A derivation of roughness correlation length for parameterizing radar backscatter models

Surface roughness is a key parameter of radar backscatter models designed to retrieve surface soil moisture (θ_S) information from radar images. This work offers a theory‐based approach for estimating a key roughness parameter, termed the roughness correlation length (L _c). The L _c is the length in centimetres from a point on the ground to a short distance for which the heights of a rough surface are correlated with each other. The approach is based on the relation between L _c and h _RMS as theorized by the Integral Equation Model (IEM). The h _RMS is another roughness parameter, which is the root mean squared height variation of a rough surface. The relation is calibrated for a given site based on the radar backscatter of the site under dry soil conditions. When this relation is supplemented with the site specific measurements of h _RMS, it is possible to produce estimates of L _c. The approach was validated with several radar images of the Walnut Gulch Experimental Watershed in southeast Arizona, USA. Results showed that the IEM performed well in reproducing satellite‐based radar backscatter when this new derivation of L _c was used as input. This was a substantial improvement over the use of field measurements of L _c. This new approach also has advantages over empirical formulations for the estimation of L _c because it does not require field measurements of θ_S for iterative calibration and it accounts for the very complex relation between L _c and h _RMS found in heterogeneous landscapes. Finally, this new approach opens up the possibility of determining both roughness parameters without ancillary data based on the radar backscatter difference measured for two different incident angles.     

Local colourings of Cartesian product graphs

A local colouring of a graph G is a function c: V(G)→? such that for each S ? V(G), 2≤|S|≤3, there exist u, v∈S with |c(u)?c(v)| at least the number of edges in the subgraph induced by S. The maximum colour assigned by c is the value χ_?(c) of c, and the local chromatic number of G is χ_?(G)=min {χ_?(c): c is a local colouring of G}. In this note the local chromatic number is determined for Cartesian products G □ H, where G and GH are 3-colourable graphs. This result in part corrects an error from Omoomi and Pourmiri [On the local colourings of graphs, Ars Combin. 86 (2008), pp. 147–159]. It is also proved that if G and H are graphs such that χ(G)≤? χ_?(H)/2 ?, then χ_?(G □ H)≤χ_?(H)+1.     

Aboveground forest biomass derived using multiple dates of WorldView-2 stereo-imagery: quantifying the improvement in estimation accuracy

ABSTRACT

The aim of this study was to investigate the capabilities of two date satellite-derived image-based point clouds (IPCs) to estimate forest aboveground biomass (AGB). The data sets used include panchromatic WorldView-2 stereo-imagery with 0.46 m spatial resolution representing 2014 and 2016 and a detailed digital elevation model derived from airborne laser scanning data. Altogether, 332 field sample plots with an area of 256 m² were used for model development and validation. Predictors describing forest height, density, and variation in height were extracted from the IPC 2014 and 2016 and used in k-nearest neighbour imputation models developed with sample plot data for predicting AGB. AGB predictions for 2014 (AGB₂₀₁₄) were projected to 2016 using growth models (AGB_{Projected_2016}) and combined with the AGB estimates derived from the 2016 data (AGB₂₀₁₆). AGB prediction model developed with 2014 data was also applied to 2016 data (AGB_{2016_pred2014}). Based on our results, the change in the 90^th percentile of height derived from the WorldView-2 IPC was able to characterize forest height growth between 2014 and 2016 with an average growth of 0.9 m. Features describing canopy cover and variation in height derived from the IPC were not as consistent. The AGB₂₀₁₆ had a bias of ?7.5% (?10.6 Mg ha^?1) and root mean square error (RMSE) of 26.0% (36.7 Mg ha^?1) as the respective values for AGB_{Projected_2016} were 7.0% (9.9 Mg ha^?1) and 21.5% (30.8 Mg ha^?1). AGB_{2016_pred2014} had a bias of ?19.6% (?27.7 Mg ha^?1) and RMSE of 33.2% (46.9 Mg ha^?1). By combining predictions of AGB₂₀₁₆ and AGB_{Projected_2016} at sample plot level as a weighted average, we were able to decrease the bias notably compared to estimates made on any single date. The lowest bias of ?0.25% (?0.4 Mg ha^?1) was obtained when equal weights of 0.5 were given to the AGB_{Projected_2016} and AGB₂₀₁₆ estimates. Respectively, RMSE of 20.9% (29.5 Mg ha^?1) was obtained using equal weights. Thus, we conclude that combination of two date WorldView-2 stereo-imagery improved the reliability of AGB estimates on sample plots where forest growth was the only change between the two dates.     

Using MODIS time series data to estimate aboveground biomass and its spatio-temporal variation in Inner Mongolia’s grassland between 2001 and 2011

It is critical to understanding grassland biomass and its dynamics to study regional carbon cycles and the sustainable use of grassland resources. In this study, we estimated aboveground biomass (AGB) and its spatio-temporal pattern for Inner Mongolia’s grassland between 2001 and 2011 using field samples, Moderate Resolution Imaging Spectroradiometer normalized difference vegetation index (MODIS-NDVI) time series data, and statistical models based on the relationship between NDVI and AGB. We also explored possible relationships between the spatio-temporal pattern of AGB and climatic factors. The following results were obtained: (1) AGB averaged 19.1 Tg C (1 Tg = 10¹² g) over a total area of 66.01 × 10⁴ km² between 2001 and 2011 and experienced a general fluctuation (coefficient of variation = 9.43%), with no significant trend over time (R² = 0.05, p > 0.05). (2) The mean AGB density was 28.9 g C m^?2 over the whole study area during the 11 year period, and it decreased from the northeastern part of the grassland to the southwestern part, exhibiting large spatial heterogeneity. (3) The AGB variation over the 11 year period was closely coupled with the pattern of precipitation from January to July, but we did not find a significant relationship between AGB and the corresponding temperature changes. Precipitation was also an important factor in the spatial pattern of AGB over the study area (R² = 0.41, p < 0.001), while temperature seemed to be a minor factor (R² = 0.14, p < 0.001). A moisture index that combined the effects of precipitation and temperature explained more variation in AGB than did precipitation alone (R² = 0.45, p < 0.001). Our findings suggest that establishing separate statistical models for different vegetation conditions may reduce the uncertainty of AGB estimation on a large spatial scale. This study provides support for grassland administration for livestock production and the assessment of carbon storage in Inner Mongolia.     

Estimating montane forest above-ground biomass in the upper reaches of the Heihe River Basin using Landsat-TM data

In this work, the results of above-ground biomass (AGB) estimates from Landsat Thematic Mapper 5 (TM) images and field data from the fragmented landscape of the upper reaches of the Heihe River Basin (HRB), located in the Qilian Mountains of Gansu province in northwest China, are presented. Estimates of AGB are relevant for sustainable forest management, monitoring global change, and carbon accounting. This is particularly true for the Qilian Mountains, which are a water resource protection zone. We combined forest inventory data from 133 plots with TM images and Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) global digital elevation model (GDEM) V2 products (GDEM) in order to analyse the influence of the sun-canopy-sensor plus C (SCS+C) topographic correction on estimations of forest AGB using the stepwise multiple linear regression (SMLR) and k-nearest neighbour (k-NN) methods. For both methods, our results indicated that the SCS+C correction was necessary for getting more reliable forest AGB estimates within this complex terrain. Remotely sensed AGB estimates were validated against forest inventory data using the leave-one-out (LOO) method. An optimized k-NN method was designed by varying both mathematical formulation of the algorithm and remote-sensing data input, which resulted in 3000 different model configurations. Following topographic correction, performance of the optimized k-NN method was compared to that of the regression method. The optimized k-NN method (R² = 0.59, root mean square error (RMSE) = 24.92 tonnes ha^–1) was found to perform much better than the regression method (R² = 0.42, RMSE = 29.74 tonnes ha^–1) for forest AGB retrieval over this montane area. Our results indicated that the optimized k-NN method is capable of operational application to forest AGB estimates in regions where few inventory data are available.     

Root-locus Equations of the Fourth Degree†

The paper is a study of the geometrical properties of root-loci for higher-order systems. Properties of general higher-plane algebraic curves are invoked to provide the broad framework of reference for systematic investigation of root-locus properties. Such an approach affords better integrated and more comprehensive understanding of the nature of the root-paths for higher-order systems.

The discussion is a composite one, encompassing all T(N,M) systems of the order N + M Δ h = 5 and 6, whose equations involve fourth-degree powers in ω: N and M denote the number of poles and zeros, respectively, of the open-loop function G(S) where, S = σ + jω.

The examples cited are illustrative of all significant features of higher-plane algebraic curves as manifest in root-locus shapes. Conditions for existence of such features in root-loci are examined and procedures for their identification and location established.     

A linear model to predict with a multi‐spectral radiometer the amount of nitrogen in winter wheat

The objective was to develop an optimal vegetation index (VI_opt) to predict with a multi‐spectral radiometer nitrogen in wheat crop (kg[N] ha^?1). Optimality means that nitrogen in the crop can be measured accurately in the field during the growing season. It also means that the measurements are stable under changing light conditions and vibrations of the measurement platform. Different fields, on which various nitrogen application rates and seeding densities were applied in experimental plots, were measured optically during the growing season. These measurements were performed over three years. Optical measurements on eight dates were related to calibration measurements of nitrogen in the crop (kg[N] ha^?1) as measured in the laboratory. By making combinations of the wavelength bands, and whether or not the soil factor was taken into account, numerous vegetation indices (VIs) were examined for their accuracy in predicting nitrogen in wheat. The effect of changing light conditions in the field and vibrations of the measurement platform on the VIs were determined based on tests in the field. VI_opt ((1+L)?(R ² _NIR+1)/(R _red+L) with L = 0.45), the optimal vegetation index found, was best in predicting nitrogen in grain crop. The root mean squared error (RMSE), determined by means of cross‐validation, was 16.7 kg[N] ha^?1. The RMSE was significantly lower compared to other frequently used VIs such as NDVI, RVI, DVI, and SAVI. The L‐value can change between 0.16 and 1.6 without deteriorating the RMSE of prediction. Besides being the best predictor for nitrogen, VI_opt had the advantage of being a stable vegetation index under circumstances of changing light conditions and platform vibrations. In addition, VI_opt also had a simple structure of physically meaningful bands.     

A commentary and correction on the article “Pansharpening by exploiting sharpness of the spatial structure”

This study clarifies the implicit potential deficiency caused by the sparse cardinality parameter k in Rong et al. (2014). In addition, k = β × W × M × N (0.9 ≤ β < 1) is suggested to avert this potential deficiency, where β is a ratio controlling the amount of sparse cardinality, W is the number of multispectral bands and M × N is the size of panchromatic image. With the choice of k suggested in this study, the low rank matrix L and sparse matrix S obtained by Go Decomposition (Zhou and Tao 2011) can be iteratively optimized and solved. Thus, instead of choosing k as W × M × N in Rong et al. (2014), the potential deficiency that L is directly obtained as an analytic solution can be averted.     

Real-time core body temperature estimation from heart rate for first responders wearing different levels of personal protective equipment

First responders often wear personal protective equipment (PPE) for protection from on-the-job hazards. While PPE ensembles offer individuals protection, they limit one's ability to thermoregulate, and can place the wearer in danger of heat exhaustion and higher cardiac stress. Automatically monitoring thermal–work strain is one means to manage these risks, but measuring core body temperature (T_c) has proved problematic. An algorithm that estimates T_c from sequential measures of heart rate (HR) was compared to the observed T_c from 27 US soldiers participating in three different chemical/biological training events (45–90 min duration) while wearing PPE. Hotter participants (higher T_c) averaged (HRs) of 140 bpm and reached T_c around 39°C. Overall the algorithm had a small bias (0.02°C) and root mean square error (0.21°C). Limits of agreement (LoA ± 0.48°C) were similar to comparisons of T_c measured by oesophageal and rectal probes. The algorithm shows promise for use in real-time monitoring of encapsulated first responders.     

Estimation of soil dispersivity using soft computing approaches

The accurate estimation of soil dispersivity (α) is required for characterizing the transport of contaminants in soil. The in situ measurement of α is costly and time-consuming. Hence, in this study, three soft computing methods, namely adaptive neuro-fuzzy inference system (ANFIS), artificial neural network (ANN), and gene expression programming (GEP), are used to estimate α from more readily measurable physical soil variables, including travel distance from source of pollutant (L), mean grain size (D ₅₀), soil bulk density (ρ _b), and contaminant velocity (V _c). Based on three statistical metrics [i.e., mean absolute error, root-mean-square error (RMSE), and coefficient of determination (R ²)], it is found that all approaches (ANN, ANFIS, and GEP) can accurately estimate α. Results also show that the ANN model (with RMSE = 0.00050 m and R ² = 0.977) performs better than the ANFIS model (with RMSE = 0.00062 m and R ² = 0.956), and the estimates from GEP are almost as accurate as those from ANFIS. The performance of ANN, ANFIS, and GEP models is also compared with the traditional multiple linear regression (MLR) method. The comparison indicates that all of the soft computing methods outperform the MLR model. Finally, the sensitivity analysis shows that the travel distance from source of pollution (L) and bulk density (ρ _b) have, respectively, the most and the least effect on the soil dispersivity.

     

Optimal control of linear retarded systems to small solutions

This paper deals with necessary optimality conditions for control of general linear retarded systems to small solutions. Here the final states at time T are required to generate small solutions of the uncontrolled system vanishing after T ? h + α, where α ≥ 0 is fixed. This generalizes the fixed final state problem where α = 0, and for α = h includes problems with fixed reduced final state Fx _T , where F is the structural operator of the delay system. Necessary optimality conditions in the form of a maximum principle are valid if a certain space called the ‘ small attainability sub-space ’ is closed in a Sobolev space. A sufficient criterion for this closedness property is indicated. Finally, an example is discussed where the maximum principle for the fixed final state problem is not valid, while the new problem with α = h can be solved by an application of the obtained results.     

On a theorem of Megan and Zabczyk

It will be proved that for any linear infinite-dimensional control system [xdot](t) = Ax(t) + Bu(t) and for any p ? [1, ∞], the implication that ‘complete stabilizabitity ? the T-controllability operator for L ^p -controls is surjective’ holds true provided A generates a strongly continuous group of bounded linear operators. This extends a theorem by Megan and Zabczyk in several directions. In particular, not necessarily separable Banach spaces are allowed, and also the class of controls which are sufficient to ensure exact controllability is restricted.     

Estimating aboveground biomass of a mangrove plantation on the Northern coast of Vietnam using machine learning techniques with an integration of ALOS-2 PALSAR-2 and Sentinel-2A data

ABSTRACT

Aboveground biomass (AGB) of mangrove forest plays a crucial role in global carbon cycle by reducing greenhouse gas emissions and mitigating climate change impacts. Monitoring mangrove forests biomass accurately still remains challenging compared to other forest ecosystems. We investigated the usability of machine learning techniques for the estimation of AGB of mangrove plantation at a coastal area of Hai Phong city (Vietnam). The study employed a GIS database and support vector regression (SVR) to build and verify a model of AGB, drawing upon data from a survey in 25 sampling plots and an integration of Advanced Land Observing Satellite-2 Phased Array Type L-band Synthetic Aperture Radar-2 (ALOS-2 PALSAR-2) dual-polarization horizontal transmitting and horizontal receiving (HH) and horizontal transmitting and vertical receiving (HV) and Sentinel-2A multispectral data. The performance of the model was assessed using root mean square error (RMSE), mean absolute error (MAE), coefficient of determination (R²), and leave-one-out cross-validation. Usability of the SVR model was assessed by comparing with four state-of-the-art machine learning techniques, i.e. radial basis function neural networks, multi-layer perceptron neural networks, Gaussian process, and random forest. The SVR model shows a satisfactory result (R² = 0.596, RMSE = 0.187, MAE = 0.123) and outperforms the four machine learning models. The SVR model-estimated AGB ranged between 36.22 and 230.14 Mg ha^?1 (average = 87.67 Mg ha^?1). We conclude that an integration of ALOS-2 PALSAR-2 and Sentinel-2A data used with SVR model can improve the AGB accuracy estimation of mangrove plantations in tropical areas.     

Optimum Control of a Certain Class of Servomechanism†

Linear control systems governed by the vector matrix differential equation x = A x + B u have been considered. It has been shown how to find the optimum control u so that the system, starting from an initial position x(0), is steered to a state specifying the first p coordinates of the system in time t _o fixed in advance, the values attained by the (n—p) coordinates being immaterial, where n is the dimension of the system. The optimization considered here is with regard to the norm of u supposed to belong to L _m E _r space.     

Factorization of disconjugate higher-order Sturm-Liouville difference operators

Using a recently proved equivalence between disconjugacy of the 2n^th-order difference equation

and solvability of the corresponding Riccati matrix difference equation, it is shown that the equation L(y) = 0 is disconjugate on a given interval if and only if the operator L admits the factorization of the form

L(y)_k+n=M^*(c_kM(y)_k)_k+n,

where M and its adjoint M* are certain n^th-order difference operators and c_k is a sequence of positive numbers.     

Reflectance spectroscopy for the assessment of soil salt content in soils of the Yellow River Delta of China

There has been growing interest in the use of reflectance spectroscopy as a rapid and inexpensive tool for soil characterization. In this study, we collected 95 soil samples from the Yellow River Delta of China to investigate the level of soil salinity in relation to soil spectra. Sample plots were selected based on a field investigation and the corresponding soil salinity classification map to maximize variations of saline characteristics in the soil. Spectral reflectances of air‐dried soil samples were measured using an Analytical Spectral Device (ASD) spectrometer (350–2500 nm) with an artificial light source. In the Yellow River Delta, the dominant chemical in the saline soil was NaCl and MgCl₂. Soil spectra were analysed using two‐thirds of the available samples, with the remaining one‐third withheld for validation purposes. The analysis indicated that with some preprocessing, the reflectance at 1931–2123 nm and 2153–2254 nm was highly correlated with soil salt content (S _SC). In the spectral region of 1931–2123 nm, the correlation R ranged from ?0.80 to ?0.87. In the region of 2153–2254 nm, the S _SC was positively correlated with preprocessed reflectance (0.79–0.88). The preprocessing was done by fitting a convex hull to the reflectance curve and dividing the spectral reflectance by the value of the corresponding convex hull band by band. This process is called continuum removal, and the resulting ratio is called continuum removed reflectance (CR reflectance). However, the S _SC did not have a high correlation with the unprocessed reflectance, and the correlation was always negative in the entire spectrum (350–2500 nm) with the strongest negative correlation at 1981 nm (R = ?0.63). Moreover, we found a strong correlation (R = 0.91) between a soil salinity index (S _SI: constructed using CR reflectance at 2052 nm and 2203 nm) and S _SC. We estimated S _SC as a function of S _SI and S _SI′ (S _SI′: constructed using unprocessed reflectance at 2052 nm and 2203 nm) using univariate regression. Validation of the estimation of S _SC was conducted by comparing the estimated S _SC with the holdout sample points. The comparison produced an estimated root mean squared error (RMSE) of 0.986 (S _SC ranging from 0.06 to 12.30 g kg^?1) and R ² of 0.873 for S _SC with S _SI as independent variable and RMSE of 1.248 and R ² of 0.8 for S _SC with S _SI′ as independent variable. This study showed that a soil salinity index developed for CR reflectance at 2052 nm and 2203 nm on the basis of spectral absorption features of saline soil can be used as a quick and inexpensive method for soil salt‐content estimation.     

Stability and absolute stability of a three-point difference scheme

Consider a three-point difference scheme −h⁻²Δ⁽²⁾y_n + q_n(h)y_n = f_n(h), n ϵ Z = {0, ±1, ±2, …}, where h ϵ (0, h₀], h₀ is a given positive number, Δ⁽²⁾y_n = y_n+1 + y_n−1, f(h) = {f_n(h)}_{n ϵ Z} ϵ L_∞(h), L_∞(h) = {f(h) : ∥f(h)∥_L∞(h) < ∞}, ∥f(h)∥_L∞(h) = sup_nϵZ ∥f_n(h)∥.We assume a unique a priori requirement 0 <- q_n(h) < ∞ for any n ϵ Z and h ϵ (0, h₀]. The main results are a criterion of stability and absolute stability of the difference scheme (1) in the space L_∞(h).     

Bound on the cardinality of a covering of an arbitrary randomness test by frequency tests

We improve a well-known asymptotic bound on the number of monotonic selection rules for covering of an arbitrary randomness test by frequency tests. More precisely, we prove that, for any set S (arbitrary test) of binary sequences of sufficiently large length L, where ∨S∨ ≤ 2 ^L(1?δ), for sufficiently small δ there exists a polynomial (in 1/δ) set of monotonic selection rules (frequency tests) which guarantee that, for each sequence t ∈ S, a subsequence can be selected such that the product of its length by the squared deviation of the fraction of zeros in it from 1/2 is of the order of at least 0.5 ln 2 L[δ/ln(1/δ)](1 ? 2 ln ln(1/δ)/ln(1/δ)).