4.1.

Forest Inventory

During the radar surveying campaigns in 2004 and 2006, forest inventories were conducted simultaneously in order to characterize the dendrometric parameters according to the annual inventory tracking. The inventory was carried out in 23 independent plots of $\sim 400{\mathrm{m}}^2$ each, in which the DBH and total tree height were measured. Allometric equations based on height, diameter, and form factor were used to calculate tree volume. Equation (1) shows the allometric equation for volume estimation and the corresponding form factors used for each range of DBH measured in the inventory.

4.2.

Radar Data Processing

The Orbisat Company provided the software to process the SAR raw data in order to obtain the images and the interferometry SAR products. The images were processed with four looks to minimize the speckle noise, and then converted to ground range and orthorectified using the corresponding DEM generated during the interferometric processing. Rombach et al.¹⁷ described the details of the data transcription, IMU processing, and SAR data processing to obtain SAR products.

The radiometry of P-band complex images was calibrated by the Radar Tools software (RAT 0.16.2 version), which provided cross talk processing, channel imbalance processing, and polarimetric calibration using a corner reflector response based on Quegan¹⁸ approach.

To calibrate the SAR data, nine corner reflectors of the same size, 1.5-m side length, were distributed between near and far ranges in the area and used for the polarimetric and interferometric calibrations. To acquire ${\sigma }^{\mathrm{o}}$ images, the corner reflector peak response was used (Zink et al.,¹⁹ Santos et al.¹) obtaining a correction factor to transform the P band amplitude images by interactive data language (IDL) routines.

The azimuth and elevation angles of the corner reflectors were calculated based on the flight plan and on the high-accuracy (5 cm) GCP geographic coordinates, the incidence angle of which did not present large variations.

The SAR data was generated using the Orbisat Company software, which allowed obtaining orthorectified SAR images using the corresponding digital elevation models. Details about the Orbisat software for data transcription, IMU and SAR data processing to obtain DEM, orthoimages, and coherences are described by Rombach et al.¹⁷

The radar cross section of each corner reflector was measured and a linearity behavior was verified. The signal to noise ratio, the antenna pattern, radar losses, and radar illumination geometry were also considered.

The corner reflector responses were also used for the phase calibration during the interferometric calibration in order to obtain the DEM in X and P bands. For polarimetric calibration, the Quegan¹⁸ methodology was used for the cross-talk processing and channel imbalance by these corner reflector responses.

4.3.

Statistical Modeling

To build the regression models, the variable selection criteria suggested by Neter et al.²⁰ were used, such as coefficient of determination ($r^2$), adjusted coefficient of determination ($r^{2}a$), Cp criterion (Neter et al.²⁰), and Stepwise criterion, using Statistica 6.0 version software.

The coefficient $r^{2}a$ employs a weighting for the extent of regression adjustment, making it more relevant than the coefficient $r^2$. Cp criterion involves the concept of total mean square error for each subset of the adjusted regression models, which considers the total error in each value set. Cp criterion can identify subsets of dependent variables in which the total mean squared error is small, i.e., when the Cp value is equal or close to the number of $p$ parameters, this model will be less biased. The forward stepwise procedure gradually adds a new variable to the model, and extracts that part whose contribution was not considered significant by an $F$-test. Equation (2) shows the Cp criterion estimation

Levene’s test was used for the heteroscedasticity evaluation (constancy of residue variance), which performs the comparison $t$-test of two subgroups of data set samples to determine whether the average absolute deviation of a subgroup differs from the other. Cook’s distance method was used to assess the existence of outliers in the data set (Neter et al.²⁰), which considers the influence of a specific observation in all other adjusted values. A sample can be considered an outlier when its percentile of $F$ distribution exceeds 20%.

For model validation of small data sets, prediction sum of squares (PRESS) criterion is used as a way to evaluate the prediction model. The test works by eliminating the $i$’th case of the data set, estimating the regression function with the remaining observations, then using the adjusted regression equation to obtain the predicted value. The quadratic sum of all these $n$ prediction errors defines the PRESS value. According Neter et al.,²⁰ the closeness between PRESS and error sum of squares (SSE) values, indicates that the MSE value can be a reasonable indicator of the model predictive ability.

5.1.

Forest Inventory

Although the whole plantation area was established in 2000, the average total height of each transect varied widely in its values after having grown for 4 and 6 years, due to the fact that it was a seminal reforestation and the supply of water and nutrients were different inside the site index, compromising the development of the Eucalyptus trees. Due to this fact, the occurrence of gaps in the plantation was between 2% and 60% for the plots, in which dead or broken trees were not considered by this criterion. The graph in Fig. 3 shows the total height behavior and its standard deviation for 2004 and 2006 of each plot inventoried.

Fig. 3

Height values of Eucalyptus stands.

The mean DBH values of trees inventoried in 2004 and 2006 are shown in Fig. 4, where the newest inventory showed higher values if compared with those of the previous inventory. The floating values happened due to the influence of the kind of plantation (seminal) as well as to the edaphic problems described previously.

Fig. 4

DBH values of Eucalyptus stands.

The volume values of trees estimated by the allometric equation [Eq. (1)] using the inventory data of 2004 and 2006 are shown in Fig. 5. It can be observed that the latest inventory showed higher values than those of the previous one, as well as a higher standard deviation.

Fig. 5

Volume values of Eucalyptus stands.

5.2.

Biophysical Models Based on Interferometric Synthetic Aperture Radar 2004 Data Acquisition

Based on the field data of the first campaign (2004), the regression models for total height and DBH, using criteria $r^2$, $r^{2}a$, Cp and Stepwise, only two variables were selected (LogHint and ${\mathrm{CohP}}_{\mathrm{vv}}$), while for the vegetation volume model, only one variable was selected (LogHint) (Table 1).

Table 1

Regression models for bitemporal INSAR acquisition.

The DBH model validation by PRESS criterion showed a value which amounted to 25.09, 45.19% being higher than the SSE (17.28), therefore, the MSE value could be used, with a final prediction error of 0.95 cm. This prediction error corresponded to 7.56% compared with the DBH average trees (12.62 cm) and 3.73% compared with the DBH maximum of 25.60 cm.

The total height model validation by PRESS, whose value amounted to 26.78, 49.78% higher than the SSE (17.88), allowed using MSE value to obtain the final prediction error of 0.995 m. Comparing the prediction error with the average tree height (18.75 m) this value corresponded to 5.31%, and if compared with the maximum tree height (23.31 m) this value corresponded to 4.28%.

The forest volume model validation by PRESS criterion (26840.1) was 19.12% higher than the SSE (22532.7), so MSE value was used for obtaining the prediction error of $33.56{\mathrm{m}}^3$. The prediction error compared to the average tree volume ($186.33{\mathrm{m}}^3$) corresponded to 18.01%, while for the maximum volume of $318.13{\mathrm{m}}^3$, it corresponded to 10.55%. Table 1 shows the obtained regression models and their determination coefficient.

Cook’s distance test verified the existence of two outlier cases for total height model and one case for DBH model, while for volume model, four cases were identified. The residues of the regression models presented a tendency to normality by the tests used, having a homoscedasticity behavior according to Levene test. The prediction errors of these models are shown in Table 2. They were estimated based on MSE values obtained for each regression model.

Table 2.

Regression models results.

Comparing the DBH regression behavior against forest inventory measures (Fig. 6), there is, on average, a good agreement with the inventoried data, not exceeding the values of the standard deviation of the field inventory, as well as a 3.73% of prediction error. Plot 4 was detected as an outlier (Table 2), and not used in the regression model.

Fig. 6

DBH model and inventory data.

The Hint showed a high variance, with very different values from the total inventory average height value exceeding in many cases, the standard deviation of the measurements, which indicates that the variable Hint alone does not satisfactorily represent the total height of the studied vegetation, as shown in the graph of Fig. 7.

Fig. 7

Total height estimated and inventory data.

Comparing the total height estimation of each plot with its corresponding inventoried data, it can be noticed that the values were similar, not exceeding the standard deviation values with a 4.28% prediction error (Fig. 8). The two outliers (plots 4 and 8) detected during the regression (Table 2), were not used in the final model.

Fig. 8

Volume estimated and inventory data.

The volume regression model which uses the LogHint matched the inventory data in some plots; however, in 8 cases their values were different from the inventoried data, exceeding the standard deviation values. Figure 8 graph shows this behavior.

The test site is composed of two plateaus with a flat relief surrounded by a clear pasture, and although plot 8 has a flat relief, its altitude is lower than that of the others, getting flooded part of the year which harms the seedling development phase and damages the development of the trees. Due to this fact, a greater variation of height and canopy size occurred, disturbing the discrimination of ground clearance by DEM in X band. It was also observed that in plot 4, despite some flaws, the height of the trees remained more uniform than that of portion 8.

The standard deviation range of the volume model with a 30 and $40{\mathrm{m}}^3/\mathrm{ha}$ range; corresponded to an estimation error in the range of 16.1% to 21.47% when compared with the average of the tree volume ($186.33{\mathrm{m}}^3/\mathrm{ha}$). However, when comparing this standard deviation range with the volume of the biggest trees ($318.13{\mathrm{m}}^3/\mathrm{ha}$), the error corresponded to a range of 9.43% to 12.57%.

5.3.

Models Performance with Interferometric Synthetic Aperture Radar 2006 Data

Using the regression model generated in 2004, the DBH estimation for 2006 showed agreement with the inventoried data in relation to the mean values for a great number of cases except 4, which showed an overestimation that surpassed the inventory standard deviation. The graph in Fig. 9 shows the obtained results.

Fig. 9

Mean values of DBH estimation and inventory data.

Similar to the DBH estimation, the tree height estimation showed four cases with an overestimation that surpassed the inventory standard deviation. The remaining 19 cases were similar to the inventory or were inside the inventory standard deviation values. The graph in Fig. 10 shows the obtained results.

Fig. 10

Mean values of tree height estimation and the inventory data.

Analyzing the behavior of the Eucalyptus mean height of each plot with its corresponding DBH, a linear relationship between these biophysical variables was observed without significant outliers, whose linear regression showed an $r^2$ of 0.987 for the 2004 inventory and 0.997 for the 2006 inventory. A progression of height and DBH responses can also be observed, indicating that the vegetative growth at the study area elapsed 2 years. The estimation values of the mean tree heights and the mean DBH, also showed a linear behavior (Fig. 11), like the inventory data, with a continuity of the tendency of the regressions; the 2004 $r^2$ estimation was 0.991 while it was 0.995 for 2006.

Fig. 11

Mean values of tree height and DBH of Eucalyptus stands.

On the other hand, when comparing the inventoried and estimated volume for these two dates, the result was linear with little dispersion, i.e., 0.948 of $r^2$, while 0.968 for inventoried volume.

Analyzing the tree volume model response, it can be noticed that inventoried and estimated values (Fig. 12) had a linear behavior with a high $r^2$ value, but an error estimation of 10.55%, acceptable for forest surveying, which considers 10% as a typical inventory error.

Fig. 12

Tree volume inventoried and estimated.

Based on the obtained regression models, a hypsometric image for the total height, DBH and volume was generated using the ${\mathrm{CohP}}_{\mathrm{vv}}$ and LogHint images, achieving a final image whose pixel values corresponded to numerical values of the regression. For the visualization of a combination of results of the regression model image data and the radiometric response in the X band was performed using the IHS technique ($I$ = intensity, $H$ = hue, $S$ = saturation), which, for all models, was defined: 50% for saturation ($S$) and the X-band amplitude image as the intensity ($I$).

The result of the IHS image referring to total height estimation using the SAR from 2004 is shown in Fig. 13(a), along with a range of colors that corresponds to the total height estimated. Color gradients with predominance on the heights of $\sim 14$ and $\sim 20\mathrm{m}$ are shown. The regions of image in blue color correspond to zero point, resulting from settlement failures.

Fig. 13

IHS ($I$ = X-band, $H$ = total height estimation, $S$ = 50%): (a) 2004 estimation and (b) 2006 estimation.

The hypsometric image of Fig. 13(b) corresponds to the total height estimated using the regression model generated in 2004 with the 2006 SAR data. Analyzing the linear regression of the 2006 inventory data and the total height model results, MSE value of 1.39 m was obtained, which represented 1.18 m, or 4.84% of prediction error, when compared with higher values of tree heights.

The hypsometric image of Fig. 14(a) was generated using the DBH regression model, and the corresponding SAR data from 2004, obtaining an image whose pixel values correspond to numerical values of the regression, with a spatial resolution of 1.9 m. It can be noticed from Fig. 14(a) that part of the test site has predominance in the DHP values $\sim 10$ to $\sim 15\mathrm{cm}$. The regions of the image in blue correspond to DHP null values, resulting from the failure of the stand.

Fig. 14

IHS ($I$ = X-band, $H$ = DBH estimation, $S$ = 50%): (a) 2004 estimation and (b) 2006 estimation.

The hypsometric image of Fig. 14(b) corresponds to the DBH estimated for 2006, using the regression model generated in 2004, and the SAR data from 2006. Using linear regression with the 2006 inventory data and the DBH model results, MSE of 1.63 cm was obtained, which represented 1.28 cm, or 6.62% of prediction error, when compared with higher values of DBH.

Using the regression model generated in 2004, it was possible to yield an IHS image of the forestry volume, whose results were shown in the hypsometric image of Fig. 15(a). For the obtained color image gradients, it can be noticed that the volume values ranged between 0 and $\sim 400{\mathrm{m}}^3/\mathrm{ha}$. The stand failures are represented in blue color, corresponding to low volume values.

Fig. 15

IHS ($I$ = X-band, $H$ = volume estimation, $S$ = 50%): (a) 2004 estimation and (b) 2006 estimation.

The hypsometric image of Fig. 15(b) is an IHS image that corresponded to the volume estimated for 2006, using the regression model generated in 2004, and the SAR data from 2006. The linear regression of the 2006 inventory data and the volume model results, presented a MSE value of $2448.01{\mathrm{m}}^3/\mathrm{ha}$, which represented $49.48{\mathrm{m}}^3/\mathrm{ha}$, or 11.96% of prediction error if compared with higher values of tree volume.

Although the planting occurred in the same period, the average total height of each transect presented a wide range of values after 4 and 6 years of development, since it is a seminal reforestation and the provision of water and nutrients varies in the area, which hinders the Eucalyptus development. Additionally the occurrence of gaps between 2% and 60% inside the plots was verified. So, this characteristic induced intermediate height measurements rather than ground height, disturbing the Hint measurement (X-band DEM minus P-band DEM) that would infer the tree heights, as illustrated in Fig. 16(a) and observed in Fig. 16(b).

Fig. 16

(a) Gaps and ground truth (red and blue lines: X-band DEM, black line: P-band DEM) and (b) Hint (X-band DEM minus P-band DEM) of the plot showing the gaps effect.

As in the regions with trees the Coh in the P-band is high, and in clear areas the P-band coherence is low, the presence of this variable in the total height and DBH models compensated the gap effects, improving the model performance as shown in graphs of Figs. 6 and 7.

When these models were used with the second acquisition data from airborne radar (2006), the estimation errors were a little higher if compared with those of 2004 used as a reference basis. One of the reasons for the slight decrease in performance can be explained by significant differences between Eucalyptus stands of different ages, which result in variations of the morphometric parameters canopy. With the growth of the individual trees tend to have a faster development of the trunk, compared to the development of the crown; i.e., the coverage ratio, salience index and living space index decrease with the growth of trees and the increasing size inside the forest stand. Thus, due to these variations of the percentage of the gaps, consequently the CohPvv and/or Hint contribution did not fit in some cases.

Nevertheless, the errors for DBH and total height models would still be suitable for forest surveying, which considers 10% as a typical inventory error. For volume prediction the error was 1.96% beyond this limit.