Derivation of Relationships between Spectral Vegetation Indices from Multiple Sensors Based on Vegetation Isolines

Abstract

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.

1. Introduction

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) [1], has been operating for the integration of information from multiple satellite ocean color sensors. SIMBIOS involves extensive efforts of collection, processing, archival, and documentation of calibration datasets from multiple sensors to ultimately eliminate incompatibilities among sensor data across multiple missions [2]. For land studies, numerous efforts have been made to continue and extend the long-term dataset acquired by the Advanced Very High Resolution Radiometer (AVHRR) sensor series with current and future sensors such as Moderate Resolution Imaging Spectroradiometer (MODIS), Vegetation (VGT), Medium Resolution Imaging Spectrometer (MERIS), and Visible Infrared Imaging Radiometer Suite (VIIRS) onboard the National Polar-orbiting Operational Environmental Satellite System Preparatory Project (NPP) platform and planned for the Joint Polar Satellite System (JPSS).

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 [3–9]. Systematic differences in VI products among sensors [10] are one such problem to be resolved. Instrument spectral band-pass filters (BPFs) are one of the factors that induce systematic differences in reflectance and VIs across sensors where this paper is aimed to contribute.

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 [11–19]. In recent years, several studies were conducted to simulate broad AVHRR bands using a weighted sum of several narrow bands of new sensors for continuity and compatibility, especially, of the NDVI [20]. Gitelson and Kaufman [21] introduced a MODIS NDVI optimization method where the MODIS red-channel was adjusted by adding the MODIS green-channel with the weight optimized to fit the broader AVHRR visible channel. Gao [22] proposed a practical technique of simulating AVHRR NIR reflectance from MODIS by combining the MODIS NIR channel with the water vapor-sensitive MODIS 0.941 μm channel. A similar approach can also be seen in their earlier work [23] for the purpose of water-vapor correction in Hyperspectral data from AVIRIS known as the atmospheric removal (ATREM) program. Gunther and Maier [24] used multiple MERIS bands to simulate AVHRR red and NIR channels with the goal of deriving an AVHRR-compatible NDVI. An assumption underlaying those approaches is that one sensor’s output can be modeled by a weighted sum of several channels.

More recently, a modeling study was reported by Trinshchenko et al. [10] to evaluate the effects of BPFs on surface reflectance and NDVI measured with AVHRR, MODIS, Global Imager, and Vegetation. They showed that surface reflectance and NDVI were both sensitive to the sensor’s BPF across various surface types and atmospheric conditions and concluded that the differences in surface reflectance and NDVI caused by the differences in BPFs are too large to neglect and, thus, need to be adjusted in particular for the purpose of consistent long-term monitoring of the Earth’s environment by multiple satellite sensors. In their study, relationships of two NDVI values measured with different sensors were modeled by a second order polynomial.

One common aspect in the studies by Gitelson and Kaufman [21], Gao [22], and Trinshchenko et al. [10] is that outputs from one sensor can be modeled by another with a certain functional form, i.e., polynomials and weighted sums. Validity of using such functional forms which describe the relationships needs to be investigated solely from a theoretical point of view. This should also serve for better understanding of the fundamentals of multi-sensor VI continuity/compatibility.

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 [25–28]. Our approach is to use vegetation isoline equations which describe the relationships between two reflectances measured at different wavelengths for both the top-of-canopy reflectances [29,30] and the top-of-atmosphere reflectances [31]. Its applications has been reported for several purposes [32–34].

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 [35] and two-band enhanced VI [36] in Section 5, and finally, EVI-like model [37] as a representative of three-band VI in Section 6. Discussions and conclusions are provided in Sections 7 and 8, respectively.

2. Approach Based on Vegetation Isolines

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 [38] such as biomass, leaf area index (LAI), fraction of green coverage, leaf angle distribution function, fraction of absorbed photosynthetically active radiation, and leaf chemical contents. Therefore, in nature, a VI should show a strong correlation to, at least, one of these attributes. Our approach of relating two VIs also assumes such implicit, strong correlations between VIs and biophysical parameters.

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 vs. MODIS band 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 (i.e., about biophysical parameters) or explicitly (e.g., about reflectances). The latter approach is taken here: We relate two VIs based on the analytical relationships of two reflectances used in VI model equation. The relationships of two reflectances at different wavelengths are represented by vegetation isoline equations derived in our previous work.

The isoline equation is written by [31]

$$\begin{array}{l}\rho \left({\lambda }_2,C_v,C_a,C_s\right)=A\left({\lambda }_1,{\lambda }_2,C_v,C_a\right)\rho \left({\lambda }_1,C_v,C_a,C_s\right)+D\left({\lambda }_1,{\lambda }_2,C_v,C_a\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+O^2\left({\lambda }_1,{\lambda }_2,C_v,C_a,C_s\right)\end{array}$$

(1)

Since details of the isoline equation and its derivation were already provided in [31], here we explain the equation briefly. The reflectances ρ at two wavelengths, λ₁ and λ₂, are related by the above equation as a function of biophysical parameters represented by C_v, atmospheric conditions, C_a, and soil conditions, C_s. A(λ₁, λ₂, C_v, C_a) and D(λ₁, λ₂, C_v, C_a) represent slope and offset of the isoline, respectively. Note that these isoline parameters are independent of the soil status, C_s, but functions of a soil line slope and offset. Thus, they are independent of the soil brightness. The last term in the isoline equation (O²) 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 [31], the contributions of those higher order terms are up to 5%, depending on the soil brightness and the optical thickness of the canopy layer. In this paper, for the purpose of brevity, the higher order interaction term will be implicitly included in the offset, D. The isoline equations are the key information that connects two VIs of different sensors and allows us to derive such a relationship analytically. The next section illustrates our approach in a general form. The generalized derivation is then applied to the case of NDVI-to-NDVI relationship and two other cases later in this work.

3. General Form of Inter-VI Relationships

3.1. Case of Two-Band VI

In this subsection, we introduce our derivation of inter-relating two 2-band vegetation indices in general form [25–28]. We do not specify the functional forms of the two VIs to be inter-related (v_a and v_b) and simply represent them as f_a and f_b for v_a and v_b, respectively, where subscripts a and b denote two sensors, namely sensor-a and sensor-b. Moreover the difference can be selections of VIs. For example, v_a and v_b (or f_a and f_b) can be a pair of the same VIs from different sensors (e.g., NDVI vs. NDVI), the same VIs from the same sensor but with different choices of channels (e.g., green-NIR vs. red-NIR VIs), or two different VIs from the same sensor or different sensors (NDVI vs. SAVI). The only limitation posed to this derivation is that the number of channels used for VIs is two. Assume that v_a is a function of two reflectances denoted by ρ_a1 and ρ_a2,

$$v_a=f_a\left({\rho }_{a1},{\rho }_{a2}\right),$$

(2)

and v_b is a function of ρ_b1 and ρ_b2,

$$v_b=f_b\left({\rho }_{b1},{\rho }_{b2}\right).$$

(3)

Our goal is to relate v_a and v_b with the help of vegetation isoline equations, which describe the relationship of any two reflectances. Specifically, the vegetation isoline equations are used to (1) eliminate reflectances from the VI equations and (2) relate two VIs.

The first isoline equation is introduced to relates ρ_a1 and ρ_a2,

$${\rho }_{a2}=g_{a12}\left({\rho }_{a1}\right).$$

(4)

Substituting ρ_a2 in Equation (2), we have

$$v_a=f_a\left({\rho }_{a1},g_{a12}\left({\rho }_{a1}\right)\right)=f_a^{*}\left({\rho }_{a1}\right).$$

(5)

Equation (5) represents the relationship between v_a and ρ_a1 with the parameters of vegetation isolines, namely slope and offset, denoted by A and D in Equation (1), respectively.

The second isoline equation is introduced to eliminate ρ_b2 from Equation (3). Using the same procedure taken to relate v_a and ρ_a1, we represent an isoline equation between ρ_b1 and ρ_b2 by

$${\rho }_{b2}=g_{b12}\left({\rho }_{b1}\right).$$

(6)

Substituting ρ_b2 in Equation (3),

$$v_b=f_b\left({\rho }_{b1},g_{b12}\left({\rho }_{b1}\right)\right)=f_b^{*}\left({\rho }_{b1}\right).$$

(7)

Solving the above equation for ρ_b1, we obtain

$${\rho }_{b1}=f_b^{*-1}(v_b).$$

(8)

The last step is to relate ρ_a1 and ρ_b1 used in v_a and v_b, respectively, by the third isoline equation. Note that this reflectance relationship is the inter-sensor relationship that connects outputs from two sensors.

$${\rho }_{a1}=g_{ba}\left({\rho }_{b1}\right).$$

(9)

Substituting ρ_b1 (Equation (8)) into Equation (9), we obtain

$${\rho }_{a1}=g_{ba}\left(f_b^{*-1}\left(v_b\right)\right).$$

(10)

Finally, substituting the above expression into Equation (5), we have the relationship of v_a and v_b in the following form,

$$v_a=f_a^{*}\left(g_{ba}\left(f_b^{*-1}\left(v_b\right)\right)\right).$$

(11)

Figure 1 summarizes the above derivation steps. The three isoline equations employed during the derivation are highlighted with boxes in the figure. The coefficients of g_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 g_b12 can be determined independently of sensor-a. The essence of inter-sensor relationship between the two sensors is then included in the function g_ba. This inter-sensor relative calibration of the two channels is the key information which relates two VIs. The choice of two channels could be either red channels of the two sensors, or NIR channels.

3.2. Case of Three-Band VI

The derivation steps in the previous subsection can be extended to the case of three-band VIs such as enhanced VI (EVI) [37]. The VI model is represented as a function of three reflectances of different wavelengths (bands) for both sensors. The number of unknown in this case is, thus, larger than the case of two-band VI. As a result, more number of isoline equations are needed. Specifically, two additional isoline equations are needed to eliminate the additional unknowns from the system of equations. The additional equations are an intra-sensor isoline for each sensor. The total number of isoline equations used during the derivation becomes five in the case of three-band VI.

In this case, v_a is a function of ρ_a1, ρ_a2, and ρ_a3,

$$v_a=f_a\left({\rho }_{a1},{\rho }_{a2},{\rho }_{a3}\right).$$

(12)

Likewise, v_b can be written as a function of ρ_b1, ρ_b2, and ρ_b3, represented by

$$v_b=f_b\left({\rho }_{b1},{\rho }_{b2},{\rho }_{b3}\right).$$

(13)

In addition to Equation (4), we need to use the following isoline equation for sensor-a,

$${\rho }_{a3}=g_{a13}\left({\rho }_{a1}\right).$$

(14)

Substituting Equations (4) and (14) into Equation (12), we obtain

$$v_a=f_a\left({\rho }_{a1},g_{a12}\left({\rho }_{a1}\right),g_{a13}\left({\rho }_{a1}\right)\right)=f_a^{*}\left({\rho }_{a1}\right).$$

(15)

Similarly, for sensor-b, we have the following isoline equation in addition to Equation (6),

$${\rho }_{b3}=g_{b13}\left({\rho }_{b1}\right).$$

(16)

Substituting Equations (6) and (16) in Equation (13), we obtain the following expression

$$v_b=f_b\left({\rho }_{b1},g_{b12}\left({\rho }_{b1}\right),g_{b13}\left({\rho }_{b1}\right)\right)=f_b^{*}\left({\rho }_{b1}\right).$$

(17)

The remaining derivation steps are exactly the same as the case of two-band VI, which are also illustrated in Figure 2. In the figure, the differences from the case of two-band VI are highlighted by the color.

4. Inter-Sensor NDVI Relationship

In this section we introduce an application of the above derivation to the case of inter-sensor NDVI relationship [25,26]. Note that, since we focus on the NDVI relationship, we assume that the reflectances of channel-1 and -2 are employed for both sensors, which is reflected in the subscripts of the reflectances for sensor-b.

The NDVI for sensor-a, v_a, is written by

$$v_a=\frac{{\rho }_{a2}-{\rho }_{a1}}{{\rho }_{a2}+{\rho }_{a1}}.$$

(18)

The first isoline equation, which relates ρ_a1 and ρ_a2 is then denoted by,

$${\rho }_{a2}=A_{a,12}{\rho }_{a1}+D_{a,12}=g_{a,12}\left({\rho }_{a1}\right),$$

(19)

where A_a,12 and D_a,12 are the isoline slope and offset, respectively, for the two channels of sensor-a. Equation (19) is an analytical expression of intra-sensor vegetation isolines that can be observed in the red-NIR reflectance space.

Using Equations (18) and (19), $f_a^{*}$ becomes

$$v_a=\frac{d_{a,12}^{-}{\rho }_{a1}+D_{a,12}}{d_{a,12}^{+}{\rho }_{a1}+D_{a,12}}=f_a^{*}\left({\rho }_{a1}\right),$$

(20)

where

$$d_{a,12}^{\pm }=A_{a,12}\pm 1.$$

(21)

Similarly, the NDVI observed by sensor-b, v_b, is written by ρ_b1 and ρ_b2 as

$$v_b=\frac{{\rho }_{b2}-{\rho }_{b1}}{{\rho }_{b2}+{\rho }_{b1}}.$$

(22)

By repeating the same procedure described above, ρ_b2 is eliminated from the equation by using the second isoline equation, or the intra-sensor isoline equation between ρ_b1 and ρ_b2,

$${\rho }_{b2}=A_{b,12}{\rho }_{b1}+D_{b,12}=g_{b,12}\left({\rho }_{b1}\right),$$

(23)

and

$$v_b=\frac{d_{b,12}^{-}{\rho }_{b1}+D_{b,12}}{d_{b,12}^{+}{\rho }_{b1}+D_{b,12}}=f_b^{*}\left({\rho }_{b1}\right),$$

(24)

where

$$d_{b,12}^{\pm }=A_{b,12}\pm 1.$$

(25)

From the relationship between ρ_b1 and v_b, we write ρ_b1 in terms of v_b,

$${\rho }_{b1}=\frac{D_{b,12}\left(1-v_b\right)}{d_{b,12}^{+}{v}_b=d_{b,12}^{-}}=f_b^{*-1}\left(v_b\right).$$

(26)

The third isoline equation that relates ρ_a1 and ρ_b1, represented by g_ba,1, is

$${\rho }_{a1}=A_{ba,1}{\rho }_{b1}+D_{ba,1}=g_{ba,1}\left({\rho }_{b1}\right).$$

(27)

Substituting Equation (26) into (27), we have

$${\rho }_{a1}=g_{ba,1}\left(f_b^{*-1}\left(v_b\right)\right)=\frac{A_{ba,1}{D}_{ba,1}\left(1-v_b\right)}{v_b{d}_{b,12}^{+}-d_{b,12}^{-}}+D_{ba,1}.$$

(28)

From Equations (20) and (28), we obtain a relationship between v_a and v_b as

$$v_a=f_a^{*}\left(g_{ba,1}\left(f_b^{*-1}\left(v_b\right)\right)\right)=\frac{h_1{v}_b-h_2}{h_3{v}_b-h_4},$$

(29)

and

$$h_1=h\left(d_{a,12}^{-},d_{b,12}^{+}\right),$$

(30)

$$h_2=h\left(d_{a,12}^{-},d_{b,12}^{-}\right),$$

(31)

$$h_3=h\left(d_{a,12}^{+},d_{b,12}^{+}\right),$$

(32)

$$h_4=h\left(d_{a,12}^{+},d_{b,12}^{-}\right),$$

(33)

where

$$h\left(\xi ,\eta \right)=\left(D_{ba,1}\xi +D_{a,12}\right)\eta -A_{ba,1}{D}_{b,12}\xi .$$

(34)

Equation (29) is an isoline-based NDVI translation equation with the four coefficients which are actually a function of the vegetation isoline parameters in similar form. The derived translation equation indicates an important fact: its coefficients may vary with optical properties of the atmosphere and vegetation layers. Note that the derivation has been performed on the first-order approximated form of vegetation isoline (Equation (1)); a more accurate translation equation could be a ratio of higher order polynomials.

5. Inter-Sensor Relationship of SAVI/EVI2-Like Two-Band Model

In this section we extend one of our early work [26] (about the case of SAVI [35]) to the case of EVI2-like model [36]. Their VI model equations can be written in a common form as the following

$$v_a=k_3\frac{{\rho }_{a2}-{\rho }_{a1}}{{\rho }_{a2}+k_1{\rho }_{a1}+k_2}.$$

(35)

where k_i’s are the coefficients unique to each index. The derivation is proceeded on the following scaled form, instead of v_a for brevity

$${{\upsilon }^{\prime }}_a=\frac{{\upsilon }_a}{k_3}=\frac{{\rho }_{a2}-{\rho }_{a1}}{{\rho }_{a2}+k_1{\rho }_{a1}+k_2}.$$

(36)

Using the first isoline equation (Equation (19)), v^′_a becomes

$${{\upsilon }^{\prime }}_a=\frac{d_{a,12}^{-}{\rho }_{a1}+D_{a,12}}{{d^{\prime }}_{a,12}^{+}{\rho }_{a1}+{D^{\prime }}_{a,12}}=f_a^{*}\left({\rho }_{a1}\right).$$

(37)

with the new definitions of $d_{a,12}^{\text{'}+}$ and D^′_a,12

$${d^{\prime }}_{a,12}^{+}=A_{a,12}+k_1,$$

(38)

and

$${D^{\prime }}_{a,12}=D_{a,12}+k_2.$$

(39)

Note that $d_{a,12}^{-}$ and D_a,12 remain the same (see Equations (21) and (19), respectively).

Similarly, using the second isoline equation, the rescaled VI value observed by sensor-b, v^′_b, is written by

$${{\upsilon }^{\prime }}_b=\frac{d_{b,12}^{-}{\rho }_{b1}+D_{b,12}}{{d^{\prime }}_{b,12}^{+}{\rho }_{b1}+{D^{\prime }}_{b,12}}=f_b^{*}\left({\rho }_{b1}\right)$$

(40)

with the definitions of $d_{b,12}^{\text{'}+}$ and D^′_b,12,

$${d^{\prime }}_{b,12}^{+}=A_{b,12}+k_1,$$

(41)

and

$${D^{\prime }}_{b,12}=D_{b,12}+k_2.$$

(42)

The inter-sensor VI relationship for SAVI/EVI2-like indices is obtained from Equations (37) and (40), and the third isoline equation (Equation (27)) symbolically as

$${{\upsilon }^{\prime }}_a=f_a^{*}\left(g_{ba,1}\left(f_b^{*-1}({{\upsilon }^{\prime }}_b\right)\right),$$

(43)

and the final form becomes

$${{\upsilon }^{\prime }}_a=\frac{h_1{{\upsilon }^{\prime }}_b-h_2}{h_3{{\upsilon }^{\prime }}_b-h_4},$$

(44)

or

$${\upsilon }_a=k_3\frac{h_1{\upsilon }_b-k_3{h}_2}{h_3{\upsilon }_b-k_3{h}_4},$$

(45)

where

$$h_1=\left(D_{ba,1}{d}_{a,12}^{-}+D_{a,12}\right){d^{\prime }}_{b,12}^{+}-A_{ba,1}{D^{\prime }}_{b,12}{d}_{a,12}^{-},$$

(46)

$$h_2=\left(D_{ba,1}{d}_{a,12}^{-}+D_{a,12}\right)d_{b,12}^{-}-A_{ba,1}{D}_{b,12}{d}_{a,12}^{-},$$

(47)

$$h_3=\left(D_{ba,1}{d^{\prime }}_{a,12}^{+}+{D^{\prime }}_{a,12}\right){d^{\prime }}_{b,12}^{+}-A_{ba,1}{D^{\prime }}_{b,12}{d^{\prime }}_{a,12}^{+},$$

(48)

$$h_4=\left(D_{ba,1}{d^{\prime }}_{a,12}^{+}+{D^{\prime }}_{a,12}\right)d_{b,12}^{-}-A_{ba,1}{D}_{b,12}{d^{\prime }}_{a,12}^{+}.$$

(49)

6. Inter-Sensor Relationship of EVI-Like Three-Band Model

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

$${\upsilon }_a=k_3\frac{{\rho }_{a2}-{\rho }_{a1}}{{\rho }_{a2}+k_1{\rho }_{a1}+k_4{\rho }_{a3}+k_2},$$

(50)

where k_i’s are the coefficients unique to each EVI-like model. By following the similar steps explained in the previous section, the derivation is proceeded using the scaled form, instead of v_a for brevity,

$${{\upsilon }^{\prime }}_a=\frac{{\upsilon }_a}{k_3}=\frac{{\rho }_{a2}-{\rho }_{a1}}{{\rho }_{a2}+k_1{\rho }_{a1}+k_4{\rho }_{a3}+k_2}.$$

(51)

The two intra-sensor isoline equations for sensor-a are Equations (19) and a newly defined relationship

$${\rho }_{a3}=A_{a,13}{\rho }_{a1}+D_{a,13}=g_{a,13}({\rho }_{a1}).$$

(52)

Using the two isoline equations, v^′_a becomes

$${{\upsilon }^{\prime }}_a=\frac{d_{a,12}^{-}{\rho }_{a1}+D_{a,12}}{{d^{\prime }}_{a,123}^{+}{\rho }_{a1}+{D^{\prime }}_{a,123}}=f_a^{*}({\rho }_{a1}),$$

(53)

with the new definitions of $d_{a,123}^{\text{'}+}$ and D^′_a,123,

$${d^{\prime }}_{a,123}^{+}=A_{a,12}+k_1+k_4{A}_{a,13},$$

(54)

and

$${D^{\prime }}_{a,123}=D_{a,12}+k_2+k_4{D}_{a,13}.$$

(55)

Note that $d_{a,12}^{-}$ and D_a,12 remain the same (see Equations (21) and (19), respectively).

Similarly, using the two intra-sensor isoline equations for sensor-b, namely Equations (23) and the following newly defined isoline

$${\rho }_{63}=A_{b,13}{\rho }_{b1}+D_{b,13}=g_{b,13}({\rho }_{b1}),$$

(56)

the rescaled VI value observed by sensor-b, v^′_b, is written by

$${{\upsilon }^{\prime }}_b=\frac{d_{b,12}^{-}{\rho }_{b1}+D_{b,12}}{{d^{\prime }}_{b,123}^{+}{\rho }_{b1}+{D^{\prime }}_{b,123}}=f_b^{*}({\rho }_{b1})$$

(57)

with the definitions of $d_{b,123}^{\text{'}+}$ and D^′_b,123,

$${d^{\prime }}_{b,123}^{+}=A_{b,12}+k_1+k_4{A}_{b,13},$$

(58)

and

$${D^{\prime }}_{b,123}=D_{b,12}+k_2+k_4{D}_{b,13}.$$

(59)

The inter-sensor VI relationship for EVI-like indices is obtained from Equations (53) and (57), and the inter-sensor isoline equation (Equation (27)). The final form becomes

$${{\upsilon }^{\prime }}_a=\frac{h_1{{\upsilon }^{\prime }}_b-h_2}{h_3{{\upsilon }^{\prime }}_b-h_4},$$

(60)

or

$${\upsilon }_a=k_3\frac{h_1{\upsilon }_b-k_3{h}_2}{h_3{\upsilon }_b-k_3{h}_4},$$

(61)

where

$$h_1=\left(D_{ba,1}{d}_{a,12}^{-}+D_{a,12}\right){d^{\prime }}_{b,123}^{+}-A_{ba,1}{D^{\prime }}_{b,123}{d}_{a,12}^{-},$$

(62)

$$h_2=\left(D_{ba,1}{d}_{a,12}^{-}+D_{a,12}\right)d_{b,12}^{-}-A_{ba,1}{D}_{b,12}{d}_{a,12}^{-},$$

(63)

$$h_3=\left(D_{ba,1}{d^{\prime }}_{a,123}^{+}+{D^{\prime }}_{a,123}\right){d^{\prime }}_{b,123}^{+}-A_{ba,1}{D^{\prime }}_{b,123}{d^{\prime }}_{a,123}^{+},$$

(64)

$$h_4=\left(D_{ba,1}{d^{\prime }}_{a,123}^{+}+{D^{\prime }}_{a,123}\right)d_{b,12}^{-}-A_{ba,1}{D}_{b,12}{d^{\prime }}_{a,123}^{+}.$$

(65)

7. Discussion

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 [21].

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, Equation (11), does not assume any particular choices in the three factors (i.e., sensors, VI equations, and spectral bands).

The NDVI translation equation, Equation (29), was derived as an example application of Equation (11). Equation (29) tells the relationship between two NDVIs at some constant biophysical and atmospheric condition, i.e., a constant LAI value and a constant atmospheric optical thickness. This is because the h functions in the NDVI translation equation are a function of LAI and atmospheric optical thickness. If the NDVI shows a one-to-one relationship with LAI, or in other words if NDVI is free from soil influences, the h functions become a function of the NDVI and atmospheric optical thickness. If the atmosphere is constant in addition, the translation equation further reduces to be a function only of NDVI. In this case, the final form of the translation equation can be approximated as a ratio of two polynomials with NDVI (of sensor-b). However, since NDVI is influenced by the soil brightness to some extent, any translation equation that uses NDVI as the only parameter cannot eliminate variations caused by the soil brightness. From practical point of view, if one can correlate soil brightness-induced variations, the ratio of two polynomials may be one choice of the form of translation equation. The orders of these polynomials and their coefficients should be determined using actual datasets.

Our results that the h functions which determine the actual form of the translation of two VIs depend on LAI (and atmospheric conditions as well) have several important implications. First, VI-to-VI relationships change with soil brightness, biophysical, and atmospheric conditions. Even after the accurate and precise atmospheric correction, the other two parameters remain undetermined. It should be noted that the relationship derived in this work as a VI translation equation eliminates the soil brightness effect by utilizing the vegetation isoline equation. However, we still need to determine at least one more parameter, such as LAI, to minimize the variations in VI relationships. Although it is desirable to have an estimation of LAI, such an estimation is often done by using the VI itself. Moreover, the purpose of computing VI is mainly to estimate such a biophysical parameter. Therefore, it is quite reasonable to assume that LAI estimation is not available before the VI computation. As a consequence, it sounds impossible to eliminate the variance by the soil brightness. Nevertheless, there is a possibility to do so by using an independent parameter alternative to LAI. One requirement for such a parameter is using another vegetation index than the target VI which shows stronger correlation with LAI. For example, utilizing one additional VI, the h functions could be approximated by the index. This possibility should be investigated solely.

8. Conclusions

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.