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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

    4.1 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