3.1. Error Analysis
From
Section 2, the influence of discrete VZAs on
Le measured with the PCA for continuous canopies is closely related to the characteristics of the
value, which vary with VZAs or LADs. The errors of
Le with the different degrees of discrete VZAs (such as the different numbers of PCA rings, i.e., four and three) are presented.
Comparisons of
among canopies with the different leaf IAs using all of the five PCA rings are shown in
Figure 5.
between
and
is closely related to the leaf IA. For horizontal and planophile leaf IA (such as leaf IA < 30° for the conical LAD (
Figure 5a) or
x > 2 for the ellipsoidal LAD (
Figure 5b)),
calculated by the discrete form of Miller’s equation is significantly larger than that calculated by Miller’s equation.
is about 10% for the planophile leaf IA, meaning that
is overestimated for the planophile leaf IAs although all five PCA rings are used. When the leaf IA is about 50° for the conical LAD or
x is about 1 (spherical LAD) for the ellipsoidal LAD,
can be ignored (lower than ±4%), meaning that the PCA has a good performance for these leaf IAs. With the increment of the leaf IA,
is negative and
is underestimated. The maximum can be up to −10%. In summary,
increases as the leaf IAs deviate from the spherical LAD.
For further analysis, the whole hemispheric space (0−90°) was divided into five sectors corresponding to the right borders of the five rings of the PCA: 0−13°, 14−28°, 29−43°, 44−58°, and 59−90°, respectively. Thus, the five sectors corresponded to the five PCA rings. We separated the results of the definite integrals in Equation (3) according to the five sectors. Here, we noted the separated result in each sector as a separated
. Thus, each sector has a separated
(noted as the separated
), and the sum of separated
is
in Equation (3). Similarly, we calculate the separated
(noted as the separated
) in Equations (4) and (16) according to the PCA rings, and the sum of separated
is
in Equations (3) and (16). Taking LAI = 3 as an example, the separated
in each sector and
in each PCA ring are compared to find the specific error (
Figure 6).
The separated
(dashed lines in
Figure 6; prefix “Dis” in the legend) in the non-last (i.e., the first four) rings are almost close to the separated
, indicating that there is no obvious error in the non-last rings, although the VZA ranges between the sectors and PCA rings are different. The separated
in the non-last PCA rings are slightly overestimated. This is because the weight coefficients (
wj in Equation (4)) of the first four PCA rings are slightly larger than the corresponding continuous item of (
). For example, the first
wj (0.034) in Equation (4) is slightly larger than the corresponding item (
, approximately 0.032) in Equation (3). This means that the errors produced by the discrete VZAs are slight in the non-last PCA rings.
The main error exists in the last PCA ring. For canopies with a spherical leaf IA, the separated in the last PCA ring is slightly underestimated. Then, the difference between in the last PCA ring and in the last sector can be offset by the first four sectors or PCA rings, thus, the total (sum of all separated ) is nearly equal to the total (sum of all separated ). However, for the horizontal and planophile leaf IAs, the separated in the last PCA ring is obviously overestimated compared with the separated ; thus, the total is significantly overestimated. For the vertical and erectophile canopies, the separated in the last PCA ring is obviously underestimated; thus, the total is underestimated. Considering that the separated in the non-last PCA rings are slightly overestimated, the magnitude of errors for these types of leaf IA are smaller than those for the horizontal and planophile leaf IAs.
- b.
Using four and three PCA rings
Fewer PCA rings inevitably leads to a higher degree of discretization of VZAs in the PCA, which must affect
measured with the PCA. Here, all of the canopy parameters (except for the number of PCA rings) and the analysis method of
are the same as those noted in the previous section. Comparison of
in
measured with the PCA among canopies with the different leaf IAs using four and three PCA rings are shown in
Figure 7. Notably, the scope of the vertical coordinates in
Figure 7 is different from that in
Figure 5, in order to achieve a more beautiful visualization.
The general change trends of the results in
Figure 7 are similar to those in
Figure 5. However, nearly all
values in
Figure 7 increase obviously with the decreasing number of PCA rings (except in the case of canopies in which the leaf IA is about 50° or a spherical LAD). For the horizontal leaf IA,
can reach up to 35% and 60% with four and three PCA rings, respectively. For the erectophile leaf IA, the PCA results can be underestimated by 15% and 27% with four and three PCA rings, respectively. Even for the leaf IA = 40° or 60°,
is also obvious when the number of PCA rings decreases to three: the
can reach up to 23% and −15% for canopies with the leaf IA = 40° and 60°, respectively. The above phenomenon is due to the error of
in the last PCA ring. Taking LAI = 3 as an example, the separated
in the last ring is overestimated by about 66% and 100% for the horizontal leaf IA using four and three PCA rings, respectively (
Figure 8).
In summary, decreasing the number of PCA rings can enhance the degree of discrete VZAs, and enlarge the magnitude of PCA error, especially for the planophile and erectophile leaf IAs.
3.2. Corrections
The above results show that the error produced by the discrete VZAs is closely related to the leaf IA. Thus, the correction for the error must be related to the leaf IA. In practice, leaf IA is often influenced by many environmental factors, such as wind and sunlight. Therefore, it is often a variable value and is not easily measured in situ. Moreover, the accurate measurement of leaf IA is in contradiction with the convenience of the PCA. Plant leaf IA type, which is an estimation of leaf IA, is often stable in a certain period of time. Here, we divided the whole hemispherical space equally into three sectors. Then, the leaf IAs were divided into three types: planophile (VZA: 0–30°), spherical-like (30–60°), and erectophile (60–90°). Linear relationships are used to correct the error produced by the discrete VZAs according to the three leaf IA types. Two corrections were made: (1) correction for only the last PCA ring (the “CL” method) and (2) the direct correction for total measured with the PCA (the “CT” method).
From the previous section, the default or original weight coefficients in the PCA rings are suitable for the spherical LAD, whereas they produce an error and need to be modified for other leaf IA types. The error of
in the last PCA ring is mainly due to the use of incorrect weight coefficients for the last PCA ring. Here, we use the regression coefficients in
Figure 9 to correct the original weight coefficients in the last PCA ring.
The linear relationships of the separated
in the last PCA ring between the discrete (
) and continuous results (
) are shown in
Figure 9 with the three different leaf IA types (
Figure 9a planophile,
Figure 9b spherical-like, and
Figure 9c erectophile) and with the decrement of the number of PCA rings from five to three (
Figure 9(a1,b1,c1) five rings,
Figure 9(a2,b2,c2) four rings, and
Figure 9(a3,b3,c3) three rings). From
Figure 9, the regression coefficients between
in the last PCA ring and
in the last sector are closely related to the leaf IA types: all of the slopes of the regression line are larger than 1 for the planophile type, but lower than 1 for the erectophile type; the regression lines for the spherical-like types are around the 1:1 line. With the decrement of the number of PCA rings, the regression lines are generally distinct from the 1:1 line. With the exception of
Figure 9(b3), all of the coefficients of determination (
R2) are larger than 0.94, indicating that the correction is reasonable and acceptable. The scattering points in
Figure 9 are concentrated around the regression lines (except for
Figure 9(b3)), indicating that the magnitude of
has little effect on the regression coefficients in most situations.
As nearly all of the values for
R2 in
Figure 9 are close to 1, we provide the new weight coefficients for the last PCA ring (
Table 3) according to the regression coefficients in
Figure 9. The only change for the new weight coefficients is the value in the last PCA ring. Generally, the values of the weight coefficients in the PCA last ring increase for the planophile type of leaf IA, but decrease for the erectophile type of leaf IA, according to
Figure 9, and change slightly for the spherical-like type of leaf IA.
In summary, the correction for each specific leaf IA is not completely necessary although the error produced by the discrete VZAs is closely related to the specific leaf IA (
Figure 6 and
Figure 8).
- b.
“CT” method
The relationships of total
between the discrete (
, sum of the separated
in all used PCA rings) and continuous result (
, sum of the separated
in all used sectors) are shown in
Figure 10 with various leaf IA types (
Figure 10a planophile,
Figure 10b spherical-like, and
Figure 10c erectophile) and with the decrement in the number of PCA rings from five to three (
Figure 10(a1–c1) five rings,
Figure 10(a2–c2) four rings, and
Figure 10(a3–c3) three rings).
All of the
R2 of the relationships in
Figure 10 are larger than 0.9, indicating that the relationships between
and
are also close, although we used the leaf IA type instead of the leaf IA. The regression coefficients are related to the leaf IA type and number of PCA rings, whereas the magnitude of
has little effect on the regression coefficients in most situations in
Figure 10. The planophile and erectophile types show a similar trend with the results in
Figure 9:
is underestimated with the PCA for the planophile-type canopies (all relationship slopes are lower than 1) but overestimated for the erectophile-type (all relationship slopes are larger than 1).
is estimated well for the spherical-like type with five and four PCA rings. Nonetheless, with the decrement in the number of PCA rings, the relationship slopes are different from 1 (about 1.14 in
Figure 10(b3)), indicating that the correction for the PCA is also needed; that is, the discrete VZA correction for the PCA is essential in most cases, especially for the planophile- and erectophile-type with four and three PCA rings.
3.3. Experiments
Planophile is a common type of leaf IA in many heliophile vegetables, such as green pepper, tomato, and soybean, when water and sunlight are sufficient. Erectophile is another common type of leaf IA in the early stage of many continuous canopies, such as rice, wheat, and some herbage.
Three crop plots—a green pepper (GP) plot (E 32°20′47″; N 117°15′15″), a corn plot (E 32°20′28″; N 117°15′03″), and a rice plot (E 32°20′53″; N 117°15′02″) —were selected to validate the influence of discrete VZAs on the PCA. Experiments were conducted during July 1–3, 2018. All of the selected plants were in early stage to avoid the influence of fruits on the interception of light in the lens. The sizes of the three plots were 30× 30 m and there were five replicates initially. A view restrictor of 270° of the PCA was used in all plots to avoid the influence of the observer. As the IAs of dead leaves on the plants were distributed irregularly, they were needed to be removed. The LAI was measured by destructive sampling. The leaf IAs were measured using a protractor and a gradienter, and are listed in
Table 4. There were significant differences in leaf IA among the three plots: the leaf IA types of the three crops obviously belonged to planophile, spherical-like, and erectophile, respectively. The GFs were measured using the LAI-2000 PCA in an overcast situation (
Figure 11). Finally,
Le in each plot was calculated in the FV2000 software (Li-Cor, Inc., Lincoln, NE) for the PCA (noted as
Le) using five, four, and three rings (noted as
Le_P5,
Le_P4, and
Le_P3, respectively,
Table 5).
is used to describe the relative error between
Le measured with the PCA and the LAI (
Figure 12).
Le measured with the different numbers of PCA rings in the three plots are listed in
Table 5. As there were no fruits and no obvious stalks in each of the three plots, foliage was the main object intercepting light. The influence of the discrete VZAs on the PCA in the three plots is similar to the results shown in
Figure 5 and
Figure 7.
Le measured with the PCA in the GP plot is larger than the true LAI, meaning that
Le is overestimated in the GP plot; on the contrary, the
Le in the rice plot is lower than the LAI, meaning
Le is overestimated in the plot.
Le measured with the PCA is closer to the true LAI in the corn plot than that in the other two plots (all
of
Le in the corn plot are lower than those in the other two plots). This means that the PCA has higher performance in the corn plot than in the other two plots. Moreover, the gap between the value of
Le and LAI increased with the decrement of the number of PCA rings. For example, the value of
Le increased from 1.29 to 1.61 with the decrement of the number of PCA rings from five to three in the GP plot. Thus, the
between
Le and LAI increased from 12.5% to 40.0% in the GP plot. Although in the corn plot with the spherical-like type of leaf IA,
can reach up to about 15%. The different performance of the PCA in the three plots validates the theoretical deduction presented in
Section 2.2.
Le corrections were made for the three crop plots using the two methods: “CL” and “CT”. (1) We changed the weight coefficients (
wj) of the last PCA ring according to
Table 3. Then,
Le was recalculated according to the new
wj in each plot (
Le is corrected as “
Le _CL” in
Figure 12). (2) We directly recalculated
Le according to the linear regression coefficients in
Figure 10 (
Le is corrected as “
Le _CT” in
Figure 12). After correction, both the
Le _CL and
Le _CT are obviously closer to the LAI than
Le measured with the PCA in all three plots (all
in the three plots are lower than 10%). In particular, with four and three rings,
Le errors produced by the discrete VZAs decrease significantly. This means that both of the two correction methods are effective for decreasing the error produced by the discrete VZAs.
The improvement due to the correction for
Le in the corn plot is not as clear as that in the other two plots (e.g.,
increases using five and four PCA rings after the “CT” correction method), although
of
Le in the corn plot is not obvious (the largest
is lower than 10%) after correction. This is due to the fact that the
Le error produced by the discrete VZAs can be neglected for the leaf IA ≈ 50° or the LAD meeting the spherical distribution (as shown in
Figure 5 and
Figure 7). In addition, the correction performed here is an approximation because corrected for
Le according to the three leaf IA types but not the specific leaf IAs. Therefore, the correction for the PCA with five or four rings is not necessary for the leaf IA ≈ 50° or the LAD meeting the spherical distribution.