Modeling Directional Brightness Temperature (DBT) over Crop Canopy with Effects of Intra-Row Heterogeneity

In order to improve the simulation accuracy of directional brightness temperature (DBT) and the retrieval accuracy of component temperature, a model considering intra-row heterogeneity to simulate the DBT angular distribution over crop canopy is proposed. At individual scale, the probability of leaf appearance is inversely proportional to the distance from central stem. On the basis of this assumption, we formulated leaf area volume density (LAVD) spatial distribution at three hierarchical scales: individual scale, row scale, and scene scale. The equations for directional gap probability and bi-directional gap probability were modified to adapt the heterogeneity of row structure. Afterwards, a straightforward radiative transfer model was built based on the gap probabilities. A set of simulated data was generated by the thermal radiosity-graphics combined model (TRGM) as the benchmark to evaluate both forward simulation and inversion ability of the new model; we compared the new DBT model against an existing model assuming row as homogeneous box. With the growth of crops, the canopy structure of row crops will gradually change from row structure to continuous canopy. The new DBT model agreed with the TRGM model much better than the homogeneous row model at the middle stage of the crop growth season. The new model and the homogeneous row model achieve similar accuracy at early stage and end stage. At the middle growth stage, the new model can improve the accuracy of soil temperature retrieval. We recommend the new DBT model as an option to improve the DBT simulation and component temperature retrieval for row-planted crop canopy. In particular, the more accurate component temperatures during the middle growth stage are fundamentally important in characterizing crop water status, evapotranspiration, and soil moisture, which are subsequently critical for predicting crop productivity.

Modeling Small-Footprint Airborne Lidar-Derived Estimates of Gap Probability and Leaf Area Index

Airborne lidar point clouds of vegetation capture the 3-D distribution of its scattering elements, including leaves, branches, and ground features. Assessing the contribution from vegetation to the lidar point clouds requires an understanding of the physical interactions between the emitted laser pulses and their targets. Most of the current methods to estimate the gap probability ( P gap ) or leaf area index (LAI) from small-footprint airborne laser scan (ALS) point clouds rely on either point-number-based (PNB) or intensity-based (IB) approaches, with additional empirical correlations with field measurements. However, site-specific parameterizations can limit the application of certain methods to other landscapes. The universality evaluation of these methods requires a physically based radiative transfer model that accounts for various lidar instrument specifications and environmental conditions. We conducted an extensive study to compare these approaches for various 3-D forest scenes using a point-cloud simulator developed for the latest version of the discrete anisotropic radiative transfer (DART) model. We investigated a range of variables for possible lidar point intensity, including radiometric quantities derived from Gaussian Decomposition (GD), such as the peak amplitude, standard deviation, integral of Gaussian profiles, and reflectance. The results disclosed that the PNB methods fail to capture the exact P gap as footprint size increases. By contrast, we verified that physical methods using lidar point intensity defined by either the distance-weighted integral of Gaussian profiles or reflectance can estimate P gap and LAI with higher accuracy and reliability. Additionally, the removal of certain additional empirical correlation coefficients is feasible. Routine use of small-footprint point-cloud radiometric measures to estimate P gap and the LAI potentially confirms a departure from previous empirical studies, but this depends on additional parameters from lidar instrument vendors.

The generating function for the Bessel point process and a system of coupled Painlevé V equations

We study the joint probability generating function for [Formula: see text] occupancy numbers on disjoint intervals in the Bessel point process. This generating function can be expressed as a Fredholm determinant. We obtain an expression for it in terms of a system of coupled Painlevé V equations, which are derived from a Lax pair of a Riemann–Hilbert problem. This generalizes a result of Tracy and Widom [C. A. Tracy and H. Widom, Level spacing distributions and the Bessel kernel, Commun. Math. Phys. 161(2) (1994) 289–309], which corresponds to the case [Formula: see text]. We also provide some examples and applications. In particular, several relevant quantities can be expressed in terms of the generating function, like the gap probability on a union of disjoint bounded intervals, the gap between the two smallest particles, and large [Formula: see text] asymptotics for [Formula: see text] Hankel determinants with a Laguerre weight possessing several jump discontinuities near the hard edge.

Spectral statistics of unitary ensembles

This article focuses on the use of the orthogonal polynomial method for computing correlation functions, cluster functions, gap probability, Janossy density, and spacing distributions for the eigenvalues of matrix ensembles with unitary-invariant probability law. It first considers the classical families of orthogonal polynomials (Hermite, Laguerre, and Jacobi) and some corresponding unitary ensembles before discussing the statistical properties of N-tuples of real numbers. It then reviews the definitions of basic statistical quantities and demonstrates how their distributions can be made explicit in terms of orthogonal polynomials. It also describes the k-point correlation function, Fredholm determinants of finite-rank kernels, and resolvent kernels.

Asymptotics of Hankel Determinants With a One-Cut Regular Potential and Fisher–Hartwig Singularities

We obtain asymptotics of large Hankel determinants whose weight depends on a one-cut regular potential and any number of Fisher–Hartwig singularities. This generalises two results: (1) a result of Berestycki, Webb, and Wong [5] for root-type singularities and (2) a result of Its and Krasovsky [37] for a Gaussian weight with a single jump-type singularity. We show that when we apply a piecewise constant thinning on the eigenvalues of a random Hermitian matrix drawn from a one-cut regular ensemble, the gap probability in the thinned spectrum, as well as correlations of the characteristic polynomial of the associated conditional point process, can be expressed in terms of these determinants.