Kauffman introduced a fundamental invariant of a virtual knot called the odd writhe. There are several generalizations of the odd writhe, such as the index polynomial and the odd writhe polynomial. In this paper, we define the n-writhe for each non-zero integer n, which unifies these invariants, and study various properties of the n-writhe.

This study describes the derivation of an expression for the relationship between red and near-infrared reflectances, called soil isolines, as an orthogonal concept for the vegetation isoline. An analytical representation of soil isoline would be useful for estimating soil optical properties. Soil isolines often contain a singular point on a dark soil background. Singularities are difficult to model using simple polynomial forms. This difficulty was circumvented in this work by rotating the original axis and employing a vegetation index-like parasite parameter. This approach produced a soil isoline model that could yield any desired level of accuracy based on the use of an index-like parameter. A technique is further introduced for approximating the removal of the parasite parameter from the relationship by truncating the higher-order terms during the derivation steps. Numerical experiments by PROSAIL were conducted to investigate the influence of the truncation errors on the accuracy of the approximated soil isoline equation. The numerical results showed that truncating terms of order greater than two in both bands, yielded negligible truncation errors. These results suggest that the derived and approximated soil isoline equations may be useful in other applications, such as the analysis and retrieval of soil optical properties. © The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

This study introduces an analytical approach to derive a relationship between values of Normalized Difference Vegetation Index (NDVI) obtained from datasets of two sensors with different band passes in the context of inter-sensor crosscalibration of such vegetation index (VI) products. Derivation of the relationship has been performed based on a concept of soil isolines. Starting from the soil isoline equations, an inter-sensor NDVI relationship was derived and represented by a system of equation with a single common parameter. In the derived form of NDVI relationship, all the coefficients were written by the soil reflectances (independent of canopy layer.) The functional form of the relationship becomes rational function of the first-order polynomials, when the soil isoline equations are approximated by the form of first-order polynomials. Those results indicate a functional form suitable to model a relationship of NDVI from two sensors

This study presents a new method that mitigates biases between the normalized difference vegetation index (NDVI) from geostationary (GEO) and low Earth orbit (LEO) satellites for Earth observation. The method geometrically and spectrally transforms GEO NDVI into LEO-compatible GEO NDVI, in which GEO’s off-nadir view is adjusted to a near-nadir view. First, a GEO-to-LEO NDVI transformation equation is derived using a linear mixture model of anisotropic vegetation and nonvegetation endmember spectra. The coefficients of the derived equation are a function of the endmember spectra of two sensors. The resultant equation is used to develop an NDVI transformation method in which endmember spectra are automatically computed from each sensor’s data independently and are combined to compute the coefficients. Importantly, this method does not require regression analysis using two-sensor NDVI data. The method is demonstrated using Himawari 8 Advanced Himawari Imager (AHI) data at off-nadir view and Aqua Moderate Resolution Imaging Spectroradiometer (MODIS) data at near-nadir view in middle latitude. The results show that the magnitudes of the averaged NDVI biases between AHI and MODIS for five test sites (0.016–0.026) were reduced after the transformation (<0.01). These findings indicate that the proposed method facilitates the combination of GEO and LEO NDVIs to provide NDVIs with smaller differences, except for cases in which the fraction of vegetation cover (FVC) depends on the view angle. Further investigations should be conducted to reduce the remaining errors in the transformation and to explore the feasibility of using the proposed method to predict near-real-time and near-nadir LEO vegetation index time series using GEO data.