Khongor Dissertation

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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

102 1


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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


4.3.1 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



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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


Chi-Squared


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


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


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


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


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


Critical


a

Simple distribution


trees

3-19 Johnson SB

0.0588 0.08623 0.34451 - - - 19.896*



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


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


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

Khongor Dissertation

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