Gap Probability Research Articles

Automated reconstruction of tree and canopy structure for modeling the internal canopy radiation regime

Understanding canopy radiation regimes is critical to successfully modeling vegetation growth and function. For instance, the vertical distribution of photosynthetically active radiation (PAR) affects vegetation growth, informative upon carbon and energy cycling. Availing upon advances in information capture and computing power, geometrically explicit modeling of forest structure becomes increasingly possible. A primary challenge however is acquiring the forest mensuration data required to parameterize these models and the related automation of modeling forest structure. In this research, to address these issues we employ a novel and automated approach that capitalizes upon the rich information afforded by ground-based laser scanning technology. The method is implemented in two steps: in the first step, geometric explicit models of canopy structure are created from the ground-based laser scanning data. These geometric explicit models are used to simulate the vertical range to first hit. In the second step, we derive canopy gap probability from full waveform laser scanning data which have been used in a number of studies for characterization of radiation transmission (Jupp et al., 2009; Yang et al., 2010) and do not require any geometric explicit modeling. The radiative consistency of the geometric explicit models from step 1 is validated against the gap probabilities of step 2. The results show a strong relationship between the radiative transmission properties of the geometric models and canopy gap probabilities at plot level (R=0.91 to 0.97), while the geometric models suggest the additional benefit to serve as a bridge in scaling between shoot level and canopy level radiation.

Impact of sensor footprint on measurement of directional brightness temperature of row crop canopies

A sensor's footprint determines the target that is observed by the sensor, and influences the angular features of the target's directional brightness temperature (DBT) at the field site. This paper describes a new radiative transfer model (FovMod) to simulate the DBT of the row crop canopy by considering the sensor's footprint in the ground measurements. The FovMod firstly divides the sensor's circular or elliptical footprint into a few small cells, and then estimates the components' fractions (e.g., leaves, sunlit soil and shaded soil) in each cell based on the gap probability theory. The canopy's DBT is finally obtained by weighting the components' brightness temperatures and their fractions using a Gaussian point spreading function (PSF) of the sensor's response. Simulation results indicate that a small footprint causes the distribution of the DBT to be strongly dominated by the row direction and a single component's temperature but little influenced by the solar position. On the contrary, a large footprint smoothes the row-space effect and causes the DBT to distribute as a uniform, continuous canopy. Comparison with a previous parallel model shows that if the diameter of the sensor's circular footprint extends to 1.5–2.0 times as large as the total width of the row crop canopy, the footprint effect is minimized, and the ground measured DBT can, theoretically, be used to evaluate the parallel model with negligible error. Finally, validations with a maize canopy demonstrated that the new model performed more accurately than the parallel model to simulate the DBT. Moreover, the FovMod also provides an opportunity to assess the measurement uncertainty caused by some unexpected changes in the sensor's footprint.

ASYMPTOTICS OF SPACING DISTRIBUTIONS AT THE HARD EDGE FOR β-ENSEMBLES

In a previous work [J. Math. Phys.35 (1994) 2539–2551], generalized hypergeometric functions have been used to a give a rigorous derivation of the large s asymptotic form of the general β > 0 gap probability [Formula: see text], provided both βa/2 ∈ ℤ≥0 and 2/β ∈ ℤ+. It is shown how the details of this method can be extended to remove the requirement that 2/β ∈ ℤ+. Furthermore, a large deviation formula for the gap probability Eβ(n; (0, x); ME β, N(λaβ/2e-2βNλ)) is deduced by writing it in terms of the characteristic function of a certain linear statistic. By scaling x = s/(4N)2 and taking N → ∞, this is shown to reproduce a recent conjectured formula for [Formula: see text], βa/2 ∈ ℤ≥0, and moreover to give a prediction without the latter restriction. This extended formula, which for the constant term involves the Barnes double gamma function, is shown to satisfy an asymptotic functional equation relating the gap probability with parameters (β, n, a), to a gap probability with parameters (4/β, n′, a′), where n′ = β(n + 1)/2 - 1, a′ = β(a - 2)/2 + 2.

THE GAP PROBABILITIES OF THE TACNODE, PEARCEY AND AIRY POINT PROCESSES, THEIR MUTUAL RELATIONSHIP AND EVALUATION

We express the gap probabilities of the tacnode process as the ratio of two Fredholm determinants; the denominator is the standard Tracy–Widom distribution, while the numerator is the Fredholm determinant of a very explicit kernel constructed with Airy functions and exponentials. The formula allows us to apply the theory of numerical evaluation of Fredholm determinants and thus produce numerical results for the gap probabilities. In particular we investigate numerically how, in different regimes, the Pearcey process degenerates to the Airy one, and the tacnode degenerates to the Pearcey and Airy ones.

Direct retrieval of canopy gap probability using airborne waveform lidar

Significant progress on quantifying state and trends in vegetation structure in savanna and woodland ecosystems has been made by integrating in situ measurements with lidar datasets. However, large area ground-based monitoring campaigns required for calibration are both costly to maintain, and reduce the generality of results. Estimation of directional gap probability (Pgap) from waveform lidar which is both direct (i.e. physically-based) and minimises or removes requirements for field calibration would be a significant advance for large area sampling. We present a new model for estimating Pgap from small footprint airborne waveform lidar data that accounts for differences in canopy (ρv) and ground (ρg) reflectivity and compare this new method with published discrete return lidar methods. We use lidar surveys acquired at multiple altitudes using RIEGL LMS-Q680i and RIEGL LMS-Q560 waveform systems over a savanna woodland in the Einasleigh Uplands bioregion of northern Queensland, Australia. The waveform model for Pgap was found to fit observed waveform data in cases where the assumption of constant ρv and ρg was satisfied. Pgap estimates from the waveform model were shown to be relatively insensitive to variation in sensor altitude. This was in contrast to other methods of estimating Pgap where differences up to ~0.15 Pgap have been observed. Comparison of lidar-derived Pgap with ground measurements showed the new waveform model produced estimates corresponding to within 5% Pgap. We suggest the waveform model to retrieve ρv/ρg and Pgap is a significant advance in retrieval of canopy structure parameters from small footprint lidar, reducing the need for local calibration, and providing direct estimates of Pgap. If the assumptions of relatively stable ρv/ρg are shown to hold across a greater range of sensor, survey, and canopy structure configurations we suggest this method may have wide practical application for retrieval of Pgap.

A directional thermal radiance model for multiple scattering over surfaces

The multiple scattering effect between heterogeneous non-isothermal surfaces is described rigorously using the configuration factor in engineering thermophysics.Based on the results a directional thermal radiance model is built,and the numerical calculation of the model is discussed.The model is integrated with the gap probability model and applied to a row structure to simulate the change of Directional Brightness Temperature(DBT).The results show that the modeled DBT agrees well with the observed DBT,especially under the condition that the gap probability is very low.It is also shows that the DBT is aggrandized because of the multiple scattering effects,whereas the change of DBT tend to average out.The temperature difference,spatial distribution,emissivity of the components can all lead to the change of DBT.That is to say,the higher of the temperature difference,the craggier of the surface and the lower of the emissivity,the more exquisite DBT changes along with the viewing zenith.The model developed in this paper can explain "hot spot effect" of thermal radiance,and it is confirmed that the existence of "hot spot" is mainly related to the spatial distribution of components' temperature.This model can be used for the study of directional effect on emissivity,the LST retrieval over urban areas and the adjacency effect of thermal remote sensing pixels.

Revisiting the vegetation hot spot modeling: Case of Poisson/Binomial leaf distributions

The accuracy of spaceborne/airborne sensor measurements in the solar domain keeps increasing over time. High resolution, multi-directional and hyperspectral image acquisitions start to be abundant. With regard to the multi-angular remote sensing data, the hot spot, i.e. the exact backscattering direction of direct sunlight together with its neighboring directions, is of special interest. Accurate hot spot models have to be used to adequately simulate the hot spot signature and to allow reliable inversion of multi-angular data. In this paper, we propose a physical hot spot model (Leaf Spatial Distribution based Model, LSDM) assuming that for a given point inside the vegetation to be sunlit (respectively, observed) it should be located within a cylinder free from leaf centers. The cylinder is oriented to the sun (respectively, sensor) direction. Assuming a leaf random, regular or clumped spatial distributions, the gap probabilities in the sun and sensor directions are expressed as a function of these cylinder volumes. Based on the same hypothesis, the bidirectional gap probability is estimated as a function of the total volume of the two cylinders. The evaluation of the needed common volume of two cylinders having different radii is reduced to calculation of some elliptic integrals. Finally, the hot spot signature is estimated based on the bidirectional gap probability distribution. Different model versions with different leaf spatial distribution functions are compared. Particularly, it is shown that compared to the random distribution, the regular (the clumped, respectively) distribution increases (decreases, respectively) the reflectance due to single scattering contribution from foliage. The proposed model is validated using the ROMC web-based tool and its better performance compared to the Semi-Discrete Model and Kuusk's model is confirmed.

Higher order analogues of Tracy–Widom distributions via the Lax method

We study the distribution of the largest eigenvalue in Hermitian one-matrix models when the spectral density acquires an extra number of k − 1 zeros at the edge. The distributions are directly expressed through the norms of orthogonal polynomials on a semi-infinite interval, as an alternative to using Fredholm determinants. They satisfy nonlinear recurrence relations which we show form a Lax pair, making contact to the string literature in the early 1990s. The technique of pseudo-differential operators allows us to give compact expressions for the logarithm of the gap probability in terms of the Painlevé XXXIV hierarchy. These are the higher order analogues of the Tracy–Widom distribution. Using known Bäcklund transformations we show how to simplify earlier equivalent results that are derived from Fredholm determinant theory, valid for even k in terms of the Painlevé II hierarchy.

Relationship between minimum gap and success probability in adiabatic quantum computing

We explore the relationship between two figures of merit for an adiabatic quantum computation process: the success probability P and the minimum gap Δmin between the ground and first excited states, investigating to what extent the success probability for an ensemble of problem Hamiltonians can be fitted by a function of Δmin and the computation time T. We study a generic adiabatic algorithm and show that a rich structure exists in the distribution of P and Δmin. In the case of two qubits, P is to a good approximation a function of Δmin, of the stage in the evolution at which the minimum occurs and of T. This structure persists in examples of larger systems.

Non-linear PDEs for gap probabilities in random matrices and KP theory

Airy and Pearcey-like kernels and generalizations arising in random matrix theory are expressed as double integrals of ratios of exponentials, possibly multiplied with a rational function. In this work it is shown that such kernels are intimately related to wave functions for polynomial (Gelfand–Dickey) reductions or rational reductions of the KP-hierarchy; their Fredholm determinant also satisfies linear PDEs (Virasoro constraints), yielding, in a systematic way, non-linear PDEs for the Fredholm determinant of such kernels. Examples include Fredholm determinants giving the gap probability of some infinite-dimensional diffusions, like the Airy process, with or without outliers, and the Pearcey process, with or without inliers.

The Scaling Limit of the Critical One-Dimensional Random Schrödinger Operator

We consider two models of one-dimensional discrete random Schrodinger operators (H_n \psi)_l ={\psi}_{l-1}+{\psi}_{l +1}+v_l {\psi}_l, {\psi}_0={\psi}_{n+1}=0 in the cases v_k=\sigma {\omega}_k/\sqrt{n} and v_k=\sigma {\omega}_k/ \sqrt{k}. Here {\omega}_k are independent random variables with mean 0 and variance 1. We show that the eigenvectors are delocalized and the transfer matrix evolution has a scaling limit given by a stochastic differential equation. In both cases, eigenvalues near a fixed bulk energy E have a point process limit. We give bounds on the eigenvalue repulsion, large gap probability, identify the limiting intensity and provide a central limit theorem. In the second model, the limiting processes are the same as the point processes obtained as the bulk scaling limits of the beta-ensembles of random matrix theory. In the first model, the eigenvalue repulsion is much stronger.

Measuring gap fraction, element clumping index and LAI in Sierra Forest stands using a full-waveform ground-based lidar

The Echidna Validation Instrument (EVI), a ground-based, near-infrared (1064nm) scanning lidar, provides gap fraction measurements, element clumping index measurements, effective leaf area index (LAIe) and leaf area index (LAI) measurements that are statistically similar to those from hemispherical photos. In this research, a new method integrating the range dimension is presented for retrieving element clumping index using a unique series of images of gap probability (Pgap) with range from EVI. From these images, we identified connected gap components and found the approximate physical, rather than angular, size of connected gap component. We conducted trials at 30 plots within six conifer stands of varying height and stocking densities in the Sierra National Forest, CA, in August 2008. The element clumping index measurements retrieved from EVI Pgap image series for the hinge angle region are highly consistent (R2=0.866) with those of hemispherical photos. Furthermore, the information contained in connected gap component size profiles does account for the difference between our method and gap-size distribution theory based method, suggesting a new perspective to measure element clumping index with EVI Pgap image series and also a potential advantage of three dimensional Lidar data for element clumping index retrieval. Therefore further exploration is required for better characterization of clumped condition from EVI Pgap image series.

Gap probabilities for the cardinal sine

We study the zero sets of random analytic functions generated by a sum of the cardinal sine functions which form an orthonormal basis for the Paley–Wiener space. As a model case, we consider real-valued Gaussian coefficients. It is shown that the asymptotic probability that there is no zero in a bounded interval decays exponentially as a function of the length.

The Canopy Semi-analytic Pgap And Radiative Transfer (CanSPART) model: Formulation and application

We present CanSPART: a model of gap probability (Pgap) based on a simple but flexible geometric vegetation canopy structure, coupled to a one-dimensional radiative transfer scheme, to account for the effects of crown structure and trunks on vertically resolved canopy radiation fluxes. The Pgap component of the model is intended for use in inverting ground-based and airborne gap-frequency data for biometric variables, while the full CanSPART model is intended for application within a one-dimensional multilayer soil-vegetation-atmosphere-transfer model. Our approach to modelling Pgap is novel because it uses an analytic approximation to the crown porosity, which makes it computationally efficient. Further, it can accommodate any distribution of crown and trunk heights and dimensions, allowing the model to be applied to complex canopy structures with multiple layers. The Pgap model is readily rewritten in terms of a clumping factor as a function of height and angle. Simulations of Pgap(θ,z) for idealised canopies compared favourably with those of two other models: the Analytical Clumped Two-Stream (ACTS) model (Ni-Meister et al., 2010) and an adaptation of the Nilson (1999) model. We test the analytic approximation to the crown porosity, also inherent in the Nilson (1999) model, and the applicability of a single clumping factor without angle nor height dependence. Both simplifications are demonstrated to be valid. Lovell et al. (2012, this issue) provide quantitative assessment of the Pgap component of CanSPART against ground-based lidar measurements from sites spanning a range of canopy structures.The radiative-transfer part of the model is an extension of the two-stream scheme, using Pgap as input and requiring the solution of a single matrix equation. In contrast to existing modified two-stream models which use a clumping factor, we account for both the primary effect of clumping (enhanced uncollided flux intensities) and the secondary effect (enhanced interception of scattered radiation). Application of CanSPART to three contrasting Australian field sites show that Pgap, the absorption of radiation by leaves, Qleaf, and albedo are sensitive to the clumping of leaves into crowns. Except for the most sparsely vegetated site, albedo predictions were significantly too high, unless both primary and secondary effects of clumping were included. This highlights the importance of accounting for the enhanced interception of radiation scattered by leaves in a clump (relative to the unclumped case) and suggests why modified two-stream canopy radiative transfer models using a clumping factor approach may systematically underestimate Qleaf and overestimate albedo.

Determinantal identity for multilevel ensembles and finite determinantal point processes

We give a simple algebraic derivation of a useful determinantal identity for multilevel ensembles such as random matrix chains and finite determinantal point processes, with applications to the calculation of point correlators, gap probabilities and Janossy densities.

The Canopy Semi-analytic Pgap And Radiative Transfer (CanSPART) model: Validation using ground based lidar

A canopy structure model based on simple geometric forms and parameterised with plot-scale statistical biometric data is used to predict gap probability (Pgap) profiles. The Semi-analytic Pgap And Radiative Transfer (CanSPART) model is flexibly formulated using distributions of tree height, crown and stem dimensions and can be used for ‘bottom-up’ or ‘top-down’ modelling of gap profiles. Gap probabilities derived from a ground-based scanning lidar are used to validate the output of the model. Direct comparisons of Pgap are made at a range of zenith angles as well as mean foliage profiles calculated from a combination of profiles at multiple angles. Test sites include one pine plantation and several eucalypt woodland sites with varying density, structural complexity, layering and clumping. It is shown that this model performs better in clumped canopies than a simple exponential model and there is agreement with the lidar data except in the near-vertical profiles where spatial variance is not fully accounted for in the measured data. Model-derived canopy gap probabilities compared with observations using a one-sample t-test are significant for 11 out of 12 test sites. Simplification of some of the model inputs was successful at some test sites, but at the most complex site the full model was needed to satisfactorily replicate the shape of the measured gap probability profiles.

Asymptotic forms for hard and soft edge general β conditional gap probabilities

An infinite log-gas formalism, due to Dyson, and independently Fogler and Shklovskii, is applied to the computation of conditioned gap probabilities at the hard and soft edges of random matrix β-ensembles. The conditioning is that there are n eigenvalues in the gap, with n≪|t|, t denoting the end point of the gap. It is found that the entropy term in the formalism must be replaced by a term involving the potential drop to obtain results consist with known asymptotic expansions in the case n=0. With this modification made for general n, the derived expansions — which are for the logarithm of the gap probabilities — are conjectured to be correct up to and including terms O(log|t|). They are shown to satisfy various consistency conditions, including an asymptotic duality formula relating β to 4/β.

The Transition between the Gap Probabilities from the Pearcey to the Airy Process—a Riemann–Hilbert Approach

We consider the gap probability for the Pearcey and Airy processes; we set up a Riemann--Hilbert approach (different from the standard one) whereby the asymptotic analysis for large gap/large time of the Pearcey process is shown to factorize into two independent Airy processes using the Deift-Zhou steepest descent analysis. Additionally we relate the theory of Fredholm determinants of integrable kernels and the theory of isomonodromic tau function. Using the Riemann-Hilbert problem mentioned above we construct a suitable Lax pair formalism for the Pearcey gap probability and re-derive the two nonlinear PDEs recently found and additionally find a third one not reducible to those.

Factorial Moments in Complex Systems

Factorial moments are convenient tools in particle physics to characterize the multiplicity distributions when phase-space resolution (Delta) becomes small. They include all correlations within the system of particles and represent integral characteristics of any correlation between these particles. In this letter, we show a direct comparison between high energy physics and quantitative finance results. Both for physics and finance, we illustrate that correlations between particles lead to a broadening of the multiplicity distribution and to dynamical fluctuations when the resolution becomes small enough. From the generating function of factorial moments, we make a prediction on the gap probability for sequences of returns of positive or negative signs. The gap is defined as the number of consecutive positive returns after a negative return, thus this is a gap in negative return. Inversely for a gap in positive return. Then, the gap probability is shown to be exponentially suppressed within the gap size. We confirm this prediction with data.

Probabilistic alignments with quality scores: an application to short-read mapping toward accurate SNP/indel detection

Recent studies have revealed the importance of considering quality scores of reads generated by next-generation sequence (NGS) platforms in various downstream analyses. It is also known that probabilistic alignments based on marginal probabilities (e.g. aligned-column and/or gap probabilities) provide more accurate alignment than conventional maximum score-based alignment. There exists, however, no study about probabilistic alignment that considers quality scores explicitly, although the method is expected to be useful in SNP/indel callers and bisulfite mapping, because accurate estimation of aligned columns or gaps is important in those analyses. In this study, we propose methods of probabilistic alignment that consider quality scores of (one of) the sequences as well as a usual score matrix. The method is based on posterior decoding techniques in which various marginal probabilities are computed from a probabilistic model of alignments with quality scores, and can arbitrarily trade-off sensitivity and positive predictive value (PPV) of prediction (aligned columns and gaps). The method is directly applicable to read mapping (alignment) toward accurate detection of SNPs and indels. Several computational experiments indicated that probabilistic alignments can estimate aligned columns and gaps accurately, compared with other mapping algorithms e.g. SHRiMP2, Stampy, BWA and Novoalign. The study also suggested that our approach yields favorable precision for SNP/indel calling.