Leaf Spectra and Weight of Species in Canopy, Subcanopy, and Understory Layers in a Venezuelan Andean Cloud Forest

We characterize the leaf spectra of tree species of an Andean cloud forest in Venezuela, grouped according to position in canopy, subcanopy and understory. We measured leaf reflectance and transmittance spectra in the 400–750 nm range using a high-resolution spectrometer. Both signals were subtracted from unity to calculate the absorbance signal. Nine spectral variables were calculated for each signal, three based on wide-bands and six based on features. We measured leaf mass per unit area of all species, and calculated efficiency of absorbance, as ratio of absorbance in photosynthetic range over leaf mass. Differences among groups were significant for several absorbance and transmittance variables, leaf mass, and efficiency of absorbance. The clearest differences are between canopy and understory species. There is strong correlation for at least one pair of band variables for each signal, and each band variable is strongly correlated with at least one feature variable for most signals. High canonical correlations are obtained between pairs of the three canonical axes for bands and the first three canonical axes for features. Absorbance variables produce species clusters having the closest correspondence to the species groups. Linear discriminant analysis shows that species groups can be sorted by all signals, particularly absorbance.


Introduction
Differences between understory and canopy environmental conditions are fundamental for forest dynamics. Among the many variables that distinguish both conditions, some are very important for physiological processes. In particular, light characteristics constitute a determinant factor in many biological processes [1][2][3][4][5][6][7].
In particular, one leaf characteristic known to vary with canopy strata is speci�c leaf mass or leaf mass per unit area (LMA). In earlier studies, leaves of species in sunny versus shady environments at different locations showed no signi�cant differences in absorbance, while showing differences in LMA [10]. Later results suggested differences in leaf spectral properties, as well as LMA, among species found in different positions of a vertical gradient (canopy, midcanopy, and understory) in a tropical forest [15]. Recent studies indicate that LMA variation is related to tree height rather than light conditions [22], that seasonal variations in leaf spectral properties are very high [23], and that physiological and morphological plasticity are essential for growth and reproduction in contrasting light environments [24].
Leaf re�ectance spectra characteristics are key to understand the potential to distinguish species in remote sensing imagery [25][26][27][28] and to provide a tool for species identi�cation [29]. All these efforts indicate that there are detectable differences in leaf spectral characteristics across species, and the environmental conditions in which they are immersed. In cloud forests of tropical mountains, the elevated values of cloudiness and complex relief impose additional restrictions to the quantity and quality of solar radiation received in different vertical strata [2,30,31]. In this paper, we evaluate leaf spectral characteristics and LMA of twenty species, located in the canopy (15-25 m height), subcanopy (5-10 m), and understory (0-5 m) of an Andean cloud forest. We study the relationship of LMA and leaf spectra, and more importantly, we look for those leaf spectral characteristics that can best discriminate among the three groups of species.
We conducted this study in order to contribute data on leaf spectral properties of tropical cloud forests that could help support remote sensing species identi�cation procedures, and to understand relationships between ecological processes and tree canopy position.

Study Area and Species
. e study site is located at 2400 m elevation asl, 8 ∘ 37 � N and 71 ∘ 10 � W, in the Monterrey area, Valle Grande, near the city of Mérida, in the Andes of Venezuela. Mean annual temperature is 13 ∘ C and annual rainfall is 2560 mm with one peak in March-May and another in August-November. Distributed in several vertical strata, the vegetation is typical of upper montane Andean cloud forest and forms a relatively open canopy of 20-25 m of height [32].
We selected 20 species common in the Andean cloud forest (Table 1); of the nine canopy species, eight are trees, and the other, Lycianthes ferruginea, is a vine with leaves and reproductive parts always present in this upper strata; Sapium stylare, one of the three subcanopy species, was an individual of a canopy species that was still growing to reach this stratum, and therefore was considered to be in partial shade.

Experimental Setup.
For each species �ve mature leaves in good condition were collected and analyzed (09 January 2001). A few minutes elapsed between collection and measurement, well within the times for leaf spectra to remain unaltered by clipping; also, measurements focused on the visible range where spectra are unaltered by clipping [33,34]. e leaves of canopy and subcanopy species were taken from a single individual, while the leaves of understory species were taken from �ve different individuals (because understory plants have fewer leaves).
In a dark room, we installed a black box with an ori�ce on one side where we placed a leaf to measure re�ection and transmission of light coming from a lamp (ELH, 120 V, 300 W) located 178 cm from the leaf. e spectrum of the lamp in the range of interest (350-800 nm) was compared to the spectrum of solar radiation at the site. Even though the spectra were not exactly the same, the differences are compensated by taking the ratio of re�ected and transmitted spectra to the spectrum of the lamp.
Two optical �bers (diameter 0.2 mm) conducted re�ected and transmitted light from the leaf to a spectrometer. e �ber optic end to measure re�ectance was placed at 2 cm from the leaf upper surface and the one for transmittance was placed at 2 cm of the lower surface of the leaf inside the black box. e �bers were connected to a portable �ber optic spectrometer (Ocean Optics SD-2000), which measures the spectrum between the ultraviolet (UV) and the near infrared (IR), in a range of 200-850 nm, using an array of 2048 diodes with an aperture slit of 100 m. e dispersion is (850 − 200)/2048 = 0.32 nm/diode, with a resolution of 12 × 0.32 = 3.8 nm (FWHM). For this study the signals were analyzed in a narrower range (400-750 nm) of interest in photosynthesis and plant responses to far red light. Data were taken simultaneously on two channels with identical speci�cations, using the two �bers; one to capture re�ected light, and the other to measure transmitted light. In both cases, the �ber�s end was bare which has a 25 ∘ �eld of view. e other end of each �ber was connected to an optical switch that allows blocking light to obtain dark response, which is subtracted from the signals in order to correct for electronic noise of the instrument.
In addition to leaf spectral characteristics, we determined LMA for �ve leaves of each species. For this purpose, we measured dry weight of the leaf blade and divided by the leaf area, which was measured using an LI-3100 Area Meter. Here we report LMA in mg/cm 2 .

Spectral Variables.
All spectral signals, reference (lamp) as well as re�ected and transmitted, were smoothed by a seven-point central moving average [31]. Re�ected and transmitted signals for the �ve leaves of each species were divided into the reference signal to obtain ratios of re�ectance and transmittance as functions of wavelength . Both ratios were also smoothed by a seven-point central moving average to obtain re�ectance and transmittance signals. Absorbance for each leaf was then calculated as As an example, , , and signals for one leaf of Guettarda steyermarkii (GS) are shown in Figure 1.
We de�ned a set of nine spectral variables for each signal ( , , and ) as summarized in Table 2. We calculated the variables for each leaf and then averaged across leaves of the same species. ree of the nine variables correspond to averages over wide wavelength bands and the other six are features, that is, averages over a narrow band ( Figure 1).
All variables are based on normalized differences and ratios reported in the literature; some of the variables have been proposed in reference to absorbance [10,15,20,35], while others have been proposed for re�ectance [19,36,37]. Instead of normalized difference we used a simple ratio because it yielded lower values of coefficient of variation across leaves. For brevity of presentation, the variables will be de�ned using the absorbance signal only. However, the calculations also apply to the and signals except that we used the inverse of the ratio in order to obtain values less than 1 ( Table 2).
For variables based on bands we will denote by 1 − 2 the mean in the band 1 − 2 calculated as the integral of absorbance in this range divided by the bandwidth where the integral is approximated by the sum of absorbance values multiplied by the interval between successive wavelengths. Here denotes the diode number for wavelength .
In all calculations, we use 705 nm, the edge of chlorophyll absorption, instead of 700 nm [36][37][38][39]. en, variable FRb, contribution of the spectrum in the far red (705-750 nm) relative to the one in the full photosynthetic range (400-705 nm), was calculated as the ratio of the integral in the �rst range (705-750 nm) over the integral in the second range (400-705 nm): e third band variable is the ratio of the average spectrum in the green band relative to the red band: When de�ning the next six variables, which are based on narrow bands or features, we will denote as the absorbance at a particular wavelength calculated as the average over a narrow band (4 nm) around this wavelength; that is to say which was approximated by because the wavelength differences were homogeneous over a small 4 nm interval. Here corresponds to the number of points in the 4 nm interval.
For example, absorbance 620 at 620 nm, wavelength at which maximum relative quantum efficiency occurs, is calculated using (7) in the interval from 618 to 622 nm. Our �rst feature variable MXf is simply 620 : Next, we de�ne PDf, the magnitude of the depression in absorbance at 550 nm which is observed in all spectra ( Figure  1). It was calculated as the ratio of average absorbance at 550 nm to the one at 660 nm: where 550 and 660 are each calculated using (7). Next, we use a concept similar to the Anthocyanin Re�ectance Index which is a difference of the inverse of re�ectance at 550 and 705 nm [19]. Instead of difference, we use a ratio of the value at 705 nm to the one at 550 nm. We will call it Anthocyanin Index (AIf): For the next variable we use the concept of Photochemical Re�ectance Index (PRI) based on a normalized index of re�ectance at 570 and 531 nm; the �rst wavelength is a reference and the second corresponds with the xanthophyll pigment which in many plants relates with light use efficiency [36,[39][40][41]. Again instead of normalized difference we use a simple ratio and de�ne it as photochemical index (PIf) Based on the modi�ed normalized difference vegetation index that uses the re�ectance at 750 and 705 nm [36], and the simple ratio (SR) of re�ectance at these wavelengths [42], we use a ratio of absorbance values at 705 and 750 nm to de�ne a far-red Index (FRf): Scienti�ca 5 In order to relate to the �uorescence peaks at 685 and 738 nm [36,43,44] we selected, as the next variable, the ratio of absorbance at 738 nm to the one at 570 nm (same reference used for PIf) as an absorbance index at the �uorescence peak (FIf): e 738 nm feature was visually appreciable in the signals for canopy species but less so for subcanopy and understory. A variable related to the peak at 685 nm was explored but not used because the signals exhibited small differences among species and groups at this wavelength. An additional composite variable, efficiency of absorbance per unit mass (EAM), combines a spectral variable PHb with LMA and was calculated as the ratio of 400-705 to the leaf weight LMA: Its units are the inverse of LMA units; that is, are given here in cm 2 /mg. We used this ratio because historically vertical variations in LMA have been associated with light conditions [10,15]. However, recent evidence indicates that vertical changes in LMA are more related to tree height [22].

Statistical Analysis.
Spectral variables were calculated using absorbance, re�ectance, and transmittance for each leaf. en, for each species the average, standard deviation, and coefficient of variation of all variables were calculated across the �ve leaves. We also calculated averages of leaf means and coefficient of variation for the three species groups, that is, canopy (C), subcanopy, (S) and understory (U). In addition, we calculated the leaf means and coefficient of variation of LMA and the leaf means of EAM. e leaf means by species were used to conduct statistical tests and multivariate analysis among the species and among the groups. All tests and analyses were conducted separately for each signal absorbance, re�ectance, and transmittance.
First, for each spectral variable we used a nonparametric analysis of variance (ANOVA, Kruskal-Wallis test) to detect differences among all three groups and a Wilcoxon test to compare each group pair. A multivariate analysis of variance (MANOVA, Wilks test) was used to determine if there were differences among the groups based on the full set of spectral variables. In addition, we conducted the same univariate tests (Kruskal-Wallis and Wilcoxon) for LMA and EAM. en, we determined whether LMA can be predicted from the set of spectral variables by stepwise multiple linear regression.
Second, we conducted multivariate analysis procedures to examine the relationships among spectral variables, among species, and among groups of species. We started with the correlation matrix to study relationships between pairs of variables and conducted principal component analysis (PCA) to examine how many potential combinations of variables could account for most of the variance among species. en, we conducted canonical correlation (CANCOR) analysis between the set of band variables and the set of feature variables to determine how well these sets explain each other. Next, we conducted hierarchical clustering using the Minkowski distance and the Ward method to examine relationships among species and compare clusters formed with the prede�ned groups. Finally, for the purpose of developing a linear combination of spectral variables that maximize differences among species groups we conducted a multiple linear discriminant analysis (LDA) using the spectral variables. All variables were standardized prior to the multivariate procedures described above.
Calculations and statistical analyses were programmed using the system version 2.10.1 [45]. e program used to calculate the discriminant function is part of the MASS package for the system [46].

Results
As exempli�ed in Figure 1, absorbance, re�ectance and transmittance signals show expected patterns for all species. Notably, for re�ectance and transmittance we see low values from 400 to 500 nm, a peak at ∼550 nm, an elbow at ∼690 nm, and a sharp increase in the 700-750 nm range (Figure 1). Such patterns produce a typical absorbance spectrum with high values from the beginning of the photosynthetic range, a valley at 550 nm, a recovery to high values at ∼690 nm, and an abrupt drop to 750 nm (Figure 1).
Absorbance values at the 550 nm depression, at the recovery past 660 nm, and the far red decline from 705 to 750 nm showed variations among canopy species and understory species (Figures 2 and 3), but less so among subcanopy species (Figure 2). e most contrasting canopy, species are CM and MQ with little and strong reduction at 550 nm respectively ( Figure 2). In the understory group, CP and PU showed the most important reduction at 550 nm, while FV, PD, and AN, showed the least important reduction at 550 nm ( Figure 3). Similar patterns are noted for transmittance; however, re�ectance spectra are very similar among species and groups (Figures 2 and 3).
Spectra averaged by group showed differences in absorbance and transmittance among groups, but very small differences in re�ectance ( Figure 4). Particularly, the 550 nm feature and the far red slope show clear differences among groups of species for absorbance and transmittance, and small differences for re�ectance. In addition, notable at 550 nm the absorbance is less for understory species, followed by canopy species and then slightly larger for subcanopy, even though these last two groups have similar values (Figure 4). At 660 nm the largest absorbance corresponds to subcanopy, being understory and canopy lower and similar to each other (Figure 4). Between 570 and 620 nm the absorbance shows the clearest difference among the groups, being larger for subcanopy, followed by canopy, and then by understory. From 700 to 750 nm the differences increase with wavelength and show a gradient from understory, to subcanopy, and canopy. In addition, two small valleys, one at 738 nm and another in between 738 and 750 nm are noticeable in all groups. e patterns just described for absorbance are also apparent in the transmittance signal but of course inverted in sign. e double valley (peaks in this case) in 738-750 nm is more accentuated (Figure 4). Group average and standard deviation of the coefficients of variation (CV) across leaves are given in Table 3. ere is relatively low variability (∼1-2%) across leaves for all absorbance variables except for far red features (FRf, FIf) which have high CV values (∼15-30%), and the far red band FRb which has intermediate variability (∼8-9%). ere is high variability (∼15-45%) across leaves for re�ectance and transmittance except for PIf which is in the range ∼2-3% and GRb which is ∼5-8%.
Leaf means of all spectral variables suggest patterns in the group differences and within-group variability. For absorbance ( Figure 5), potentially signi�cant differences among groups are not evident except for FRb, AIf, FRf, and FIf. Differences in group averages of all variables are very small for re�ectance� moreover, the variability within groups is large for all variables (Figure 6). Potentially signi�cant   differences in group averages (and with lower within-group variability) are more evident in a few transmittance variables; PHb, FRb, and MXf ( Figure 7). However, there is frequent occurrence of extreme values for canopy species in most variables. ere are no signi�cant ( ) differences among all three groups or pairs of groups in re�ectance (Table 4) except between canopy and understory species for AIf. Differences among all three groups (Table 4) were found to be signi�cant in absorbance for FRb, AIf, FRf, and FIf, and in transmittance for PHb, MXf, and PDf. is suggests the importance of far red absorbance and the photosyntheticrange transmittance in separating groups. Subcanopy species had signi�cant differences with understory species only. �ven though these two groups exhibit signi�cant differences only in MXf absorbance, they have signi�cant differences in transmittance forPHb, FRb, MXf, and FIf. e clearest differences are between canopy and understory species. ese groups show signi�cant differences in absorbance for FRb, PIf, FRf, and FIf (all related to far red except for PIf), in re�ectance for AIf, and in transmittance for PDf ( Table 4). �one of the signals are signi�cantly different  Weight and the composite variable (LMA and EAM) exhibit differences among groups (Table 5). Group average of LMA is highest for canopy, followed by understory and lowest for subcanopy. Variability across leaves is relatively high, with CV being highest for canopy, followed by subcanopy, and being lowest for understory. Signi�cant differences are found among all groups (Kruskal-Wallis -value 0.023 and 0.01 for LMA and EAM respectively) and between canopy and understory for both LMA ( 0.021) and EAM, ( 0.008), and between canopy and subcanopy for EAM ( 0.036). Stepwise regression analysis of LMA as a function of spectral variables generated the best predictor based on all variables (except PHb) for absorbance, on four variables (MXf, AIf, PIf, and FIf) for re�ectance, and on only two variables (FRb, and PDf) for transmittance. e adjusted 2 for these regressions are 0.64, 0.52, and 0.90 for absorbance, re�ectance and transmittance, respectively. e -values indicate a very signi�cant trend in all cases, and particularly for transmittance (0.007, 0.004, and 1.7 × 10 −9 respectively). e -values for the coefficients of the regression for transmittance are very signi�cant ( 4 × 10 −9 ). However, all coefficients for absorbance have -values exceeding 5%, and those for re�ectance are in the 1�5% range. CM, having a LMA much larger than any other species generates a very large 5% con�dence interval at high values of LMA.
Correlation analysis (Table 6) shows strong correlation coefficients (>0.85) for at least one pair of band variables for each signal: GRb-FRb in absorbance, PHb-FRb in both re�ectance and transmittance. Also, one feature variable FIf is correlated with other feature variables: with FRf for absorbance, with MXf and FRf in re�ectance, and with MXf, FRf, and AIf in transmittance. Each band variable is strongly correlated with at least one feature variable for all signals, except GRb in transmittance. e pair PHb-MXf has strong correlation for all signals, whereas PIf show low correlation values with other variables for all signals.
ree principal components explain more than 95% of the variance for absorbance and transmittance, and nearly 90% for re�ectance (Table 7, PCA). Very high canonical correlations are obtained between pairs of the three canonical axes for bands and the �rst three canonical axes for features ( Table 7, CANCOR). e �rst pair of axes have correlation values larger than 0.995, the second pair of axes has values larger than 0.95, and the third pair has values larger than 0.89. e correlations are always highest for absorbance ( Table 7, CANCOR). e square of the SVD (singular value decomposition) terms obtained by the LDA and their ratios show much higher discrimination power for the �rst axis (LD1) compared to the second (LD2) for absorbance ( 3.5x) Scienti�ca 9 T 5: LMA by species, group average for LMA and EAM, and coefficient of variation (CV, in %) across leaves for LMA.    and re�ectance (∼8x), but nearly the same for transmittance (∼1.1x) ( Table 7, LDA). Dendrograms from the cluster analysis were cut at a distance (height) generating three clusters. Of all signals, absorbance produces species clusters having the closest correspondence to the pre-de�ned species groups (Figure 8). Cluster 1 (lemost box) consists mostly of canopy species and two of the subcanopy species (SS and SG); whereas clusters 2 and 3 (middle and rightmost boxes) are mostly understory species with a couple of canopy species (LY and MQ) and the other subcanopy species (PB). e only understory species included in cluster 1 is FV, which joins SS at low height. However, two canopy species are included in clusters 2 and 3; LY in cluster 2, and MQ in cluster 3. Both join these clusters at higher nodes and thus are relatively dissimilar to all the understory species.
LDA results are similar for all three signals. As an example Figure 9 shows the results for absorbance. We can clearly appreciate differences among groups. e �rst axis (LD1) discriminates between canopy and understory species. Subcanopy species are located at intermediate positions of this axis, but separated from the other two groups by the second axis (LD2). All canopy species are in the negative part of LD1 whereas understory species have positive values. IN is at the extreme of canopy species, whereas PD and MM are at the extreme of understory species (Figure 9). For the sake of space we do not include LDA coefficients but we observed that with relatively higher values GRb, PDf, FRf, and FIf contribute to LD1, while FRb, GRb, PDf, and FIf contribute the most to LD2.

Discussion
Absorbance, re�ectance, and transmittance spectra display patterns similar to the ones reported for other tropical forests [10,15,20]. Absorbance spectra showed variations among species in the 550 nm depression and the far red decline (Figures 2 and 3). ese differences are noticeable for canopy and understory species, but less so for subcanopy species. Absorbance in the 500-705 nm range is less for understory species, followed by canopy species and subcanopy ( Figures  2 and 3). is is evident at 550 nm con�rming previous results in rain forests [15]. From 705 nm the largest absorbance corresponds to canopy species, which is a different result for three of the four species measured at La Selva, Costa Rica [15]. Canopy species presented higher values of LMA, compared to subcanopy and understory species. is result con�rms vertical differences of LMA observed in all forests [22]. Our LMA values are in the same range as those reported for La Selva [15] but lower than those reported for a cloud forest in Puerto Rico [20]. is �nding may suggest less severe light or water restrictions in our site when compared to the cloud forest in Puerto Rico. ere is substantial variability in LMA as indicated by relatively large values ∼ 10-17% of CV (Table 5). LMA differences are signi�cant only when comparing canopy and understory species; however there are signi�cant differences in �AM between canopy and subcanopy as well.
Several spectral variables show differences among the groups of species, mostly those variables related to far red and weight ( differences between subcanopy and understory, and PIf which shows differences between canopy and understory ( Table 4). In general, the difference between subcanopy and canopy groups and between subcanopy and understory are less appreciable than the differences between canopy and understory species (Table 4). For absorbance, variables related to far red, FRb, FRf, and FIf all display a gradient from high to low for canopy, subcanopy and understory leaves ( Figure 5). For FRf and FIf this is due to a lesser absorbance decline from 705 to 750 nm for subcanopy and canopy when compared to understory ( Figure 4). For FRb however, the pattern is an indication of increasing absorbance at 550 nm from understory to canopy and subcanopy.
EAM allows differentiation among canopy and subcanopy groups, and canopy and understory, but not between subcanopy and understory. EAM is lower for canopy species, which is due to larger values of LMA, thus con�rming previous results in other forests [10,15]. Historically, this �nding is interpreted in terms of lower plant's cost to produce the leaf mass needed to achieve required absorbance. However, recent evidence indicates that vertical changes in LMA are due to tree height because of different water restrictions at higher canopy levels [22].
It is interesting that only two transmittance variables explain 90% of LMA. e regression coefficients indicate that estimated LMA decreases with FRb and increases with PDf. is �nding would suggest that leaves with higher mass will transmit less light and thus suppress growth in lower forest strata. However, upon further scrutiny of this result we found that this relation may not be robust because of high leverage by CM. We ran the regression analysis again aer removing CM from the data set. e adjusted 2 for the best predictors declined to 0.38, 0.30, and 0.60 for absorbance, re�ectance, and transmittance, respectively. Furthermore, four variables are required by the best predictor based on transmittance and only two (GRb and MXf) had -values < 0.05.
Strong correlation between PHb with MXf, indicates the importance of 620 nm in explaining absorbance over the photosynthetic range. Similarly, the strong correlation between PDf with GRb, indicates that the depression at 550 nm (relative to the mid of the red-band) explains most of the difference between the red and green bands. Other two strong correlations correspond to the variables in the far red, FRb with FIf, and FRf with FIf. ree principal components suffice to account for more than 90% of the variance suggesting co-linearity among many variables. Our approach was to select variables based on ratios of well-known bands and features. Variables could be selected by pattern recognition methods and other features may be found. For example, when separating leaves of trees and lianas in tropical dry forests, based on re�ectance spectra, as many as 10-100 features are selected [47].
Absorbance variables can generate one species cluster related to canopy species and two other clusters related to understory species (Figure 8). ese two clusters join at a slightly higher distance and can be considered as one cluster related mostly to understory species. However, subcanopy species do not form a separate cluster, but mix within the other clusters; SS and SG with the canopy cluster and PB with the understory clusters. is �nding con�rms that species differences are much more marked between canopy and understory, and that subcanopy tends to be similar to canopy. It should be noted that we treated SS as a subcanopy species because leaves were taken from an individual in midcanopy, but normally trees of this species reach the canopy.
As shown by the LDA, the �rst axis separates species along a gradient from canopy to understory (le to right in Figure 9). Satisfyingly, subcanopy species are intermediate along this axis. Of the three subcanopy species, SS has positive values and is closest with understory species, particularly to FV (also suggested by the cluster analysis). e other two, SG and PB, have negative values and are closest to the canopy species, especially to GS and LY (but recall that PB came closest to understory in the cluster analysis).
Only one LDA axis would suffice except for slight differences of subcanopy species with extremes of the other two groups. Further separation of subcanopy species with respect to the other two groups is provided by the second LDA axis. However, PB remains close to other groups along this axis as well. In this case, M1 (canopy group) and AN (understory group) have a position approaching the subcanopy group. ere is relatively good separation among species themselves along the two axes; although we did not address differentiation at the species level, our results indicate that it may be feasible.