Inversion of a forest reflectance model to estimate structural canopy variables from hyperspectral remote sensing data

The potential of canopy reflectance modelling to retrieve simultaneously several structural variables in managed Norway spruce stands was investigated using the “Invertible Forest Reflectance Model”, INFORM. INFORM is an innovative extension of the FLIM model, with crown transparency, infinite crown reflectance and understory reflectance simulated using physically based sub-models (SAILH, LIBERTY and PROSPECT). The INFORM model was inverted with hyperspectral airborne HyMap data using a neural network approach. INFORM based estimates of forest structural variables were produced using site-specific ranges of stand structural variables. A relatively simple three layer feed-forward backpropagation neural network with two input neurons, one neuron in the hidden layer and three output neurons was employed to map leaf area index (LAI), crown coverage and stem density.To identify the optimum 2-band spectral subset to be used in the inversion process, all 2-band combinations of the HyMap dataset were systematically evaluated for model inversion. Field measurements of structural variables from 39 forest stands were used to validate the maps produced from HyMap imagery. Using two HyMap wavebands at 837 nm and 1148 nm the obtained accuracy of the LAI map amounts to an rmse of 0.58 (relative rmse = 18% of mean, R² = 0.73). With HyMap data resampled to Landsat TM spectral bands and using two “optimum” bands at 840 nm and 1650 nm, rmse was 0.66 and relative rmse 21%. In contrast to approaches based on empirical relations between spectral vegetation indices and structural variables, the main advantage of the inversion approach is that it does not require previous calibration.     

Initial-interval driven biasing of nodal concentration along a fixed curve

Element size transitioning in the construction of spatial meshes for finite element models is often controlled by biasing the concentration of nodes, towards one end or the other, along each of a set of curves in the model. A simple, common and efficient scheme to implement such nodal concentration biasing along a given curve is to require that the nodal spacings δ_i be (sequence) terms bⁱδ₀ of a geometric series. Current practice takes the parameter value b, or its equivalent, as an independent input, so that the initial nodal spacing δ₀ must be a computed output. This is the most straightforward approach, but the lack of direct control over the value δ₀ is a significant shortcoming. In an element size transitioning scenario, δ₀ is often a parameter for which the model builder/analyst has independent quantitative information. It may represent the a priori known thickness of a thin bond or weld, for example. A more rational choice for these cases, proposed by this paper, is a scheme for which δ₀ is an independent input parameter instead of b. The parameter b is computed by a convergence-guaranteed algorithm for which the existence of b as a single-valued function of its input is proven.     

Early detection of plant stress by digital imaging within narrow stress-sensitive wavebands

Digital images of soybean canopies [Glycine max (L.) Merrill] were obtained within selected narrow wavebands (6–10 nm bandwidths) to determine their capability for early detection of plant stress. Images and physiological measurements of stress were acquired 2 days, 4 days, and 7 days following application of control, drought, and herbicide [(3,4-dichlorophenyl)-1, 1-dimethylurea, or DCMU] treatments. As a result of frequent rainfall, drought stress never occurred. However, exposure to herbicide rapidly induced plant stress. By day 4, the ratio of variable to maximum leaf fluorescence (F_v/F_m) decreased and leaf water potentials (ψ_w) increased in the herbicide treated soybean, indicating damage to the photosynthetic apparatus and stomatal closure. Also, Munsell leaf color had increased from approximately 5GY 4.6/5.7 to a lighter green-yellow value. Canopy reflectances at 670 nm, 694 nm, and in the 410–740 nm band (R_vis), as well as reflectance at 694 nm divided by reflectance at 760 nm (R₆₉₄/R₇₆₀), detected stress simultaneously with the physiological measurements and increased consistently with stress through day 7. Reflectances at 420 nm and 600 nm, together with R₆₀₀/R₇₆₀ and R_vis/R₇₆₀, did not increase until leaves were yellow or brown and wilted and canopies had begun to collapse on day 7. None of the reflectance or reflectance ratio images detected stress prior to visible color changes. This was attributed primarily to the rapid inducement of chlorosis by the herbicide. Reflectance in narrow wavebands within the 690–700 nm region and its ratio with near-infrared reflectance should provide earlier detection of stress-induced chlorosis compared with broad band systems or narrow bands located at lesser wavelengths.     

Large area mapping of southwestern forest crown cover, canopy height, and biomass using the NASA Multiangle Imaging Spectro-Radiometer

A rapid canopy reflectance model inversion experiment was performed using multi-angle reflectance data from the NASA Multi-angle Imaging Spectro-Radiometer (MISR) on the Earth Observing System Terra satellite, with the goal of obtaining measures of forest fractional crown cover, mean canopy height, and aboveground woody biomass for large parts of south-eastern Arizona and southern New Mexico (> 200,000 km²). MISR red band bidirectional reflectance estimates in nine views mapped to a 250 m grid were used to adjust the Simple Geometric-optical Model (SGM). The soil-understory background signal was partly decoupled a priori by developing regression relationships with the nadir camera blue, green, and near-infrared reflectance data and the isotropic, geometric, and volume scattering kernel weights of the LiSparse–RossThin kernel-driven bidirectional reflectance distribution function (BRDF) model adjusted against MISR red band data. The SGM's mean crown radius and crown shape parameters were adjusted using the Praxis optimization algorithm, allowing retrieval of fractional crown cover and mean canopy height, and estimation of aboveground woody biomass by linear rescaling of the dot product of cover and height. Retrieved distributions of crown cover, mean canopy height, and aboveground woody biomass for forested areas showed good matches with maps from the United States Department of Agriculture (USDA) Forest Service, with R² values of 0.78, 0.69, and 0.81, and absolute mean errors of 0.10, 2.2 m, and 4.5 tons acre^- 1 (10.1 Mg ha^- 1), respectively, after filtering for high root mean square error (RMSE) on model fitting, the effects of topographic shading, and the removal of a small number of outliers. This is the first use of data from the MISR instrument to produce maps of crown cover, canopy height, and woody biomass over a large area by seeking to exploit the structural effects of canopies reflected in the observed anisotropy patterns in these explicitly multiangle data.     

A new approach for mapping of Biological Soil Crusts in semidesert areas with hyperspectral imagery

Biological Soil Crusts (BSCs), consisting of cyanobacteria, algae, microfungi, lichens and bryophytes in varying proportions, live within or immediately on top of the uppermost millimeters of soil, where they form a more or less firm aggregation of soil particles and organisms. They mainly occur in soils of arid and semi-arid regions, which cover more than 35% of the earth's land surface and are assumed to play a major role as primary producers, C- and N-sinks and soil stabilizers.

In order to establish a methodology for mapping of BSCs, their spectral characteristics with respect to different crust types were analyzed. The resulting reflectance spectra of different crust types had a shallow absorption feature centered around 680 nm in common, in which they differed from the spectra of bare soil.

In October 2004, hyperspectral CASI data with a spatial resolution of 1 m were recorded in conjunction with field spectroscopic measurements in the Succulent Karoo, South Africa. Available spectral indices for Biological Soil Crusts were tested on the image but did not produce satisfying classifications. Therefore, an alternative approach was established based on spectral field data, field observations and the hyperspectral dataset. The newly developed Continuum Removal Crust Identification Algorithm (CRCIA) is based on small and narrow spectral characteristics, that were extracted by continuum removal and subsequently expressed as a set of logical conditions. Using this method, 16.2% of the area which covers 12 km² of gently sloping hills with some granite outcrops were classified as BSCs. Validation of the classification resulted in a Kappa index of 0.831.

In a next step, the methodology will be tested with regard to scale-dependent effects and applied to images covering areas with additional types of BSCs and soil to develop a robust and generally applicable method.     

Remote sensing of native and invasive species in Hawaiian forests

Detection and mapping of invasive species is an important component of conservation and management efforts in Hawai'i, but the spectral separability of native, introduced, and invasive species has not been established. We used high spatial resolution airborne imaging spectroscopy to analyze the canopy hyperspectral reflectance properties of 37 distinct species or phenotypes, 7 common native and 24 introduced tree species, the latter group containing 12 highly invasive species. Airborne Visible and Infrared Imaging Spectrometer (AVIRIS) reflectance and derivative-reflectance signatures of Hawaiian native trees were generically unique from those of introduced trees. Nitrogen-fixing trees were also spectrally unique from other groups of non-fixing trees. There were subtle but significant differences in the spectral properties of highly invasive tree species in comparison to introduced species that do not proliferate across Hawaiian ecosystems. The observed differences in canopy spectral signatures were linked to relative differences in measured leaf pigment (chlorophyll, carotenoids), nutrient (N, P), and structural (specific leaf area; SLA) properties, as well as to canopy leaf area index. These leaf and canopy properties contributed variably to the spectral separability of the trees, with wavelength-specific reflectance and absorption features that overlapped, but which were unique from one another. A combination of canopy reflectance from 1125–2500 nm associated with leaf and canopy water content, along with pigment-related absorption features (reflectance derivatives) in the 400–700 nm range, was best for delineating native, introduced, and invasive species. There was no single spectral region that always defined the separability of the species groups, and thus the full-range (400–2500 nm) spectrum was highly advantageous in differentiating these groups. These results provide a basis for more detailed studies of invasive species in Hawai'i, along with more explicit treatment of the biochemical properties of the canopies and their prediction using imaging spectroscopy.     

The delineation of tree crowns in Australian mixed species forests using hyperspectral Compact Airborne Spectrographic Imager (CASI) data

In mixed-species forests of complex structure, the delineation of tree crowns is problematic because of their varying dimensions and reflectance characteristics, the existence of several layers of canopy (including understorey), and shadowing within and between crowns. To overcome this problem, an algorithm for delineating tree crowns has been developed using eCognition Expert and hyperspectral Compact Airborne Spectrographic Imager (CASI-2) data acquired over a forested landscape near Injune, central east Queensland, Australia. The algorithm has six components: 1) the differentiation of forest, non-forest and understorey; 2) initial segmentation of the forest area and allocation of segments (objects) to larger objects associated with forest spectral types (FSTs); 3) initial identification of object maxima as seeds within these larger objects and their expansion to the edges of crowns or clusters of crowns; 4) subsequent classification-based separation of the resulting objects into crown or cluster classes; 5) further iterative splitting of the cluster classes to delineate more crowns; and 6) identification and subsequent merging of oversplit objects into crowns or clusters. In forests with a high density of individuals (e.g., regrowth), objects associated with tree clusters rather than crowns are delineated and local maxima counted to approximate density. With reference to field data, the delineation process provided accuracies > ∼70% (range 48-88%) for individuals or clusters of trees of the same species with diameter at breast height (DBH) exceeding 10 cm (senescent and dead trees excluded), with lower accuracies associated with dense stands containing several canopy layers, as many trees were obscured from the view of the CASI sensor. Although developed using 1-m spatial resolution CASI data acquired over Australian forests, the algorithm has application elsewhere and is currently being considered for integration into the Definiens product portfolio for use by the wider community.     

On extended P4-reducible and extended P4-sparse graphs

A graph G was defined in [16] as P₄-reducible, if no vertex in G belongs to more than one chordless path on four vertices or P₄. A graph G is defined in [15] as P₄-sparse if no set of five vertices induces more than one P₄, in G. P₄-sparse graphs generalize both P₄-reducible and the well known class of p₄-free graphs or cographs. In an extended abstract in [11] the first author introduced a method using the modular decomposition tree of a graph as the framework for the resolution of algorithmic problems. This method was applied to the study of P₄-sparse and extended P₄-sparse graphs.

In this paper, we begin by presenting the complete information about the method used in [11]. We propose a unique tree representation of P₄-sparse and a unique tree representation of P₄-reducible graphs leading to a simple linear recognition algorithm for both classes of graphs. In this way we simplify and unify the solutions for these problems, presented in [16–19]. The tree representation of an n-vertex P₄-sparse or a P₄-reducible graph is the key for obtaining O(n) time algorithms for the weighted version of classical optimization problems solved in [20]. These problems are NP-complete on general graphs.

Finally, by relaxing the restriction concerning the exclusion of the C₅ cycles from P₄-sparse and P₄-reducible graphs, we introduce the class of the extended P4-sparse and the class of the extendedP₄-reducible graphs. We then show that a minimal amount of additional work suffices for extending most of our algorithms to these new classes of graphs.     

Invasive species detection in Hawaiian rainforests using airborne imaging spectroscopy and LiDAR

Remote sensing of invasive species is a critical component of conservation and management efforts, but reliable methods for the detection of invaders have not been widely established. In Hawaiian forests, we recently found that invasive trees often have hyperspectral signatures unique from that of native trees, but mapping based on spectral reflectance properties alone is confounded by issues of canopy senescence and mortality, intra- and inter-canopy gaps and shadowing, and terrain variability. We deployed a new hybrid airborne system combining the Carnegie Airborne Observatory (CAO) small-footprint light detection and ranging (LiDAR) system with the Airborne Visible and Infrared Imaging Spectrometer (AVIRIS) to map the three-dimensional spectral and structural properties of Hawaiian forests. The CAO-AVIRIS systems and data were fully integrated using in-flight and post-flight fusion techniques, facilitating an analysis of forest canopy properties to determine the presence and abundance of three highly invasive tree species in Hawaiian rainforests.

The LiDAR sub-system was used to model forest canopy height and top-of-canopy surfaces; these structural data allowed for automated masking of forest gaps, intra- and inter-canopy shadows, and minimum vegetation height in the AVIRIS images. The remaining sunlit canopy spectra were analyzed using spatially-constrained spectral mixture analysis. The results of the combined LiDAR-spectroscopic analysis highlighted the location and fractional abundance of each invasive tree species throughout the rainforest sites. Field validation studies demonstrated < 6.8% and < 18.6% error rates in the detection of invasive tree species at  7 m² and  2 m² minimum canopy cover thresholds. Our results show that full integration of imaging spectroscopy and LiDAR measurements provides enormous flexibility and analytical potential for studies of terrestrial ecosystems and the species contained within them.     

Properties of Noncentral Dirichlet Distributions

Let X₁,…, X_r+1 be independent random variables, X_iGa (a_i, θ, δ_i), i = 1,…, r + 1. Define and V_i = X_i/X_r+1, i = 1,…, r. Then, (U₁,…, U_r) and (V₁,…, V_r) follow noncentral Dirichlet Type 1 and Type 2 distributions, respectively. In this article several properties of these distributions and their connections with the uniform, the noncentral multivariate-F and the noncentral multivariate-t distributions are discussed.     

Triple positive solutions for some second-order boundary value problem on a measure chain

In this paper, we study the existence of three positive solutions for the second-order two-point boundary value problem on a measure chain,

where f:[t₁,σ(t₂)]×[0,∞)×R→[0,∞) is continuous and p:[t₁,σ(t₂)]→[0,∞) a nonnegative function that is allowed to vanish on some subintervals of [t₁,σ(t₂)] of the measure chain. The method involves applications of a new fixed-point theorem due to Bai and Ge [Z.B. Bai, W.G. Ge, Existence of three positive solutions for some second order boundary-value problems, Comput. Math. Appl. 48 (2004) 699–707]. The emphasis is put on the nonlinear term f involved with the first order delta derivative x^Δ(t).     

Analytic numerical solution of coupled semi-infinite diffusion problems

In this paper, we consider coupled semi-infinite diffusion problems of the form u_t(x, t)− A² u_xx(x,t) = 0, x> 0, t> 0, subject to u(0,t)=B and u(x,0)=0, where A is a matrix in , and u(x,t), and B are vectors in . Using the Fourier sine transform, an explicit exact solution of the problem is proposed. Given an admissible error and a domain D(x₀,t₀)={(x,t);0≤x≤x₀, t≥t₀ > 0, an analytic approximate solution is constructed so that the error with respect to the exact solution is uniformly upper bounded by in D(x₀, t₀).     

Ratios of leaf reflectances in narrow wavebands as indicators of plant stress

Ratios of leaf reflectances that were measured within narrow wavebands (2nm) were evaluated as indicators of plant stress. Wavebands used in ratio computation were based on earlier studies that determined the wavelength regions in which reflectance was most affected by 8 stress agents among 6 plant species. Several ratios, such as reflectance at 695 nm divided by reflectance at 670 nm (R695/R670), were affected by some but not all stress agents. However, R695/R420, R605/R760, R695/R760 and R710/R760 were significantly greater (p≤0·05) in stressed compared with non-stressed leaves for all stress agents. The ratios that most strongly indicated plant stress were reflectance at 695 nm divided by reflectance at 420 nm or 760 nm.     

Biinfinite words with maximal recurrent unbordered factors

A finite non-empty word z is said to be a border of a finite non-empty word w if w=uz=zv for some non-empty words u and v. A finite non-empty word is said to be bordered if it admits a border, and it is said to be unbordered otherwise. In this paper, we give two characterizations of the biinfinite words of the form ^ωuvu^ω, where u and v are finite words, in terms of its unbordered factors.

The main result of the paper states that the words of the form ^ωuvu^ω are precisely the biinfinite words w=a₋₂a₋₁a₀a₁a₂ for which there exists a pair (l₀,r₀) of integers with l₀<r₀ such that, for every integers ll₀ and rr₀, the factor a_la_l0a_r0a_r is a bordered word.

The words of the form ^ωuvu^ω are also characterized as being those biinfinite words w that admit a left recurrent unbordered factor (i.e., an unbordered factor of w that has an infinite number of occurrences “to the left” in w) of maximal length that is also a right recurrent unbordered factor of maximal length. This last result is a biinfinite analogue of a result known for infinite words.     

Analytic-numerical solutions with a priori error bounds for time-dependent mixed partial differential problems

The aim of this paper is double. First, we point out that the hypothesis D(t₁)D(t₂) = D(t₂)D(t₁) imposed in [1] can be removed. Second, a constructive method for obtaining analytic-numerical solutions with a prefixed accuracy in a bounded domain Ω(t₀,t₁) = [0,p] × [t₀,t₁], for mixed problems of the type u_t(x,t) − D(t)u_xx(x,t) = 0, 0 < x < p, t> 0, subject to u(0,t) = u(p,t) = 0 and u(x,0) = F(x) is proposed. Here, u(x,t) and F(x) are r-component vectors, D(t) is a C^{r × r} valued analytic function and there exists a positive number δ such that every eigenvalue z of (1/2) (D(t) + D(t)^H) is bigger than δ. An illustrative example is included.     

Leaf chlorophyll content retrieval from airborne hyperspectral remote sensing imagery

Hyperspectral remote sensing has great potential for accurate retrieval of forest biochemical parameters. In this paper, a hyperspectral remote sensing algorithm is developed to retrieve total leaf chlorophyll content for both open spruce and closed forests, and tested for open forest canopies. Ten black spruce (Picea mariana (Mill.)) stands near Sudbury, Ontario, Canada, were selected as study sites, where extensive field and laboratory measurements were carried out to collect forest structural parameters, needle and forest background optical properties, and needle biophysical parameters and biochemical contents chlorophyll a and b. Airborne hyperspectral remote sensing imagery was acquired, within one week of ground measurements, by the Compact Airborne Spectrographic Imager (CASI) in a hyperspectral mode, with 72 bands and half bandwidth 4.25-4.36 nm in the visible and near-infrared region and a 2 m spatial resolution. The geometrical-optical model 4-Scale and the modified leaf optical model PROSPECT were combined to estimate leaf chlorophyll content from the CASI imagery. Forest canopy reflectance was first estimated with the measured leaf reflectance and transmittance spectra, forest background reflectance, CASI acquisition parameters, and a set of stand parameters as inputs to 4-Scale. The estimated canopy reflectance agrees well with the CASI measured reflectance in the chlorophyll absorption sensitive regions, with discrepancies of 0.06%-1.07% and 0.36%-1.63%, respectively, in the average reflectances of the red and red-edge region. A look-up-table approach was developed to provide the probabilities of viewing the sunlit foliage and background, and to determine a spectral multiple scattering factor as functions of leaf area index, view zenith angle, and solar zenith angle. With the look-up tables, the 4-Scale model was inverted to estimate leaf reflectance spectra from hyperspectral remote sensing imagery. Good agreements were obtained between the inverted and measured leaf reflectance spectra across the visible and near-infrared region, with R² = 0.89 to R² = 0.97 and discrepancies of 0.02%-3.63% and 0.24%-7.88% in the average red and red-edge reflectances, respectively. Leaf chlorophyll content was estimated from the retrieved leaf reflectance spectra using the modified PROSPECT inversion model, with R² = 0.47, RMSE = 4.34 μg/cm², and jackknifed RMSE of 5.69 μg/cm² for needle chlorophyll content ranging from 24.9 μg/cm² to 37.6 μg/cm². The estimates were also assessed at leaf and canopy scales using chlorophyll spectral indices TCARI/OSAVI and MTCI. An empirical relationship of simple ratio derived from the CASI imagery to the ground-measured leaf area index was developed (R² = 0.88) to map leaf area index. Canopy chlorophyll content per unit ground surface area was then estimated, based on the spatial distributions of leaf chlorophyll content per unit leaf area and the leaf area index.     

A limit characterization for the number of spanning trees of graphs

In this paper we propose a limit characterization of the behaviour of classes of graphs with respect to their number of spanning trees. Let {G_n} be a sequence of graphs G₀,G₁,G₂,… that belong to a particular class. We consider graphs of the form K_n−G_n that result from the complete graph K_n after removing a set of edges that span G_n. We study the spanning tree behaviour of the sequence {K_n−G_n} when n→∞ and the number of edges of G_n scales according to n. More specifically, we define the spanning tree indicator ({G_n}), a quantity that characterizes the spanning tree behaviour of {K_n−G_n}. We derive closed formulas for the spanning tree indicators for certain well-known classes of graphs. Finally, we demonstrate that the indicator can be used to compare the spanning tree behaviour of different classes of graphs (even when their members never happen to have the same number of edges).     

Heaps and heapsort on secondary storage

A heap structure designed for secondary storage is suggested that tries to make the best use of the available buffer space in primary memory. The heap is a complete multi-way tree, with multi-page blocks of records as nodes, satisfying a generalized heap property. A special feature of the tree is that the nodes may be partially filled, as in B-trees. The structure is complemented with priority-queue operations insert and delete-max. When handling a sequence of S operations, the number of page transfers performed is shown to be O(∑_{i = 1}^S(1/P) log_(M/P)(N_i/P)), where P denotes the number of records fitting into a page, M the capacity of the buffer space in records, and N_i, the number of records in the heap prior to the ith operation (assuming P 1 and S> M c · P, where c is a small positive constant). The number of comparisons required when handling the sequence is O(∑_{i = 1}^S log₂ N_i). Using the suggested data structure we obtain an optimal external heapsort that performs O((N/P) log_(M/P)(N/P)) page transfers and O(N log₂ N) comparisons in the worst case when sorting N records.     

Constructible functions in cellular automata and their applications to hierarchy results

We investigate time-constructible functions in one-dimensional cellular automata (CA). It is shown that (i) if a function t(n) is computable by an O(t(n)−n)-time Turing machine, then t(n) is time constructible by CA and (ii) if two functions are time constructible by CA, then the sum, product, and exponential functions of them are time constructible by CA. As an application, it is shown that if t₁(n) and t₂(n) are time constructible functions such that lim_n→∞ t₁(n)/t₂(n) = 0 and t₁(n)n, then there is a language which can be recognized by a CA in t₂(n) time but not by any CA in t₁(n) time.