^{1}

^{⋆}

^{2}

^{1}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (

An analytical form of relationship between spectral vegetation indices (VI) is derived in the context of cross calibration and translation of vegetation index products from different sensors. The derivation has been carried out based on vegetation isoline equations that relate two reflectance values observed at different wavelength ranges often represented by spectral band passes. The derivation was first introduced and explained conceptually by assuming a general functional form for VI model equation. This process is universal by which two VIs of different sensors and/or different model equations can be related conceptually. The general process was then applied to the actual case of normalized difference vegetation index (NDVI) from two sensors in a framework of inter-sensor continuity. The derivation results indicate that the NDVI from one sensor can be approximated by a rational function of NDVI from the other sensor as a parameter. Similar result was obtained for the case of soil adjusted VI, enhanced VI, and two-band variance of enhanced VI.

Satellite observations play an important role for global monitoring and characterization of the atmosphere and terrestrial environment in relation with Earth system science. In these studies, multi-year to multi-decadal observation records are often required to detect changes that have occurred or are occurring, and to identify trends and causes of those changes. In order to continue long-term satellite observations effectively, an interdisciplinary approach in the framework of interagency and international cooperation is indispensable.

In the field of ocean studies, a NASA program, Sensor Intercomparison and Merger for Biological and Interdisciplinary Oceanic Studies (SIMBIOS) [

Generation of such long-term observation records by merging multiple sensor data requires a considerable amount of scientific investigations on data continuity and compatibility due to differences in both hardware and software configurations, e.g., sensor/platform characteristics and processing algorithms [

Sensor calibration, including both pre- and post-launch absolute calibration and sensor cross calibrations, has been a major issue and intensively studied by numerous researchers [

More recently, a modeling study was reported by Trinshchenko

One common aspect in the studies by Gitelson and Kaufman [

This study is aimed at analytically providing a theoretical justification on and a functional form of inter-sensor VI relationship. In order to ensure the integrity of the derivation steps, a part of this work is devoted to provide comprehensive overview of our preliminary results [

In the reminder of this paper, we first introduce our basis of VI-to-VI relationships along with a brief review of the vegetation isoline equations in Section 2. We then introduce a derivation process of a relationship between two-band VIs in a general form in Section 3. The generalized process is then extended to the case of three-band VIs in the same section. The next three sections are devoted to explain its applicability by considering actual VI models. The first case is the relationship of two NDVIs computed from two pairs of reflectance measured at different pairs of wavelength and present the derived functional form in Section 4. The derivation is further applied to the soil adjusted VI [

Numerous kinds of VIs have been proposed by many researchers and routinely used for various purposes. Although those indices vary in form to some extent, the ultimate goal is to design an index sensitive only to vegetation biophysical and biochemical attributes [

From the data continuity point of view, finding NDVI-to-NDVI relationships across multiple sensors is the issue to be addressed, especially for continuation of AVHRR broader bands observations with narrower bands of new sensors. Hence, in this study, we assume the conditions that either of two VIs, for which to derive an VI-to-VI relationship, do not have a derived explicit relationship with any biophysical parameters and that the spectral band passes used to compute VIs are not identical among sensors (e.g., AVHRR channel 1

In order to derive a relationship between two VIs of different sensors under such conditions, we need to relate any two variables used in the VI model equations either implicitly (

The isoline equation is written by [_{1} and _{2}, are related by the above equation as a function of biophysical parameters represented by _{v}_{a}_{s}_{1}, _{2}, _{v}, C_{a}_{1}, _{2}, _{v}, C_{a}_{s}^{2}) represents the contributions of the higher order interactions between the two boundary layers, namely, the atmospheric and canopy layer, and the canopy and soil layer. As was discussed in [

In this subsection, we introduce our derivation of inter-relating two 2-band vegetation indices in general form [_{a}_{b}_{a}_{b}_{a}_{b}_{a}_{b}_{a}_{b}_{a}_{a}_{1} and _{a}_{2},
_{b}_{b}_{1} and _{b}_{2},

Our goal is to relate _{a}_{b}

The first isoline equation is introduced to relates _{a}_{1} and _{a}_{2},
_{a}_{2} in _{a}_{a}_{1} with the parameters of vegetation isolines, namely slope and offset, denoted by

The second isoline equation is introduced to eliminate _{b}_{2} from _{a}_{a}_{1}, we represent an isoline equation between _{b}_{1} and _{b}_{2} by
_{b}_{2} in

Solving the above equation for _{b}_{1}, we obtain

The last step is to relate _{a}_{1} and _{b}_{1} used in _{a}_{b}_{b}_{1} (_{a}_{b}

_{a12} depend only on the band characteristics of sensor-a, so they can be determined independently of sensor-b. This can be understood as a relative calibration of two channels of sensor-a (intra-sensor relative calibration). In the same manner, the coefficients of _{b}_{12} can be determined independently of sensor-a. The essence of inter-sensor relationship between the two sensors is then included in the function _{ba}

The derivation steps in the previous subsection can be extended to the case of three-band VIs such as enhanced VI (EVI) [

In this case, _{a}_{a}_{1}, _{a}_{2}, and _{a}_{3},
_{b}_{b}_{1}, _{b}_{2}, and _{b}_{3}, represented by

In addition to

Similarly, for sensor-b, we have the following isoline equation in addition to

The remaining derivation steps are exactly the same as the case of two-band VI, which are also illustrated in

In this section we introduce an application of the above derivation to the case of inter-sensor NDVI relationship [

The NDVI for sensor-a, _{a}_{a}_{1} and _{a}_{2} is then denoted by,
_{a,}_{12} and _{a,}_{12} are the isoline slope and offset, respectively, for the two channels of sensor-a.

Using

Similarly, the NDVI observed by sensor-b, _{b}_{b}_{1} and _{b}_{2} as
_{b}_{2} is eliminated from the equation by using the second isoline equation, or the intra-sensor isoline equation between _{b}_{1} and _{b}_{2},
_{b}_{1} and _{b}_{b}_{1} in terms of _{b}

The third isoline equation that relates _{a}_{1} and _{b}_{1}, represented by _{ba,}_{1}, is
_{a}_{b}

In this section we extend one of our early work [_{i}_{a}

Using the first isoline equation (^{′}_{a}^{′}_{a,}_{12}
_{a,}_{12} remain the same (see

Similarly, using the second isoline equation, the rescaled VI value observed by sensor-b, ^{′}_{b}^{′}_{b,}_{12},

The inter-sensor VI relationship for SAVI/EVI2-like indices is obtained from

In this section we further extend the derivation steps to the case of three-band VI, specifically to relate EVI-like indices from two sensors. The general form of EVI-like model can be written by
_{i}_{a}

The two intra-sensor isoline equations for sensor-a are ^{′}_{a}^{′}_{a,}_{123},
_{a,}_{12} remain the same (see

Similarly, using the two intra-sensor isoline equations for sensor-b, namely ^{′}_{b}^{′}_{b,}_{123},

The inter-sensor VI relationship for EVI-like indices is obtained from

In general, there are three factors to consider for monitoring of vegetation status with optical sensors. One factor is the sensor spatial and temporal resolutions to meet specific monitoring requirements. However, a particular choice of the spatial and temporal resolutions comes with a particular sensor with its specific spectral bandpasses. For example, MODIS is one choice for high temporal resolution monitoring while ASTER is a choice of sensor for high resolution monitoring; however, these sensors’ red and NIR spectral bandpasses are different. Another factor is the index formulation (NDVI, SAVI, etc.), whether to meet required accuracy in capturing biophysical parameters of interest. The last factor is the band selection. While red and NIR spectral bands have been the most widely used combination, a green channel for example may be an alternative to the red as reported in [

If we limit our discussions to the spectral aspect, it is the combination of bandpasses and VI equations that characterize VI-to-VI relationships. The derivation and derived equations relating two VIs presented in this study should be applicable to many possible combinations of various bandpasses and VI equations. The reason is that the general form of relationships,

The NDVI translation equation,

Our results that the

An analytical form of relationship between spectral vegetation indices (VI) has been derived by using three of the vegetation isoline equations for the two-band case, and five isoline equations for the three-band case. The derivation steps were first introduced and explained conceptually by assuming a general functional form of VI model equation. This universal technique of the derivation with the isoline equations was then applied to the case of inter-NDVI relationship to identify functional form that is suitable to approximating the relationship. It was found that a rational function with two linear polynomials is appropriate, from analytical point of view, to approximate the relationship between the NDVIs from two sensors of different BPFs. The derivation technique was also applied to the SAVI/EVI2-like form and EVI-like three-band form of VI model equation. The derived expressions also indicate that the same functional form as the case of NDVI is relevant to model the relationships of such VIs.

Since this study clearly indicates the fact that the inter-VI relationships can be written by a rational function, one can chose such a functional form as a good candidate to model or determine the relationships, in practice, between the VIs from actual satellite data of two different sensors. In this connection, the derived expressions are useful from practical point of view. Further investigations are definitely needed to numerically demonstrate the validity of the derived expressions prior to practical applications of cross calibration of actual satellite data in the framework of continuity and compatibility studies.

This work was supported by JSPS KAKENHI 21510019 (HY) and a NASA grant NNX11AH25G (TM).

Illustration of the derivation steps for the case of two-band VI.

Illustration of the derivation steps for the case of three-band VI. The differences from the case of two-band VI are denoted by red.