Above-Ground Biomass Models of Caragana korshinskii and Sophora viciifolia in the Loess Plateau, China

Abstract

The quantification of above-ground biomass is based on the calculation of carbon storage, which is important for the balance of carbon cycling. However, the allometric models of shrubs for calculating the above-ground biomass of shrubs in the Loess Plateau are scarce. In order to solve this issue, this study analyzed some highly correlated variables, including height (H), branch diameters (D), canopy volume (C_v), canopy area (C_a), and then built a regression model to predict the above-ground biomass in two common shrubs (Caragana korshinskii and Sophora viciifolia) in the Loess Plateau, China. The results show that the above-ground biomass of these two shrubs can be accurately predicted by H and D, and then we can use allometric model (y = ax^b) to calculate shrub above-ground biomass (including leaf biomass and branch biomass). Furthermore, the correlation between leaf biomass and branch biomass in Caragana korshinskii and Sophora viciifolia indicates that the components of above-ground biomass are closely related to each other. In addition, there is a strong linear relationship (p < 0.01) between the observed and estimated biomass values, which confirms the data accuracy of the above-ground biomass estimation models. In summary, these two biomass estimation models provide an accurate way to estimate the quantification of carbon for shrubs in the Loess Plateau.

1. Introduction

Above-ground biomass, the amount of organic matter in dead and living material, is a key plant property resulting in the balance of carbon cycling [1,2,3,4,5]. The shrub biomass model is an important method for estimating shrub biomass. In the past decades, above-ground biomass has received more attention in terms of carbon storage in ecosystems [4,5,6]. Accordingly, there are a multitude of methods for above-ground biomass estimation by harvesting individuals with morphometric variables through an allometric model (including trees and shrubs) [4,7,8]. Furthermore, advanced techniques and methods are widely used to determine above-ground biomass [9,10]. For example, height (H), branch diameters (D), crown volume (C_v), and crown cover (C) may be employed in biomass estimation models as the independent variables [11,12,13,14,15]. As for trees, stem-related variables, specifically height (H) and diameter (D), have proven to accurately predict individual biomass [4,10,14]. In contrast, shrub biomass is an important component for the total biomass in a terrestrial ecosystem. In multi-stemmed shrub biomass estimations, each stem is usually considered as a separate shrub, thus, by applying the standard relationship between D and H, one can estimate the shrub biomass [16,17,18]. Furthermore, many scientists estimate shrub biomass using the same formula as with tree biomass estimation, which is confirmed in the literature [4,10,14]. However, there is little literature to report the above-ground biomass estimation for shrubs because of the lack of methodology and difficulty in calculation.

In the Loess Plateau, above-ground biomass has greatly increased because of the Grain for Green Project from 1999, and much more attention has been placed on developing generalized allometric models for shrubs [15,19,20]. In particular, there are several common shrub species in this region where allometric models have not yet been developed. Therefore, developing more accurate regression equations for shrub biomass estimation is essential for estimating carbon storage in this region. In this way, the choice of fitting methods and models is the most important issue to be solved [9,10]. Previous studies have reported that the biomass equations can be separated into additive and non-additive equations [21,22,23]. In additive equations, the estimation for the biomass components is calculated to predict the total biomass [4,14]. In contrast, in non-additive equations, the biomass components are neglected, and the biomass components are measured separately [23,24]. Therefore, the calculation of shrub biomass (n = 57) should be based on the same method and variables, the most appropriate regression equation, and regression coefficient test for biomass components (n = 58). This study aimed to find a scientific method for quantifying shrub biomass, which is easy to use in the field, and produces reliable results for forecasting carbon storage in the Loess Plateau.

In this study, two common shrubs (Caragana korshinskii and Sophora viciifolia) were investigated in the Loess Plateau, and some ecological variables were tested to predict shrub biomass. The aim was: (1) to develop equations for biomass components based on easily measured variables; (2) to calculate parameters for estimating biomass components by fitting regression models; and (3) to test the constructed models in the field. Although this study included only two shrub species, it can provide a basis for the standardization of useful variables to predict shrub biomass in the Loess Plateau.

2. Materials and Methods

2.1. Study Site

The study area, Yanhe River catchment (36°23′–37°17′ N, 108°45′–110°28′ E), north of Shaanxi Province, is located in the middle of the Yellow River catchment. Hills cover 90% of the region (which is 7687 km² in total area). The Yanhe River catchment has a periodic flooding and heavy seasonal rainfall with a semi-arid climate. The average annual rainfall from 1970–2000 was about 497 mm, and there are distinct rainy and dry seasons. The average annual temperature is between 7 °C to 9 °C. The rainy season is from July to October, and August accounts for 23% of the total annual rainfall. Most of this area lies at an altitude between 900 to 1500 m. The topography of this area is characterized by cliffs with steep slopes (about 30%–40%). According to the Chinese Soil Taxonomy (Chinese Soil Taxonomy Research Group, 2001), the main soil type is typical Haplic-ustic Cambisol in this area. The dominant vegetation types are forest land, shrubland, natural grassland, and cropland, and the most abundant plants are medium or small shrubs, including Caragana korshinskii (non-native shrub artificial planted shrub) and Sophora viciifolia (native shrub) [25,26].

2.2. Sampling Design

The field surveys were conducted in the middle of August 2016 in the Loess Plateau, and two representative multi-branched shrubs, Caragana korshinskii and Sophora viciifolia, were selected across the whole study area (Figure 1). Keeping the environmental and climatic conditions as constant as possible, six sites were selected in this study (

Table 1. The basic characteristics of sample plots.

Shrubs	Site	Sample Content	Longitude and Latitude	Slope/(%)	Altitude/(m)	Soil Temperature/(℃) 0–10 cm 10–20 cm		Accompanying Species
Caragana korshinskii	1	93	36°47′30″ N	SE	1151	21.8 ± 1.7 a	20.1 ± 0.6 b	Lespedeza daurica, Artemisia giraldii, Patriniascaniosaefolia
			109°16′10″ E	15~23
	2	161	36°49′01″ N	SW	1329	22.6 ± 0.8 a	20.7 ± 1.3 b	Lespedeza daurica, Artemisia giraldii, Patriniascaniosaefolia
			109°14′46″ E	10~20
	3	161	36°49′04″ N	NE	1347	27.4 ± 1.9 a	25.6 ± 0.8 b	Lespedeza daurica, Artemisia giraldii, Melilotussuavcolen
			109°14′43″ E	23~35
	4	160	36°43′54″ N	NE	1219	20.8 ± 1.2 a	20.7 ± 1.1 a	Artemisia vestita, Lespedeza daurica, Artemisia giraldii
			109°15′30″ E	9~18
	5	158	36°45′36″ N	SW	1301	21.2 ± 0.5 a	19.9 ± 0.3 b	Artemisia vestita, Lespedeza daurica, Artemisiascoparia
			109°16′01″ E	10~20
	6	128	36°44′49″ N	SW	1236	22.0 ± 0.8 a	20.5 ± 0.6 b	Artemisia vestita, Artemisiascoparia, Artemisia frigida
			109°16′13″ E	9~13
Sophora viciifolia	1	116	36°47′20″ N	S	1256	24.3 ± 5.1 a	22.1 ± 2.3 b	Lespedeza daurica, Bothriochloaischaemum, Stipabungeana
			109°16′30″ E	19~35
	2	180	36°47′35″ N	W	1248	23.4 ± 2.1 a	21.2 ± 1.9 b	Lespedeza daurica, Bothriochloaischaemum, Stipabungeana
			109°16′41″ E	14~20
	3	139	36°47′58″ N	SE	1271	25.5 ± 3.2 a	23.3 ± 1.7 b	Lespedeza daurica, Bothriochloaischaemum, Lespedeza daurica
			109°16′47″ E	8~12
	4	165	36°44′42″ N	NE	1283	25.2 ± 1.8 a	24.4 ± 1.3 b	Lespedeza daurica, Bothriochloaischaemum, Thyme
			109°14′35″ E	11~19
	5	96	36°44′56″ N	NW	1269	27.9 ± 0.6 a	25.3 ± 0.4 b	Lespedeza daurica, Artemisia giraldii, Bothriochloaischaemum
			109°14′01″ E	8~23
	6	132	36°45′03″ N	NE	1156	26.9 ± 0.9 a	26.1 ± 1.1 a	Artemisia giraldii, Patriniascaniosaefolia, Lespedeza daurica
			109°15′11″ E	18~30

Note: Values followed by lowercase letters within columns are significantly different for 0.05 using Tukey’s method. The same applies below.

) and each site was 50 × 50 m². There were five quadrats in each plot, as shown in the Figure 1, and the distance among the sample plots was about 1 km. During the field sampling, each quadrat was set as 5 × 5 m². Finally, more than 800 leaves and branches of these two shrubs were harvested. More than 100 individual shrubs were chosen by grid method to cover represent each species in the field. The two common shrubs in the Loess Plateau were selected Each branch was considered as being the main upright portion of the shrubs [27].

Individual shrubs were harvested at ground level, and fresh subsamples of leaves and branches were cut down and stored in sealed plastic bags. Then, the samples were oven-dried in forced-air ovens at 80 °C until constant weight. The water weight was subtracted from the individual fresh mass to get the leaf and branch biomass (Table 1).

According to above-ground biomass, several variables were quantified to be used to estimate leaf and branch biomass: the maximum height (H_max), defined as the distance between the highest crown point and the ground surface; number of branches emerging from the root; the maximum (D₁) and minimum (D₂) canopy diameters were measured in 12 transects of 100 m × 100 m. With these easy and measurable variables, canopy volume (C_v) and canopy area (C_a) were calculated as [28]:

$$C_a=\frac{\pi }{4\left(D_1{D}_2\right)}\cdot CF=e^{(SEE\times SEE/2)}$$

(1)

$$C_v=C_a{H}_{\max }$$

(2)

2.3. Biomass Equations

Regression analyses were used to examine the relationship among above-ground biomass components and the measured variables, including diameter (D), height (H), canopy volume (C_v), and canopy area (C_a). A graphical and statistical analysis indicated that D, D²H explained most of the variation in biomass, and the optimum biomass components for the regression equation model was calculated by least-squares procedures using D and D²H. Compared with regression coefficients, the following equations were presented for describing the relationship between above-ground biomass components:

$$W=a+bx+c{x}^2$$

(3)

$$W=a{x}^2$$

(4)

$$W=a+b\ln x$$

(5)

In addition, the optimum statistical model was selected according to the Akaike Information Criterion (AIC) which penalizes the number of parameters [29,30]. To reflect the accuracy of the statistical model, the predictive mean squared error (MSE) was chosen to be the alternative statistic. Besides, the total variance of equations was conventionally determined by the summation of squared residuals. Thus, the standard error of the estimation (SEE), the coefficient of determination (R²), and F value were used to make the goodness fitting. The equations were originally described in previous reports. The parameter as follows [31,32]:

$$R^2=1-\frac{RSS}{TSS}$$

(6)

$$y_2=\left[{\displaystyle \sum _{i=1}^n{y}_1}\right]/n$$

(7)

$$RSS={{\displaystyle \sum _{i=1}^n(y_1-y_2)}}^2$$

(8)

$$TSS={{\displaystyle \sum _{i=1}^n(y_1-y_2)}}^2$$

(9)

$$F=\frac{TSS-RSS}{RSS/(n-2)}$$

(10)

where y₁ refers to the estimated biomass (g·m⁻²); y_i refers to the observed biomass (g·m⁻²); y₂ refers to the mean of the observed value of y_i (g·m⁻²); and n refers to the number of observations.

Finally, in order to examine the constructed model in the field, correction factors (CF) were computed so as to be applied to above-ground biomass in the constructed model, and CF were calculated as follows [31,32]:

$$above-groundbiomass=CF\times e^{\ln (above-groundbiomass)}$$

(11)

$$CF=e^{(SEE\times SEE/2)}$$

(12)

where SEE refers to the standard error of the estimation; CF refers to correction factor.

2.4. Statistical Analysis

SPSS 20.0 (SPSS Inc., Chicago, IL, USA) was used for all analyses. Before the parametric tests, a homogeneity of variance test was completed. Analysis of variance (ANOVA) was used to analyze the differences between the Caragana korshinskii and Sophora viciifolia; multiple comparisons were used to determine significant analysis by Tukey’s test (p < 0.01 or p < 0.05). Average and standard deviation were determined to fit to a Gaussian distribution using the Kolmogorov–Smirnov test. In Gaussian distribution, the kurtosis of residuals and the standardized values in a N (0,1) distribution must be guaranteed. Finally, all the above-ground biomass components data were randomly divided into two groups with a Chow Test, which was a validation method to determine whether the above-ground biomass components data had the same regression model compared with the random sub-samples.

3. Results

3.1. Correlation Analysis of Biomass Variables

The observations of this study indicate that the biomass variables of Caragana korshinskii were significantly higher than those of Sophora viciifolia (p < 0.05) except for height (

Table 2. Ecological parameters for the Caragana korshinskii and Sophora viciifolia.

Shrubs	Site	Height/(cm)	Diameter/(mm)	Canopy Area/(cm2 104)	Canopy Volume/(cm3 106)	Coverage/(%)	Leaf Biomass/(g)	Branch Biomass/(g)
Caragana korshinskii	1	156 ± 32	5.2 ± 0.6	2.91 ± 0.21	4.54 ± 0.56	58 ± 6	358 ± 35	589 ± 39
	2	137 ± 21	4.8 ± 0.5	2.32 ± 0.58	3.18 ± 0.89	67 ± 9	321 ± 29	512 ± 35
	3	125 ± 35	4.6 ± 0.9	1.66 ± 0.35	2.08 ± 0.52	83 ± 8	305 ± 16	456 ± 29
	4	113 ± 19	3.2 ± 0.3	1.34 ± 0.26	1.51 ± 0.41	92 ± 12	294 ± 32	478 ± 34
	5	109 ± 26	3.4 ± 0.4	1.22 ± 0.10	1.33 ± 0.32	85 ± 5	287 ± 18	423 ± 25
	6	151 ± 14	5.1 ± 0.8	2.77 ± 0.43	4.18 ± 0.78	62 ± 8	389 ± 42	563 ± 37
	CV/(%)	14.79	19.81	35.91	48.96	18.73	12.29	12.76
	Mean	132 ± 19 a	4.4 ± 0.9 a	2.04 ± 0.73 a	2.80 ± 1.37 a	75 ± 14 a	326 ± 40 a	503 ± 64 a
	F	89.69	156.98	178.32	135.69	103.56	158.96	147.89
Sophora viciifolia	1	123 ± 11	4.2 ± 0.2	1.15 ± 0.09	1.41 ± 0.45	23 ± 3	324 ± 36	456 ± 52
	2	96 ± 23	2.9 ± 0.6	0.83 ± 0.12	0.79 ± 0.21	35 ± 5	257 ± 23	356 ± 43
	3	108 ± 34	3.1 ± 0.5	0.90 ± 0.25	0.97 ± 0.19	38 ± 9	269 ± 18	324 ± 51
	4	145 ± 25	4.8 ± 0.8	1.92 ± 0.08	2.78 ± 0.23	20 ± 6	395 ± 25	423 ± 41
	5	112 ± 18	3.5 ± 0.4	1.09 ± 0.15	1.22 ± 0.35	43 ± 8	298 ± 31	389 ± 38
	6	97 ± 16	3.1 ± 0.3	0.86 ± 0.14	0.83 ± 0.14	39 ± 8	278 ± 17	362 ± 31
	CV/(%)	16.81	20.79	36.47	56.05	28.23	16.69	12.51
	Mean	114 ± 18 a	3.6 ± 0.7 b	1.13 ± 0.41 b	1.33 ± 0.75 b	33 ± 9 b	304 ± 51 a	385 ± 48 b
	F	96.38	156.41	147.85	125.87	214.79	134.57	126.84

Note: Values followed by lowercase letters within columns are significantly different for 0.05 using Tukey’s method. The same applies below.

), which presented no significant difference between Caragana korshinskii and Sophora viciifolia (p > 0.05). In addition, the correlation analysis indicates that there was a significant correlation between biomass variables of Caragana korshinskii and Sophora viciifolia shrubs (

Table 3. Correlation analysis of each variable of Caragana korshinskii and Sophora viciifolia.

Shrubs	Item	Height	Diameter	Canopy Area	Canopy Volume	Leaf Biomass	Branch Biomass
Caragana korshinskii	Height	1.000
	Diameter	0.724 *	1.000
	Canopy area	0.523 *	0.321	1.000
	Canopy volume	0.469	0.456	0.759 **	1.000
	Leaf biomass	0.826 **	0.856 **	0.223	0.566 *	1.000
	Branch biomass	0.879 **	0.891 **	0396	0.547 *	0.899 **	1.000
Sophora viciifolia	Height	1.000
	Diameter	0.789 **	1.000
	Canopy area	0.587 *	0.563 *	1.000
	Canopy volume	0.436	0.525 *	0.689**	1.000
	Leaf biomass	0.875 **	0.845 **	0.368	0.698 **	1.000
	Branch biomass	0.786 **	0.756 **	0.423	0.513 *	0.903 **	1.000

*, p < 0.05; **, p < 0.01.

).

Positive correlations were observed between H, D, leaf biomass, and branch biomass of Caragana korshinskii and Sophora viciifolia shrubs (p < 0.05). There were significantly positive correlations between canopy volume (C_v), leaf biomass, and branch biomass (p < 0.05), while canopy area (C_a) had no significant positive correlations with leaf biomass or branch biomass (p > 0.05). Furthemore, H and D had significantly positive correlations with leaf biomass and branch biomass of the Caragana korshinskii and Sophora viciifolia shrubs (p < 0.05). Thus, H and D were selected to estimate the leaf biomass and branch biomass for the biomass equations.

The response curves of leaf biomass to branch biomass were linear relationship (R² = 0.7753, p < 0.001; R² = 0.8331, p < 0.001) (Figure 2), which might give an explanation for the correlations between leaf biomass and branch biomass of the Caragana korshinskii and Sophora viciifolia shrubs.

The confidence interval (CI) and average for above-ground biomass components was 77.53% and 23.8 g for Caragana korshinskii, and 83.31% and 13.2 g for Sophora viciifolia. In other words, leaf and branch biomass components accounted for 77.53% and 83.31%, respectively. However, the line of slope of Caragana korshinskii was above 1.0 and the line of slope of Sophora viciifolia was below 1.0. Therefore, the major part of the biomass of the Caragana korshinskii shrubs was concentrated in the branches and for the Sophora viciifolia shrubs it was concentrated in the leaves.

In addition, the relationship between D, D²H and above-ground biomass was calculated by unit regression (Figure 3 and Figure 4). The results show the positive exponential model between D and above-ground biomass, and the positive linear relationship between D²H and above-ground biomass. Furthermore, the SEE of the above-ground biomass is randomly distributed on the zero level with the increasing D and D²H, which means that the heteroscedasticity of the data was weaker, and it was suitable for estimation model analysis, which indicated that D or D²H with the biomass equations could accurately estimate the above-ground biomass.

3.2. Allometric Model for Biomass

On the basis of the independent variables, to build as many models as possible, the same methodology and test were used to find the optimum model (

Table 4. Biomass regression models of Caragana korshinskii and Sophora viciifolia.

Shrubs	Outliers/%	Equation Parameters
		Variate	Formula	Equations	a d	b d	c	R2	Adjust R2	SEE	F
Caragana korshinskiia	13.6	xc = D	1	W = a + bx + cx2	13.424	−6.203	1.308	0.729 ***	0.701	35.154	560.709
	9.2	x = D2H		W = a + bx + cx2	4.282	0.005	0.007	0.770 ***	0.625	33.368	968.813
	6.8	x = D	2	W = ax b	0.864	1.739		0.765 ***	0.632	31.682	1117.664
	10.7	x = D2H		W = ax b	0.076	0.703		0.898 ***	0.771	8.369	1748.984
	8.5	x = D	3	W = a + blnx	−18.354	22.196		0.695 ***	0.524	35.389	483.002
	9.3	x = D2H		W = a + blnx	−49.752	9.039		0.775 ***	0.605	32.347	777.264
Sophora viciifoliab	6.2	x = D	1	W = a + bx + cx2	0.001	0.125	2.727	0.683 ***	0.598	31.157	130.596
	3.8	x = D2H		W = a + bx + cx2	0.150	0.153	1.371	0.829 ***	0.772	7.492	1453.608
	13.6	x = D	2	W = ax b	0.199	4.607		0.727 ***	0.658	29.568	152.494
	5.3	x = D2H		W = ax b	0.745	0.080		0.868 ***	0.753	6.987	2614.672
	4.9	x = D	3	W = a + blnx	2.226	4.653		0.681 ***	0.602	34.714	145.606
	8.2	x = D2H		W = a + blnx	6.025	−27.358		0.805 ***	0.647	11.493	1575.173

Note: ^a Caragana korshinskii sample content = 861; ^b Sophora viciifolia sample content = 828. Models for all dependent variables are of these three forms, where y is the dependent variable (above-ground biomass), ^c x is the predictor variable (D or D²H), and ^d a and b are constants in the equation. Percentage of outliers, R², F-value of ANOVA test, and average and standard deviation of residuals are shown. *, p < 0.05; **, p < 0.01; ***, p < 0.001.

). The correlations between above-ground biomass and various independent variables were generally greater. However, the ability to predict the biomass of Caragana korshinskii was better than that of the Sophora viciifolia shrubs (with the higher correlation coefficient), and so this statistical test was applied to these equations.

In all the equations, the Caragana korshinskii and Sophora viciifolia shrubs have the same regression model across all samples. R² values of the exponential model were higher than the other models with the lower SEE, and the predictability (ANOVA test) was significant (p < 0.05). In the multiple-variables model, H and D were selected, as expected, the optimum model was also model (2), explaining about 90% of the above-ground biomass of the Caragana korshinskii and Sophora viciifolia shrubs (p < 0.001). In addition, the exponential model (2) had the highest R² values, although R² values were generally high in models (1) and (3). In most cases, R² values of D²H were higher than D when models (2) were used to investigate the Caragana korshinskii and Sophora viciifolia shrubs. Thus, model (2) was the best equation to estimate the above-ground biomass using D²H with the higher R², Adjust R², F, and the lower SEE. Furthermore, the allometric model of the Caragana korshinskii shrubs was W = 0.076x^0.703, the allometric model of the Sophora viciifolia shrubs was W = 0.745x^0.080.

3.3. Biomass Model Verification

In this part, these independent variables were firstly checked to see whether they could improve the fitness of the regression model. Secondly, the “intercept” was tested to see whether it had a statistically significantly difference from zero using the t-test. If this was not the case, we removed the “intercept” from our models using the statistical points. Finally, some of the “intercepts” were recommended to remain even though the value was not significantly different from zero, because they help to make a precise prediction. Thus, the mean error (ME), mean absolute error (MAE), total relative error (TRE), mean sybranchatic error (MSE), and mean absolute percentage error (MPSE) were calculated by three models, and all these parameters were low, which indicated that the allometric model had an ecological significance according to one-way ANOVA (

Table 5. Testing of accuracy of biomass regression models of Caragana korshinskii and Sophora viciifolia.

Shrubs	Equations	Goodness Indicator of Fitting
		MEa/g	MAE b/g	TREc/%	MSEd/%	MPSEe/%
Caragana korshinskii	W = a + bx + cx2	13.256	43.257	2.369	15.036	32.568
	W = ax b	0.803	15.369	−0.063	−1.058	15.036
	W = a + blnx	9.125	32.025	8.263	8.214	23.256
Sophora viciifolia	W = a + bx + cx2	10.039	34.177	−0.159	3.692	12.035
	W = axb	0.236	19.256	−1.256	2.369	9.285
	W = a + blnx	5.123	23.054	5.032	8.038	29.877

Note: Diameter at breast height versus total tree height models for Caragana korshinskii and Sophora viciifolia used to develop the allometric models. ^a Mean while minimum values of mean error (ME), ^b mean absolute error (MAE), ^c total relative error (TRE), ^d mean sybranchatic error (MSE), and ^e mean absolute percentage error (MPSE).

).

In order to test the biomass estimation model in the field, the observed and estimated biomass values of the Caragana korshinskii (n = 125) and Sophora viciifolia (n = 129) shrubs were compared (Figure 5). The results show that the response curves of the observed and estimated values of the Caragana korshinskii and Sophora viciifolia shrubs had a linear relationship (R² = 0.8499, p < 0.001, SEE = 26.3856; R² = 0.9007, p < 0.001, SEE = 24.1357), which indicated that the estimated values were quite close to the observed values, suggesting that this allometric model could be used to estimate the above-ground biomass of the Caragana korshinskii and Sophora viciifolia shrubs in the Loess Plateau.

4. Discussion

Now, constructing an empirical model to estimate forest biomass from easily measurable indicators (height, DBH, etc.) is the most effective and environmentally friendly way to evaluate forest biomass and carbon storage [33,34,35]. This study also used this method and made a contribution to above-ground biomass prediction by developing the equation models. Variables for the above-ground biomass components (leaf and branch) with three procedures were estimated: (1) estimating leaf and branch biomass by using the same method; (2) adding the optimum regression models of each leaf and branch biomass; and (3) the joint generalized least squares was used to iteratively force the sum of the coefficients to satisfy the equation models [21,22,23,24,36]. Finally, the estimated and observed values were used to test the biomass estimation model.

The results of this study show that that the biomass variables of Caragana korshinskii were significantly higher than those of Sophora viciifolia (p < 0.05) except for height (Table 2), which may be because Caragana korshinskii and Sophora viciifolia have different growth characteristics, namely, they have different transpiration rate and water use efficiency [37,38]. A previous study found that it was not ideal to use a single variable to simulate the biomass regression equation. Only the proper combination of each variable can estimate the biomass of shrubs accurately [39]. In the present study, H and D were selected to estimate the leaf biomass and branch biomass by the best fitting biomass equations for Caragana korshinskii and Sophora viciifolia, which confirmed that a single variable cannot reflect the above-ground biomass.

In this study, the biomass equation models presented a cross-validated R², which indicated the suitability to calculate the above-ground biomass of shrubs. Therefore, we can conclude that the allometric model of the Caragana korshinskii shrubs was W = 0.076 x^0.703, and the allometric model of the Sophora viciifolia shrubs was W = 0.745x^0.080 (Table 4). These results are in agreement with the other studies [4,10,15], which also certify the allometric model for above-ground biomass estimation.

Furthermore, other studies have shown that the optimal biomass estimation models of different shrub species were different, while the biomass estimation models for the same species were mostly the same; this is due to the differences between species [40]. However, in this study, the biomass models of Caragana korshinskii and Sophora viciifolia were both based on the form y = ax^b, which may be due to Caragana korshinskii and Sophora viciifolia both belonging to the leguminous family.

The fitting of the allometric model is based on logarithmic transformation of variables, which may produce a higher homogeneity of variance [4,10]. Furthermore, the present study found that the harvested biomass of the percentage of outliers in all samples (measured) was highly significant (p < 0.001) (Table 5). The percentage of outliers was more than 10% in a few cases, so in future studies, increasing the sample scale might act to reduce the number of outliers, though there was a normal level of outliers. In addition, as regards the progress of these two shrubs’ growth, the exponential biomass–size relationships may be transformed into linear models [12,14,32]. In view of these considerations, other important variables related to the growth of shrubs should be considered in above-ground biomass estimation [12,14].

5. Conclusions

This study explored the above-ground biomass models of two common shrubs (Caragana korshinskii and Sophora viciifolia) in the Loess Plateau, China. For both shrubs, models based on y = ax^b were suitable to estimate above-ground biomass using the height and crown diameter. In addition, the response curves of the observed and estimated values of the Caragana korshinskii and Sophora viciifolia shrubs also confirmed that our above-ground biomass estimation models were suitable. Because of the impact of climate, soil, terrain, and other factors on the growth of shrubs, especially in arid and semi-arid areas, the growth of shrubs can be greatly affected by the soil moisture status because of the great variation in precipitation resources in interannual or annual distribution; therefore, the same species may have different estimation models under different site conditions. Hence, the above-ground biomass models for the Caragana korshinskii and Sophora viciifolia shrubs may be unsuitable in other regions. In the future, studies in this field should attempt to use environmental factors as the explanatory variables.