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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The need for an efficient and standard technique for optimal spectral sampling of hyperspectral data during the inversion of canopy reflectance models has been the subject of many studies. The objective of this study was to investigate the utility of the discrete wavelet transform (DWT) for extracting useful features from hyperspectral data with which forest LAI can be estimated through inversion of a three dimensional radiative transfer model, the Discrete Anisotropy Radiative Transfer (DART) model. DART, coupled with the leaf optical properties model PROSPECT, was inverted with AVIRIS data using a look-up-table (LUT)-based inversion approach. We used AVIRIS data and ^{2} = 0.77) than the original spectral bands (RMSE = 0.60, R^{2} = 0.47). The results indicate that the discrete wavelet transform can increase the accuracy of LAI estimates by improving the LUT-based inversion of DART (and, potentially, by implication, other terrestrial radiative transfer models) using hyperspectral data. The improvement in accuracy of LAI estimates is potentially due to different properties of wavelet analysis such as multi-scale representation, dimensionality reduction, and noise removal.

Leaf area index (LAI) of vegetation canopies controls and moderates different climatic and ecological functions [

Physically-based models range in complexity from simple nonlinear models to complex numerical radiative transfer models. One-dimensional (1-D) canopy radiative transfer models are among the more commonly used because of their ease of implementation and computational efficiency [

The Discrete Anisotropy Radiative Transfer (DART) model is a widely used 3-D model [

For the estimation of vegetation properties such as LAI, a model has to be inverted against measured reflectance data. Several model inversion techniques are available [

Model inversion requires that for a unique determination of

The discrete wavelet transform (DWT) has been increasingly used in recent years in the processing of hyperspectral images for a variety of purposes [

The advantages of DWT in LUT inversion are threefold. First, it transforms the correlated set of spectral features into wavelet features at different scales; the information carried at different scales is less correlated because of the orthogonal nature of transformation [

The main objective of this study was to assess the utility of the Haar discrete wavelet transform as a feature extraction technique for accurately estimating forests LAI through LUT inversion of DART using airborne hyperspectral data (AVIRIS). We posited that the transformation of hyperspectral reflectance into wavelet features and subsequent LUT inversion in the wavelet domain might provide improved estimates of LAI. We also examined three different sets of wavelet features and their effect on inversion accuracy.

A wavelet transform enables signal (data) analysis at different scales or resolutions by creating a series of shifted and scaled versions of the mother wavelet function [_{α,b}

We refer to the running difference, or fluctuation, as the detail vector,

The subsignals of the original signal define the first level of the Haar transform, usually referred to as 1-level. As such, the approximation coefficients and detail coefficients from the first level can be referred to as a1 and d1, respectively. Computation of approximation and detail coefficients for subsequent levels is achieved by recursively applying ^{4}, four levels of Haar transforms can be computed.

Mallat

The study area comprises a range of broadleaf deciduous forest types within the state of Wisconsin, USA (

Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) data used in this study (Flight ID: f080713t01 and f080714t01) were acquired in July 2008 on NASA’s ER-2 aircraft at an altitude of 20 km, yielding a pixel (

AVIRIS image preprocessing involved manual delineation of clouds and cloud-shadows, cross-track illumination correction, and conversion to top-of-canopy (TOC) reflectance via atmospheric correction. Redundant bands due to overlap of detectors were also removed. Cross-track illumination effects arise from a combination of flight path orientation and relative solar azimuth. We removed this effect by developing band-wise bilinear trend surfaces, ignoring all cloud/shadow-masked pixels, and trend-normalizing the images by subtracting the illumination trend surface and adding the image mean. Atmospheric correction of the cross-track illumination corrected images to TOC reflectance employed the ACORN5b™ software (Atmospheric CORrection Now; Imspec LLC, USA). Due to the low ratio of signal to noise at both spectral ends (366 nm–395 nm and 2,467 nm–2,497 nm), and in bands around the major water absorption regions (1,363 nm–1,403 nm and 1,811 nm–1,968 nm) those wavelength regions were dropped, resulting in a final total of 184 bands. The pixel spectra corresponding to the centers of the plot locations as described in the next section were extracted for the final 184 atmospherically corrected AVIRIS bands.

Optical measurements of effective LAI (_{e}_{e}_{e} is the effective LAI, Ω is the clumping index and α is the woody-to-total leaf area ratio (_{e} (1/Ω)), in which ^{2} of ground area). In this study, we calculated LAI from _{e}_{e}/Ω).

Hemispherical images were collected at 18 plots (60 m × 60 m) characterized by broadleaf decidous forest types. All measurements were made during the peak of the summer growing seasons in 2008 and 2010 during uniformly overcast skies or during dusk or dawn when the sun was hidden by the horizon. Images were collected in the JPEG format at the highest resolution (2,560 × 1,920 pixels) to maximize the detection of small canopy gaps. We followed the protocol of Zhang _{e} at nine subplot locations: the plot center, 30 m from the plot center in each of the four cardinal directions, and the mid-point of each 30 m transect. Images were processed using DHP-TRAC software to estimate _{e} and Ω, using a nine ring configuration but selecting only the first six rings for analysis to minimize the impacts of large zenith angles on the _{e} retrievals and the calculation of LAI [

One broadleaf plot of our 18 total was removed from the analysis based on Cook’s Distance, which identifies influential observations on the basis of how a linear function changes when a certain observation is deleted [

We used DART version 4.3.3 to simulate canopy reflectance for different parameter combinations. The DART model was run in a UNIX environment using a computer with 32 processors (eight Quad-Core AMD Opteron TM 8356 Processor) and 64 GB of physical random access memory (RAM).

DART requires users to build a computer representation of a relevant earth landscape scene such as a forest, an urban area, and water body. Ideally, the scene area should be large and should include important details about the landscape. For instance, a forest scene should be built with a large number of trees, and the resolution of the scene (cell) should be high enough to represent canopy elements such as leaves, twigs, and branches. This level of details leads to unacceptable computational time requirements for a LUT building process. Thus, a scene of reasonable size and detail needs to be determined that allows one to operate DART with an acceptable accuracy level. In this study, simulations were conducted using a simple forest representation. A repetitive forest landscape pattern made up of four trees with ellipsoid shaped crowns was chosen; previous studies have found this representations to be optimal [

Construction of a LUT of reflectance values for a large number of bands with a 3-D model such as DART requires an unacceptable computational time. We used an efficient strategy to populate the LUT grid with a sufficient number of DART simulations by identifying the realistic combinations of different DART parameters and their suitable ranges. The process to develop the LUT is described below.

Initially, a preliminary LUT with a wide range and coarse increments for parameters was constructed for only four narrow AVIRIS bands (550 nm, 1,139 nm, 1,692 nm, and 2,208 nm). PROSPECT [^{2}), leaf dry matter content (DM, g/cm^{2}), and equivalent water thickness (EWT, cm). Together with the PROSPECT parameters, the parameters that were systematically varied (free parameters) were leaf area index (LAI), leaf angle distribution (LAD), soil reflectance (SL), and crown cover (CC). The range for some of the parameters (Cab, LAI, DM, and EWT) were set based upon the databases of leaf measurements spanning all common and rare tree species collected by Serbin [

The simulations in the LUT were then compared with the image reflectance to find the optimal ranges for individual parameters. The corresponding maximum and minimum reflectance values in image-extracted spectra were determined for each simulated band. The maximum and minimum values in the individual band were combined to form a maximum and minimum reflectance (max/min) spectrum. To allow for potential model/measurement uncertainties, the space was expanded by five percent in both directions. A search filter was applied to find the LUT simulations confined within the max/min space. The model inputs that led to simulations (candidate solutions) in the resulting reduced LUT were examined to infer information about optimal parameter ranges and realistic parameter combinations. Combinations of parameter values that produced similar output to image reflectance were determined as the range for the rest of model parameters for different combinations of CC, DM, and EWT. We selected the four narrow bands in the preliminary LUT as representative of the four distinct regions of the vegetation reflectance spectrum. The selection of different subsets of the candidate bands did not significantly alter our result. Once the ranges and realistic combinations were determined, a simple sensitivity analysis (SA) was performed as in Santis

If we had considered all the parameter combinations, 134,400 (16 * 10 * 7 * 2 * 5 * 4 * 3) simulations for each AVIRIS band would have been required. After searching for realistic parameter combinations, only 5,960 simulations were needed in the infrared bands and 1,920 in the visible bands. Simulations were done separately for visible and infrared bands in order to reduce computation time, as reflectances in the two regions were sensitive to different sets of PROSPECT parameters. Canopy reflectances in the visible bands were simulated using fixed values for EWT (0.007 cm) and DM (0.007 g/cm^{2}). Similarly, a constant value for Cab (30 μg/cm^{2}) was used for the infrared bands. Common parameter values in both parts of the spectrum were later used to append the two tables.

When computing the DWT, two input parameters are required: the choice of mother wavelet and the level of decomposition. We chose the Haar mother wavelet as it is the simplest of all available mother wavelets, and recent investigations have illustrated its effectiveness in hyperspectral data analysis [

LUT inversion required similarity matching between wavelet coefficients from plot spectra (measured) and the wavelet coefficients from simulated spectra (modeled). Similarity was assessed using a least root mean square error (RMSE) comparison of the simulated and measured coefficients according to _{measured} is a wavelet coefficient of measured reflectance in test dataset and _{modeled}

In DWT, a large number of coefficients usually contain very low values (near-zero), and do not retain any useful information. For example, only 42 out of total 184 wavelet coefficients had 99.99% of the total energy (equivalent to variance) corresponding to original reflectance in one of the plots. The rest of the coefficients (142) had very low values (cumulatively containing only 0.01% of total energy) and might cause inversion bias. Hence, we repeated the inversion procedure but discarded the low coefficients. For each measured wavelet spectrum, the amount of retained energy by an individual coefficient was calculated as the square of its value. The wavelet coefficients were then sorted in descending order of their energies. Two sets of coefficients, which together contained 99.99% and 99.0% of the energy of original spectrum, were created. During inversion, similarity measurements between the wavelet coefficients and simulated spectra (

For comparison purposes, LUT inversion was also performed using the original spectral bands (SPECTRAL BANDS) which required assessing the similarity between plot and simulated spectra. As before, this was carried out using a least root mean square error (RMSE) comparison, in the case of the measured and modeled spectra, according to _{measured}_{modeled}

LUT inversions were performed independently for all wavelet coefficients (ALL COEFFICIENTS), high energy wavelet coefficients (ENERGY SUBSET 1 and ENERGY SUBSET 2), and untransformed original spectral reflectance (SPECTRAL BANDS). All three sets of LUT were sorted from minimum to maximum according to the cost function (RMSE).

The final results are summarized in ^{2} between model inverted LAI and field measured LAI for ALL COEFFICIENTS, ENERGY SUBSET 1, ENERGY SUBSET 2, and SPECTRAL BANDS respectively.

The results show that the LAI estimated from ENERGY SUBSET 1 provided the greatest accuracy (RMSE = 0.46 and R^{2} = 0.77), followed by the LAI estimated from ENERGY SUBSET 2 (RMSE = 0.56 and R^{2} = 0.49). Among the different numbers of solutions, the median calculated from 30 solutions provided the best estimate of LAI, the differences were greatest for q = 10 (RMSE = 0.62) and q = 30 (RMSE = 0.46) with ENERGY SUBSET 1. The relations between the measured and model estimated LAI based on the median of 30 solutions from ENERGY SUBSET 1, and SPECTRAL BANDS are shown in

The results indicate that the discrete wavelet transform (DWT) can be a useful tool for estimating forest LAI using hyperspectral data by inverting DART. A simple wavelet decomposition of modeled and measured hyperspectral data by Haar wavelets followed by inversion using high energy wavelet coefficients provided better estimates of LAI than those provided by inversion using the original spectral reflectance.

According to Proposition 3 in Verstraete

The results of this study show that the wavelet transformation of hyperspectral bands can be a useful alternative to the band selection techniques employed by previous studies. Wavelet features in this study were less correlated than the original reflectance bands owing to the orthogonal nature of the transformation. DWT also afforded dimensionality reduction, as 99.99% of the total energy was concentrated in few coefficients. Moreover, DWT provided a multi-resolution representation of the original signal in which the effects of different parameters on canopy reflectance can possibly be distributed over wavelet coefficients of different scales. A narrow band in the red edge region of the reflectance spectrum might carry signatures of different parameters including leaf pigments, LAI, etc. After wavelet decomposition, the effect of leaf pigments might be relegated to a small-scale wavelet coefficient, whereas the effect of LAI might be more pronounced in one of the large scale wavelet coefficients (as LAI generally tends to affect reflectance over a larger part of the spectrum than leaf pigments). Such decoupling of the effects over wavelet coefficients at multiple scales and consequently greater independence among parameters is a useful property for inversion. The greatest accuracy obtained by the LUT inversion using ENERGY SUBSET 1 might be attributed to one or all of these benefits offered by DWT.

The poor estimates of LAI by ALL COEFFICIENTS compared to ENERGY SUBSET 1 and ENERGY SUBSET 2 demonstrate how essential it is to select optimal wavelet features. The selection of wavelet features employed in this study was relatively straightforward as only a small fraction of coefficients retained significant information (as measured by energy). The ENERGY SUBSET 1 discarded the low coefficients and only included the top coefficients cumulatively adding up to 99.99% of total energy. By doing so, the latter represented the original reflectance with a limited number of coefficients without causing significant errors for signal representations (loss of 0.01% of total energy). One of the important properties of DWT is that the energy of the Gaussian noise component of the signal will usually be dispersed as relatively small coefficients [

To understand the effect of different thresholds, we repeated the analysis using a new threshold of 99.0% energy. This led to a decrease in the best accuracy of LAI estimates from RMSE = 0.46 to RMSE = 0.56. With 99.0% energy, only 13 coarse-scale coefficients were retained (three approximations and others from scale level 4, 5 and 6), whereas, the 99.99% threshold level retained both coarse scale and fine scale detail coefficients from level 1 to 6. These coefficients from different levels are functions of scale and position (fine detail

The results also demonstrate that the different number of solutions estimated LAI with different accuracies. The differences were greatest between

We chose the Haar mother wavelet in this study, as it is the simplest of various mother wavelets, and has been found useful for hyperspectral data analysis in previous studies [

The plot LAI was estimated using DHP following the protocol of Zhang

The main objective of this study was to investigate the utility of the discrete wavelet transform (DWT) for estimating forest leaf area index (LAI) via inversion of the Discrete Anisotropy Radiative Transfer (DART) model using airborne hyperspectral data (AVIRIS). The DWT transforms the hyperspectral data into wavelet features at a variety of spectral scales. The multiscale features detect and isolate variation in the reflectance continuum not detectable in the original reflectance domain such as amplitude variations over broad and narrow spectral regions. We employed the DWT on reflectance spectra obtained from hyperspectral data to improve estimation of LAI in temperate forests. The model inversion was performed with three different datasets, the original reflectance bands, the full set of wavelet extracted features, and two wavelet subsets containing 99.99% and 99.0% of the cumulative energy of the original signal. The results show that the LAI estimated from the wavelet subset with 99.99% of the cumulative energy provided the greatest accuracy. Our results suggest that the Haar discrete wavelet transform can be an effective tool for accurate DART model inversion. This study focused only on comparing multiscale DWT coefficients to the original spectral bands. The work could be extended by applying sensitivity analyses to assess the effect of variation in LAI on wavelet features at different positions and scales. Such an effort might also help identify features that are more sensitive to LAI and less to background signals caused by soil or crown cover. The theory of wavelet transforms is still developing, and more wavelet families are being introduced. The exploration of the utility of other wavelets in model inversion is well warranted since different families of wavelets are suited to different signals and applications.

The authors declare no conflict of interest.

Study area: (

Tree scenes with variable tree ground covers from 100% down to 70%. Discrete Anisotropy Radiative Transfer (DART) works with an infinite scene comprised of a repetitive pattern of these tree scenes.

Observed

Observed

Reconstructed (with Haar coefficients having cumulative 99.0% energy)

Reconstructed (with Haar coefficients having cumulative 99.99% energy)

Reconstructed (with db3 coefficients having cumulative 99.99% energy)

Parameters, their ranges and increments for the final simulation. The final column shows the total number of values used for each parameter. The parameters are leaf area index (LAI), leaf equivalent water thickness (EWT), leaf dry matter content (DM), leaf structural parameter (N), leaf chlorophyll a + b concentration (Cab), canopy cover (CC) and leaf angle distribution (LAD). The three LADs used were planophile, plagiophile, and erectophile.

LAI (m^{2}/m^{2}) |
2.75 | 6.75 | 0.25 | 16 |

EWT (cm) | 0.003 | 0.0183 | 0.0017 | 10 |

DM (g/cm^{2}) |
0.001 | 0.0132 | 0.0017 | 7 |

N | 1.75 | 2.25 | 0.5 | 2 |

Cab (μg/cm^{2}) |
20 | 60 | 10 | 5 |

CC (%) | 70% | 100% | 10% | 4 |

LAD | - | - | - | 3 |

Results from LUT inversion. The first column shows the total number of solutions selected by least RMSEr to calculate the median LAI value. The other columns show the RMSE and R^{2} between measured and estimated LAI by inversion using ALL COEFFICIENTS, ENERGY SUBSET 1, ENERGY SUBSET 2, and SPECTRAL BANDS respectively.

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^{2} |
^{2} |
^{2} |
^{2} | |||||

10 | 0.82 | 0.32 | 0.62 | 0.63 | 0.62 | 0.44 | 0.79 | 0.32 |

20 | 0.69 | 0.43 | 0.50 | 0.75 | 0.60 | 0.49 | 0.65 | 0.46 |

30 | 0.69 | 0.42 | 0.46 | 0.77 | 0.56 | 0.49 | 0.60 | 0.47 |

40 | 0.71 | 0.34 | 0.49 | 0.70 | 0.58 | 0.41 | 0.62 | 0.41 |

50 | 0.70 | 0.35 | 0.56 | 0.58 | 0.58 | 0.47 | 0.62 | 0.40 |