Often spruce forests grow in riverside of mountain valley and either grow by pure
stand or mixed with larch in Mongolia (Tsedendash, 2007). Forest fire frequently occurred during
spring due to its dry climate in Mongolia. Spruce seedlings are fire intolerant. The study plot ofspruce-larch mixed forest grows separately from main forest in river valley. Therefore, spruce trees
successfully develop in river valley. But, still spruce trees are easy to die in river valley after
nutrient layer of soil is washed away by flooding and their thin roots are exposed in sun.
Spruce forests are less occur in the country and separated within each other as patches in
Mongolia. However, mixed spruce-larch forests are common in Western Khentii (Dugarjav, 2006).
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The spruce forest grows along the southern boundary of its areal distribution in Khentii ridge.
Spruce forest productivity and natural regeneration become low into transition zone. So, the spruce
forests in transition zone are in developing regress succession. Spruce trees mix with pine and larch
forests and sometimes only juvenile spruce trees occur in forests. This can be explained by the
forest composition change. However, this is also a sign of the transition zone effect. Spruce and
larch forests belong to cool taiga and areal distributions follow permafrost region. And as a
consequence of global warming and anthropogenic negative impact, the areal distribution is now
reducing. Therefore, elimination of their distribution caused by dry steppe effect and recent general
dryness.
Studying spruce-larch mixed forest size structure may offer better understanding about what
is happening in the mixed stand in transition zone. This understanding could lead us to develop a
proper forest management policy for uneven aged mixed forest.
Strong intensity disturbances or high intensity regeneration may change regular D structure to
bimodal structure. If disturbance happens repeatedly in same forest, then the D distribution would
forms multimodal shape. In forestry, the probability density functions, such as Weibull (Bailey and
Dell, 1973; Lin et al., 2007), Gamma (Nelson, 1964), Burr (Lindsay et al., 1996), Johnson SB
(Hafley and Schreuder, 1977) etc. are well known and broadly used. These distribution functions
have unimodal shape and are weak in multimodal or irregular shaped distributions.
Mixture model is better at modeling multimodal forest horizontal and vertical structural size
distributions. Jaworski and Podlaski (2011) found the usefulness and applicability (Zhang and Liu,
2006) of two-component and three-component models for describing D distributions in mixed stand
(Shunzhong et al., 2006). A frequency distribution made up of two or more component distributions
is defined as a mixture distribution (Liu et al., 2002; Zasada and Cieszewski, 2005; Zhang et al.,
2001; Zhang and Liu, 2006).
Researchers focused on understanding the structure, function, and productivity of larch
forests(Danilin, 1995; Osawa et al, 2010; Tsogt, 1993), and D distribution of larch forests (Khongor
et al., 2011a and b), but spruce-larch mixed forest D and H distributions have not been studied in
Mongolia. Therefore, it is necessary to model and explain D and H distributions of spruce-larchmixed forests.
The objective of this study was to investigate the suitability of either simple or mixture
functions in modeling the D and H distributions of spruce-larch mixed forest in river valley of
forest-steppe zone.
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3.3 Material and Methods3.3.1 Field measurement
Study plot was selected from Gachuurt village 10 km to North East in river valley surrounded
by grassland, in green zone Ulaanbaatar, Mongolia and the geographical location is 4800'30.9'' N
and 10712'49.5'' E with altitudinal H 1495 m above sea level. The forest trees appear in multi
cohort and double stratum. Composition of the stand is mixed spruce and larch with few birch and
willow. 60% of live trees is spruce, 40% is larch and less than 1% is birch and willow. Birch and
willow trees are not included in the study. Spruces in the first stratum are 160 years old and larches
are 80-100 years old. And in the second stratum, both species are 50 years old. There are few larch
trees are 180-200 years old. The inventory was conducted summer in 2009. Composition of the
stand is mixed spruce and larch with few birch and willow. For D and H distribution modeling only
live trees were analyzed (Table 3.1). Plot size is 0.16 ha (40x40 m). This forest stand is so small and
there are some these kind of spruce-larch mixed forests grow along that river valley but occurrence
of formed stand trees along the river valley are distant around 100 m to 1 km. The distance between
study plot and the nearest forest is 374 m in mountain slope. It is pure larch forest. The forest is
natural stand, and no management activity had been taken previously. But,because livestock
grazing do not give chance to survive young seedlings around the study area, even if seedlings were
regenerated.
Table 3.1 D and H characteristics of the spruce-larch forest in the study siteNumber of
observation
Diameter (cm) Height (m)
Mean Min-max Mean Min-max
Spruce149 live
95 dead
9.8
3.0
2.2-30.2
1.0-18.5
9.2 3.0-18.5
Larch99 live
12 dead
13.1
7.5
2.2-24.2
2.2-25.9
12.8 3.0-19.0
3.3.2 Data analysis
Choosing appropriate distribution model is problematic. In order to do that first you have tounderstand the distribution shape and depend on the shape of the distribution you could choose the
model. There are similar distributions but sensitive in specific subtle skewness and kurtosis. So, it
will ask you to know about distribution models characteristics. However, statistical software
packages provide letting us to choose distribution models without prior knowledge.
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There is only few distribution models used in forestry and have their biological meanings and
vast amount of distribution models which used in engineering field have no biological meaning. But,
sometimes distribution models which have no biological meaning could describe forest structural
distributions better than broadly used one according to their goodness of fit test. In that case we
could use those distribution functions to model forest structures with higher fitness.
We have chosen the best fitted distribution function among Burr, Dagum and Johnson S B
functions (Appx 1) according to their goodness of fit test for modeling dbh and h distributions
(Table 3.2 and 3.3). The goodness-of-fit test descriptions are given in heading 1.3.2, chapter 1.
3.3.3 A flexible mixture model for dbh structure modeling
A typical modeling method of the dbh structure of a forest stand is single distribution
modeling or unimodal approach (SDM approach). It assumes that the dbh distribution has a single
peak and can be fitted with a particular distribution function. If the tally dbh histogram has
obviously more than one peak, the dbh structure or distribution is irregular and can be divided into
several sub-groups. A modeling of irregular dbh distribution is mixture (multi-single) distribution
modeling or multimodal approach (MDM approach). It applies first a single model to fit each of the
sub-group dbh distribution and then integrates all of the single models into a multi-single model or
mixture model. It is supposed that a simple distribution model is potentially not able to cover the
high diverged dbh histogram while mixture models have better estimation ability because a mixture
model can cover the detailed outstanding features of the dbh variation.
The proposed multimodal approach is a four-step modeling process. First, to examine the
breakpoint(s) or gap(s) that occurred in the whole-range dbh histogram of a sample plot. A
continuous pdf curve is theoretically a common feature of any arbitrary distribution function. Thus
an apparent discontinuous point existing in the curve can be defined as a breakpoint. Next, the
number of sub-groups is determined by the number of breakpoints plus one. The corresponding dbh
of a breakpoint is assigned as a boundary of a sub-group pdf curve. Third, a sub-group is considered
as a histogram with a particular distribution and is fitted with each of the five theoretical pdf
functions. A suitable pdf function is suggested based on the goodness of fit test. Finally, a mixturemodel for the whole range dbh histogram is integrated directly by the suggested single model of
each sub-group histogram.
Suppose a multimodal dbh distribution is composed of k sub-groups and each of the sug-
group has m dbh classes. Then the generalized mixture distribution model, f(x) can be expressed as
Eq. 3.1,
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k
j
m
i ijijj
k
j jjxfnwxfwxf
1 11)()()( Eq. 3.1
where ijn is the number of stems of the ith dbh class in jth sub-group (adopted from Zhang and Liu
2006). By the definition of probability, the area under the pdf curve of a single distribution function
equals 1. So, the total number of stems would be scaled to 1. Once a single distribution model is
derived successfully, the predicted number of stems in each diameter classes of that specific sub-
group can be determined by multiplying the estimated probability and the observed total number in
that sub-group.
3.3.4 Model comparison
RMSE and 2 tests were used for comparing how closely predict by unimodal and mixture
distribution models for a given D and an H distributions. RMSE is a measure of the differences
between values predicted by a model and the values observed from the measurement.
3.4 ResultsThe stand is assumed to consist of two individual generations which are spruce and larch trees.
It is feasible to say that there are two generations of trees in both species according to D and H
structures of stand in Fig. 3.1.By D class, the first generation of spruce trees are 14-30 cm and larch
trees are 10-26 cm, and the second generation of spruce trees are 2-12 cm and larch trees are 2-8 cm
(Fig. 3.1b and 3.1c). By H class, the first generation of spruce trees are 13-19 m and larch trees are
9-19 m, and second generation of the spruce trees are 3-12 m and larch trees are 3-8 m (Fig. 3.1e
and 3.1f). In Fig. 2a and 2d we can see that the first generation of both spruce and larch stems
clustering distributed around 12-30 cm and the second generation stems clustering distributed
around 2-11 cm by D class; by H class the first generation clustering distributed around 12-19 m
and the second generation of stems clusteing distributed around 3-11 m. If we look separately,
species by species, both spruce and larch D and H distributions are also bimodal. But, their stem
clustering appeared in different places in the D clustering. D distribution of spruce stems is right
tailed. It is very similar to reverse J shape distribution. The major stem clustering is around 4-10 cm
and then slowly decreasing until 30 cm except gape at 12 cm. H distribution of spruce trees is right
tailed and the first generation of stems clustered at two distinct positions which are 14 m and 18 m.
The second generation of spruce trees D distribution structure seen that the right tail is heavier than
left tail. Both D and H right tailed distributions show that regeneration status of spruce trees is good.
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For instance, we can see in Fig. 1b and 1e D distribution that there are many smaller trees and big
stems which gradually declined in number.
D distribution of larch trees is left tailed, and stem clustering is around 4-8 cm and 14-16 cm.
However even though larch trees regeneration peaked at 6 cm, it is still smaller than first generation.
H distribution of larch trees is left tailed and also both humps are left tailed. The first generation of
larch stems holds the most proportion of all the stems. From both D and H distribution shape, larch
trees regeneration status is not so good (Fig. 3.1c and 3.1f).
The parameter estimates of the simple and mixture models are given in Table 2 and 3, Noting
that spruce-larch forest D and H distributions are bimodal. The predicted frequencies by D and H
classes were obtained from simple and mixture model. Plus, single and mixed species are tested in
these models. The predictions from each model were compared with the observed frequencies. The
RMSE, and 2 test were computed for each model and each condition (spruce-larch mixed, spruce
and larch) (Table 3.4). The observed frequency distribution (histograms) and the simple and
mixture prediction curves are illustrated for each condition (Fig. 3.1).
The simple models were definitely not flexible enough to fit the distribution at all. These
simple functions missed the second peak as well as the valley between the two peaks. However all
models passed their test statistics for Kolmogorov-Smirnov, 2 and Anderson-Darling (Table 3.2
and 3.3). The mixture models fit to the plot data better, while single mode models were not enough
to characterize the distribution (Table 3.4). Fig. 1 shows that simple models under-predict small and
large trees and over-predict middle sized trees.
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Fig. 3.1 Model comparison for the Spruce-Larch mixed forest. The histogram represents the
observed diameter at breast height (dbh) and height (h) distributions with simple distribution model
(solid line), mixture distribution model (dashed line). a) dbh of spruce-larch mixed trees fit by
simple distribution, Burr (4P) and fit by mixture distribution, Burr (4P) and Johnson S B.b) dbh of
spruce trees fit by simple distribution, Burr (4P) and fit by mixture distribution Burr (4P) and
Johnson SB. c) dbh of larch trees fit by simple distribution, Dagum (3P) and fit by mixture
distribution, double Johnson SB. d) h of spruce-larch mixed trees fit by simple distribution Johnson
SB and fit by mixture distribution double Johnson SB. e) h of spruce trees fit by simple distribution,
Dagum (3P) and fit by mixture distribution Dagum (4P) and Johnson SB. f) h of larch trees fit by
simple distribution, Dagum (3P) and fit by mixture distribution Dagum (3P) and Johnson SB.
0
50
100
150
200
250
300
350
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Numberofstems(N/ha)
Diameter (cm)
0
40
80
120
160
200
240
3 4 5 6 7 8 9 10111213141516171819
Numberofstems(N/ha)
Height (m)
0
50
100
150
200
250
300
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Numberofstems(N/ha)
Diameter (cm)
0
30
60
90
120
150
180
3 4 5 6 7 8 9 10111213141516171819
Numberofstems(N/ha)
Height (m)
0
30
60
90
120
150
180
2 4 6 8 10 12 14 16 18 20 22 24
Number
ofstems(N/ha)
Diameter (cm)
0
20
40
60
80
100
120
3 4 5 6 7 8 9 10111213141516171819
Number
ofstems(N/ha)
Height (m)
a d
c
b e
f
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Table 3.2 Parameter estimates of diameter at breast height distribution models for the spruce-larch mixed, spruce and larch trees (=0.05)
Species and model type
Diameter at
breast height
range (cm)
Distribution
Parameters
Kolmogorov-Smirnov Chi-Square Anderson-Darling
StatisticCritical
valueP value Statistic
Critical
valueP value Statistic
a
Simple distribution
model of spruce-larch mixed
trees
2-30 Burr (4P) 0.08474 0.08623 0.05353 21.071* 14.067 0.00367 2.278
Mixture distribution
model of spruce-larch mixed
trees
2-11 Burr (4P) 0.05083 0.11015 0.80804 10.628 14.067 0.15569 0.48412
12-30 Johnson SB 0.04745 0.1319 0.96166 2.3259 12.592 0.88742 0.27905
Simple distribution
model of spruce trees2-30 Burr (4P) 0.08483 0.11125 0.22104 23.76 14.067 0.00126* 1.5427
Mixture distribution
model of spruce trees
2-12 Burr (4P) 0.05504 0.13067 0.88093 1.3878 12.592 0.96659 0.29153
13-30 Johnson SB 0.07312 0.20517 0.96614 0.97413 11.07 0.96463 0.14632
Simple distribution
model of larch trees2-24 Dagum (3P) 0.05769 0.13469 0.87759 3.8462 12.592 0.69748 0.37686
Mixture distribution
model of larch trees
2-10 Johnson SB 0.10927 0.22425 0.75696 1.5339 7.8147 0.67448 0.39013
11-24 Johnson SB 0.06094 0.15755 0.93698 3.7463 12.592 0.71096 0.27327
a- the critical value of Anderson-Darling statistic used is 2.5018 for all distributions at =0.05
* - assumption rejected at =0.05
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Table 3.3 Parameter estimates of tree height distribution models for the spruce-larch mixed, spruce and larch trees (=0.05)
Species and model typeTrees height
range (m)
Distribution
Parameters
Kolmogorov-Smirnov Chi-Square Anderson-Darling
StatisticCritical
valueP value Statistic
Critical
valueP value Statistic
a
Simple distribution
model of spruce-larch mixed
trees
3-19 Johnson SB
0.0588 0.08623 0.34451 - - - 19.896*
Mixture distribution
model of spruce-larch mixed
trees
3-11 Johnson SB 0.07333 0.11396 0.4102 5.8852 14.067 0.55322 0.78883
12-19 Johnson SB 0.06157 0.1319 0.79315 5.723 12.592 0.45492 0.27435
Simple distribution
model of spruce trees3-19 Dagum (3P) 0.06995 0.11125 0.43954 6.9528 14.067 0.43381 1.1546
Mixture distribution
model of spruce trees
3-12 Dagum (4P) 0.06203 0.12663 0.74418 4.864 12.592 0.56137 0.61725
13-19 Johnson SB 0.07644 0.21273 0.96341 - - - 4.1025*
Simple distribution
model of larch trees3-19 Dagum (3P) 0.06385 0.13469 0.79034 2.3502 12.592 0.88484 0.3465
Mixture distribution
model of larch trees
3-8 Burr (3P) 0.10555 0.2749 0.93632 0.22348 5.9915 0.89428 0.35917
9-19 Johnson SB
0.06748 0.14868 0.8301 3.5696 12.592 0.73468 0.28456
a- the critical value of Anderson-Darling statistic used is 2.5018 for all distributions at =0.05
* - assumption rejected at =0.05
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Table 3.4 RMSE and 2 test of the simple and the mixture models for the spruce-larch mixed,
spruce and larch trees (N/ha)
Species and model typeDiameter Height
RMSE Chi-Square test Bias RMSE Chi-Square test Bias
Spruce-Larch model 17.81 130.00 4.34 13.94 -3.50 -3.38
Mixture Spruce-Larch model 5.97 22.08 0.48 9.39 38.68 1.38
Spruce model 10.12 131.52 4.14 8.63 134.68 2.86
Mixture Spruce model 5.22 21.11 1.45 7.02 53.97 1.58
Larch model 6.65 40.35 0.53 4.76 41.59 0.26
Mixture Larch model 5.00 13.37 -0.05 3.77 13.15 0.18
For each condition, mixture models produced satisfactory fitting results. All mixture
distribution models fit to observed D and H frequency distributions better than simple distribution
according to the RMSE and 2 tests (Table 3.4) and yield similar predictions across tree Ds and Hs.
On the other hand, the simple model did not fit the forest D and H distribution well and definitely
missed the valley portion of the distribution (Fig. 3.1).
3.5 DiscussionFor a mixed conifer forest, size distribution is often a better predictor of future forest
composition (Veblen, 1992). In this sense, we can guess the future stand composition on Fig. 3.2.
Species composition remained relatively constant in two strata. The current size structure suggests
that the stem composition would be remained relatively constant in the future. Understory size
classes support this conclusion. Comparing with larch, many of the spruce trees were died and most
of them took place in small D class around 3 cm (Table 3.1). However, there might be changes in
the stem composition of the study plo