Fitting Non-Linear models for Tree Volume Estimation in a Nigerian Strict Nature Reserve, South-Western, Nigeria

Abstract

The study tested the efficacy of nonlinear models for tree volume estimation in a complex tropical natural ecosystem. Data for this study were collected from the four permanent sample plots (PSP) located in Strict Nature Reserve in Akure forest reserve, each plot covering an area of 0.25ha. All living trees within the range of specified dbh (>10 cm) were measured within all the permanent sample plots.

The data were pooled together and sorted according to family: Annonaceae, Meliceae, Sterculiaceae, and Ulmaceae.

Six nonlinear models were fitted using curve expert for the volume models and ranked according to their best of fit using the AIC, standard error and significance at 5% level of probability. A total of 266 trees were observed in the four sampling plots but the study made use of the 171 trees comprising 17 species distributed among 4 families.

Meliaceae and Sterculiaceae had the highest number of species (6 spp. each) while the most abundant species was Mansoniaaltisima, this was followed by Celtiszenkerii. These species have 45 and 42 individual trees, respectively.

The assessment criteria using AIC and standard error showed that the entire models are suitable for volume estimation in the study area. The non-linear models showed a reasonable variation depending on family.

The result showed that Weibull, Gompertz Relation and Logistic Power models were the most consistent model and the models gave the best predicted volume when compared with the observed volume for each family in the study area but Ratkowsky model ranked the best of the six models generated when data from each family were combined.

The validation result showed that all the models were of good fit. Very low percentage biases and RMSE were obtained. The student t-test showed that there were no significant differences between the observed and predicted volume. All the models are very good for tree volume estimation in the study area. They are therefore recommended for further use in this ecosystem and similar ones.

Introduction

Today, substantial focus has been given to forestry for the development of estimation schemes to predict volume at individual tree level and the whole stand. This is due to its economic impact. Periodic inventories are often needed forest industries and other organization as it helps to determine the quantity of wood utilization. The main reason for conducting forest inventory either in the plantation or natural forest ecosystem is to estimate the timber volume of the plots installed in the entire stand (Adekunle 2007). Estimating tree volume is important for forest management due to its effectiveness in assessment of growing stock, timber valuation, distribution of area allocated for harvest and decision taking on the use and management of the forest.

Natural forest ecosystem is characterized by dense canopy closure which makes it practically difficult, inefficient and costly to measure all the predictor variables for every tree in each plot. To overcome this problem, the use of volume equation with dbh and height as predictor variables is developed (Adekunle 2007). Despite the increasing use of biomass and density, volume is the most widely used traditional measure for tree quantity Koiralaetal., (2017).

The most predictive variables for volume estimation are diameter at breast height and total tree height, occasionally also some measures of tree form. Numerous allometric tree volume models have been developed for different tree species and forest types in Europe (Zianisetal. 2005), North America (Ter-Mikaelian and Korzukhin 1997; Jenkins etal. 2004; Zhou and Hemstrom 2010) and Australia (Griersonetal. 2000; Keith etal. 2000), but relatively few models have been developed for tropical forests types in sub-Saharan Africa, as indicated in the review by Henry etal. (2011). Moreover, the reliability of forest volume estimates using the volume models that do exist for tropical forests is questionable because there are many species, tree sizes and geographic areas that are not covered by these models (Hofstad 2005; Henry etal. 2011). The study aim to develop model for estimating volume of tropical hardwood species in a strict nature reserve.

Materials and Methods

The study was carried out in Queen’s plot, Akure forest reserve which is located in a tropical rainforest of the Southwestern Nigeria in Akure South Local Government of Ondo State. It is composed of diverse tree species of varying sizes. Akure forest reserve covers an area of 69.93km2. It lies between latitude 7017’ 46” and longitude 50 01’ 48”E. It is a lowland tropical rain forest type with distinct wet and dry season. The mean annual temperature is about 26oC (min 19oC and max 34oC).

The reserve has a rainfall of 1500mm with bimodal rainfall pattern. The rainy season lasts for 9 months annually between March and November while dry seasons last for 3 months between December and February. The natural vegetation is the high forest composed of many varieties of hardwood timber such as Mansoniaaltissima, Cola gigantea, e.t.c. It lies on an altitude of 279m above sea level. Owena River flows from north to south across the reserve into the Atlantic Ocean which makes the soil to be well drained. The underlying rock of the forest reserve is crystalline, and referable to the basement complex.

Data Collection

Data is collected from four permanent sample plot (POP), located in the strict nature reserve in Akure forest reserve. It is under the management of FRIN. Each plot has a total size of 0.25ha in which total enumeration of DBH and height was carried out. All living trees within the range of specified dbh (>10 cm) were measured within all the permanent sample plots. The data were pooled together and sorted according to family: Annonaceae, Meliceae, Sterculiaceae, and Ulmaceae. The results were presented in form of tables and chart. The following tree growth variables were obtained on the field: diameter at breast height, diameter at the base, diameter at the middle, diameter at the top and total height.

Data analysis

Basal area estimation

The basal Area (BA) of individual trees was estimated using the formula according to Huschetal., (2003) in equation 1:

Where: BA= Basal area (m²), D= Diameter at breast height (cm), π= Pie (3.142).

Volume estimation

The volume of all trees in the sample plots was computed using the Newton’s formula:

Where V = Volume (m³), H = Total height (m), Db = Diameter at the base (cm), Dm = Diameter at the middle (cm), Dt = Diameter at the top (cm) and π = 3.142

Correlation coefficient calculation

Spearman correlation was carried out to examine the type of relationship between the tree growth variables

Fitting of the Volume Models

For the purpose of modeling, individual tree growth variable belonging to respective family in the sample plot were used. The tree growth variables in each family were as well pooled together to generate model that could be used for volume estimation in tropical ecosystem of southwest Nigeria and in other places with similar vegetation and environmental factor (Adekunle 2007)

Six nonlinear models were fitted using curve expert for the volume models and ranked according to their best of fit using the AIC, standard error and significance at 5% level of probability.

Logistic Power

V = a/(1+(x/b)**c)

Gompertz Relation

V = a*exp(-exp(b-c*x))

MMF

V = (a*b + c*x^d)/(b + x^d)

Weibull

V = a - b*exp(-c*x^d)

Logistic

V = a/(1 + b*e^(-cx))

Ratkowsky model

V = a / (1+exp(b-c*x))

Where a, b, c and d are the estimated parameters, V is the volume in (m3), x is the Dbh (cm) while exp. is the exponential.

Assessment of the models

Akaike’s Information Criteria (AIC) are commonly used for model selection and comparison Hoetingetal., (2006). The candidate model with the lowest AIC is selected as the best model as reported by Scaranelloetal., (2012) and Adesoye (2014). Other criteria include the use of standard error of estimate and residual plots that shows how the error of estimate is evenly distributed.

Validation of the Models

The model validation was done by comparing the models output values with observed values on the field, this makes curve expert professional suitable for modeling volume as the software has model validation criterion in built. The validation process examines the usefulness or validity of the model, Marshall and Northway (1993). Residual plot was also used for model validation as it shows whether there is an upward or downward trend to the data as well as showing whether the residuals bias upward or downward.

The best-fitted models for each family and pooled data were selected for predicting volume from diameter at breast height. The predictive capacity of the models was tested by considering the significance difference between the observed and predicted value using student t-test as well as bias percentage and root mean square error. The result indicating no significant differences between the observed volume data and predicted volume values was considered the best.

Results

A total of 266 trees were observed in the four sampling plots but the study made use of the 171 trees in the four families that were selected for this study. The species and family names of trees encountered in the sample plots with their individual relative abundance are presented in Table 1. Table 2 shows the various family and number of species in respective family, the result shows that Meliaceae and Sterculiaceae had the highest number of species (6 spp. each). On the whole, 17 Nigerian tropical tree species distributed among 4 families and 171 individuals were present in the study area. The most abundant species was Mansoniaaltisima, this was followed by Celtiszenkerii.

These species have 45 and 42 individual trees, respectively. Summary of the growth variables as well as the species and family names are listed in Table 3. The Sterculiaceae family has the minimum and maximum diameter at breast height of 12.4 cm and 105 cm respectively. The minimum height was 15.00 cm found in the Ulmaceae family while the maximum height of 81.7 cm was found in Sterculiaceae family. The standard error for diameter at breast height ranges from 1.27 to 7.28 while the standard error for total height ranges from 0.86 to 4.11. Table 3 showed the correlation matrix between the tree growth variables. The result revealed that that there was a strong relationship between the growth parameters for each family and the pooled data

Table 1: Species encountered in the study area and their relative abundance in sample plot

Species

Abundance

Anonidium mannii

20

Cleistopholispatens

1

Monodoramyristica

1

Entandrophragmaangolensis

3

Entandrophragma cylindrical

1

Entandrophragmautile

8

Trichiliaheudelotii

1

Trichiliamonadefia

2

Trilepciousmadgascalear

2

Cola gigantea

14

Cola hispida

4

Mansoniaaltisima

45

Sterculiaoblongata

1

Sterculiarhinopetala

9

Triplochitonscleroxylon

14

Celtislibri

3

Celtiszenkerii

42

Total

171

Table 2: Tree species distribution into families in the study area

Family

No of species

Annonaceae

3

Meliaceae

6

Sterculiaceae

6

Ulmaceae

2

Total

17

Table 3: Descriptive statistics of tree growth variables for families and the pooled data

DBH (cm)

Height (m2)

Families

Mean

SD

Min

Max

Mean

SD

Min

Max

Annonaceae

36.32

2.76

13.50

83.5.0

28.3

1.87

15.5

57.00

Meliacea

39.33

7.28

16.00

97.00

34.48

4.11

19.40

79.00

Sterculiaceae

28.48

1.64

12.40

105.00

39.59

1.52

19.00

81.70

Ulmaceae

25.66

1.34

12.50

55.00

33.95

1.45

15.00

58.00

Pooled

32.03

1.27

11.00

142.00

36.38

0.86

11.00

81.70

Table 4: Correlation Matrix of tree growth variables for families and the pooled data

Family

variables

DBH(cm)

BA(m2)

H(m)

V(m3)

Annonaceae

DBH(cm)

1.00

BA(m2)

0.97

1.00

H(m)

0.68

0.74

1.00

V(m3)

0.90

0.98

0.82

1.00

Meliaceae

DBH(cm)

1.00

BA(m2)

0.99

1.00

H(m)

0.83

0.80

1.00

V(m3)

0.95

0.95

0.90

1.00

Sterculiaceae

DBH(cm)

1.00

BA(m2)

0.95

1.00

H(m)

0.64

0.52

1.00

V(m3)

0.88

0.91

0.66

1.00

Ulmaceae

DBH(cm)

1.00

BA(m2)

0.98

1.00

H(m)

0.72

0.70

1.00

V(m3)

0.87

0.91

0.78

1.00

Pooled

DBH(cm)

1.00

BA(m2)

0.95

1.00

H(m)

0.67

0.59

1.00

V(m3)

0.92

0.93

0.72

1.00

DBH= Diameter at breast height, BA= Basal Area, H= Height, V= Volume

Result of the Non-Linear Models

The study observes the volume models suitable enough to define the relationship between diameter at breast height and volume in strict nature reserve. The models considered were Logistics, Gompertz Relation, Logistic Power, Ratkowsky model, Richards and Weibull model and were determined to be good models in describing volume of trees in the forest reserve.

Akaike Information Criterion (AIC) value and standard error was used to rate the best model. The standard error and residuals were similarly used to express the goodness of a particular curve fit, the standard error of estimate with a smaller standard error represent a better curve fit. Since all the residuals of the modeled functions have the data spread on the positive and negative sides of the plot show a constant breadth and the horizontal independent variables do not follow any systematic trend.

In the Annonaceae Family, the result indicate that Logistic, Ratkowsky model and Weibull model gave a good fit and as a result, They are very adequate for tree volume estimation as shown in table 5 with their AIC values as; -39.45, -39.46 and -36.98 while the standard error was; 0.39, 0.39 and 0.40 respectively. Meanwhile Logistics model gave the best line fit because of the lowest in AIC and standard error values. The t-test result carried out for the observed and predicted volume indicates that the t-statistics of 0.04 is less than the t-critical level of 1.72, meaning there is no significant difference (p>0.05) between the observed and predicted volume as shown in table 6.

Result of the volume models for Meliaceae indicates that Gompertz Relation, Logistic Power, and Ratkowsky model gave the lowest AIC value and standard error of 2.28, 2.28, 2.28 and 1.02, 1.02, 1.02 respectively which make it the best model that could estimate volume in the study area. The validation of the result shows that t-statistic of 1.63 is less than t-critical of 1.74, meaning there is not a significant difference between the observed volume and predicted volume.

Logistic Power, Gompertz Relation and Weibull model were adjudged to be good fitted models for Sterculiaceae family in volume estimation as shown in figure 5 with their AIC values as 37.19, 38.86 and 39.09 and their standard error as 1.23, 1.23 and 1.24 respectively as shown in table 5. Meanwhile, Logistic Power gives the best model because of its low AIC value and standard error; this was however confirmed by the result of the residuals with a probability of 96.30% with the independent variable spreading across the horizontal line. The t-test result carried out for the observed and predicted volume indicates that the t-statistics of 0.28 is less than the t-critical level of 1.66, meaning there is no significant difference (p>0.05) between the observed and predicted volume as shown in table 18.

For the Ulmaceae family, Gompertz Relation, Logistic Power, and Weibull model gave good fit in calculating the volume using diameter as the predicting variable. The AIC value and standard error of Gompertz Relation was -64.52 and 0.48 respectively, see table 5. Result of the validation indicate that there is no significant difference at (p>0.05) in the observed volume and predicted volume as seen in table 6.

Ratkowsky Model, Weibull Model, Gompertz Relation, Logistic Power, MMF, and Logistic were adjudged the best fit for the prediction of volume. However, Ratkowsky model was return the best model due to its lowest AIC value and standard error; 29.56 and 1.08 respectively, table 5.

The t-test result carried out for the observed and predicted volume indicates that the t-statistics of 0.28 is less than the t-critical level of 1.66, meaning there is no significant difference (p>0.05) between the observed and predicted volume as shown in table 6. In addition, the residual plot showed a good fit as the data were evenly distributed which confirmed the goodness of fit of the model, figure 1.

Generally, the percentage biases when the output of the each model was compared with the observed volume are very small (less than 30%) for all the models. The Annonaceae, percentage biases ranges from 0.15 to 0.19%. Meliaceae ranges from -3 to 8%, For Sterculiaceae, percentage biases are from -27 to 2%, For Ulmaceae, percentage biases ranges from -51 to 1%. For the entire forest, percentages biases range from -3 to 5%.

Table 5: Models, Model functions, models parameters and selection criteria to estimate volume (m3) for each family and pooled data

Family

Models

Parameters Estimate

Fit Statistics

a

b

c

d

AIC

Std. Error

Annonaceae

Logistic

24.00

207.00

0.07

-39.45

0.40

Ratkowsky

24.00

5.33

0.07

-39.45

0.40

Wiebull

23.40

22.90

2.91E-08

3.92

-36.98

0.40

Gompertz

360.00

2.14

0.01

-38.92

0.40

Logistic Power

6300

688.00

-2.86

-36.94

0.42

MMF

-4.38

14.10

84.60

1.30

16.56

1.36

Meliceae

Gompertz Relation

14.90

98.50

1.51

2.28

1.02

Logistic Power

14.90

65.60

-0.85

2.28

1.02

Ratkowsky

14.90

85.30

1.29

2.28

1.02

Logistic

14.90

1.13E+13

0.45

2.37

1.02

Wiebull

17.10

17.10

4.41E-06

2.85

22.83

1.78

MMF

-2.08

1130

66.8

1.28

26.72

1.99

Sterculiaceae

Logistic Power

35.30

102.00

-2.54

35.19

1.23

Gompertz Relation

27.10

1.86

0.03

36.86

1.23

Weibull

22.60

22.50

9.63E-06

2.59

39.09

1.24

Ratkowsky

19.50

4.42

0.06

39.47

1.25

MMF

-2.41

1030

66.50

66.97

1.45

Logistic

-3.56E+08

-5.34E+08

0.03

77.01

1.55

Ulmaceae

Gompertz Relation

85.3

2.03

0.02

-64.52

0.48

Logistic Power

9080

882

-2.63

-64.52

0.48

Weibull

0.06

0.06

2.49E-07

2.77

-62.39

0.49

Logistic

14.3

150

0.09

-63.18

0.49

Ratkowsky

14.2

5.01

0.09

-63.18

0.49

MMF

-1.39

1040

33.5

1.34

-43.45

0.6

Pooled Data

Ratkowsky

17.4

4.73

0.07

29.56

1.08

Weibull

17.3

17

8.71E-07

3.24

32.68

1.09

Gompertz

24.3

1.97

0.03

34.08

1.1

Logistic Power

26.9

85.1

-2.91

35.61

1.11

MMF

-2.59

1020

70.8

1.23

117.47

1.4

Logistic

-3.12E08

-5.45E08

0.03

125

1.44

Table 6: Validation result of the non-linear models with Bias, Bias percentage, RMSE, and T-test

Model

Bias

RMSE

Bias %

Observed Volume

Predicted Volume

T-stat

P-value

Remark

Annonaceae

Logistic

-0.0015

0.37

-0.15

1.94

1.95

0.04

0.48

ns

Ratkowsky

-0.00014

0.37

-0.014

Weibull

0.0019

0.36

0.19

Gompertz Relation

0.06

1.30

6.00

3.24

3.00

1.63

0.06

ns

Meliacea

Logistic Power

0.08

1.25

8.00

Ratkowsky

-0.03

1.27

-3.00

Sterculiaceae

Logistic Power

0.02

1.21

2.00

1.84

1.81

0.28

0.38

ns

Gompertz Relation

-0.27

1.41

-27.00

Weibull

0.004

1.21

0.40

Ulmaceae

Gompertz Relation

-0.08

0.76

-8.00

1.07

1.16

1.22

0.11

ns

Logistic Power

0.01

0.47

1.00

Weibull

-0.51

1.01

-51.00

Pooled data

Ratkowsky

0.05

1.09

5.00

1.79

1.71

0.99

0.17

ns

Weibull

0.02

1.08

2.00

Gompertz Relation

-0.03

1.09

-3.00

Annonaceae

Meliaceae

Sterculiaceae

Ulmaceae

Pooled Data

Figure 1: Residual results per each model for the families and the pooled data at Dbh (cm) and Volume (m3)

Discussion

Tree species diversity obtained in this study area is typical of tropical rainforestexosystem (Adekunle 2007). Seventeen Nigerian tropical tree species distributed among 4 families and 171 individuals were encountered in the study area. The ecosystem is rich in species and diversity. Species in this ecosystem are very useful as timber, enrichment of soil fertility, creation of microclimate and supply of many non-timber forest products.

The tree growth variable measured in the study area showed that the mean dbh value of Annonaceae, Meliaceae, Sterculiaceae, Ulmaceae and the pooled data were 36.32cm, 39.33, 28.48, 25.66, 32.03 respectively as shown in table 3. This shows that most of the trees encountered in this study area are below the minimum merchantable size of 48cm stipulated by logging policy of southwestern Nigeria (Adekunle 2006).

It was observed from the correlation matrix that positive linear relationship exists between the variables tested at individual family and the pooled data. This result is in accordance with the findings of Adekunle (2007) who observed a positive linear relationship between the tree growth variable. There was a strong relationship between Dbh and volume for respective family and the pooled data with a correlation value of 0.90, 0.95, 0.88, 0.87, and 0.92 for Annonaceae, Meliaceae, Sterculiaceae, Ulmaceae, and the pooled data. This agreed with the findings of Akindele and Lemay (2006) who reported that volume is linearly related to Dbh or basal area in a curvilinear manner.

The effectiveness of non-linear model for estimating volume in the tropical forest ecosystem was also considered in this study. Logistic, Gompertz Relation, Ratkowsky, MMF, Logistic Power, and Weibull model were considered and suitable for describing the volume-diameter relationship in the study area. The assessment criteria (AIC and standard error) showed that the entire model are suitable for volume estimation in the study area. The non-linear models showed a reasonable variation depending on family.

The result showed that Weibull, Gompertz Relation and Logistic Power model are the most consistent model and the models gave the best predicted volume when compared with the observed volume for each family in the study area but Ratkowsky model ranked the best of the six models generated when data from each family were combined. The residual plots for the models generally showed an even spread of residuals above and below the zero line with no systematic trend.

At the bottom left corner of each residual plot, the observed number of runs is listed, as well as the likelihood which shows that the observed number of runs of the models fitted the data correctly (i.e. the residuals are randomly distributed around the curve. The result of the student t-test showed that there was no significant difference (p>0.05) in the models’ outputs. All the non-linear models developed in this study were discovered to be very adequate for yield estimation in the study area and they are recommended for further use.

Conclusion

The study tested the efficacy of nonlinear models for tree volume estimation in tropical natural ecosystem. The tree growth data were obtained from four permanent sample plots located in the strict nature reserve. One hundred and seventy one trees comprising 17 species distributed among 4 families were involved in model generation and validation. Curve expert professional suitable for modeling were used as the software has model validation criterion in built. All the models developed in this study were discovered to be very adequate for yield estimation and are recommended for tree volume estimation in tropical natural forest ecosystem of southwest Nigeria and in any similar ones

Reference

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