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Keywords:

  • Kramers-Kronig analysis;
  • leaf absorption;
  • leaf optical modeling;
  • leaf radiative transfer;
  • leaf refractive index;
  • variational function fitting

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. KK Relations of Complex Refractive Index
  5. 3. A Testing With Pure Liquid Water
  6. 4. VFF-KK Analysis of Leaf Refractive Index
  7. 5. Analysis Results and Discussion
  8. 6. Summary and Concluding Remarks
  9. Acknowledgments
  10. References
  11. Supporting Information

[1] Kramers-Kronig dispersion relations are the inherent cross-constraints on the real and imaginary parts of the complex refractive index of dielectric materials. These relations are utilized in this study for estimating leaf refractive index from leaf absorption spectra. A mixing model is developed for calculating the leaf refractive index with respect to the variation of leaf mass compositions and absorptive properties. The model is based on the variational function fitting method of Kramers-Kronig analysis. The validity of the model is analyzed with pure liquid water measurements and well-established leaf optical measurements. It is found that the model has a very good performance in modeling the spectral details of leaf refractive index and provides an efficient physical framework for improving and extending the leaf-level optical property model.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. KK Relations of Complex Refractive Index
  5. 3. A Testing With Pure Liquid Water
  6. 4. VFF-KK Analysis of Leaf Refractive Index
  7. 5. Analysis Results and Discussion
  8. 6. Summary and Concluding Remarks
  9. Acknowledgments
  10. References
  11. Supporting Information

[2] In canopy radiative transfer models, leaf dielectric constant εleafis a key parameter coupling leaf bio-traits with leaf radiative properties. The biophysical and biochemical characteristics of leaves, such as specific leaf area (SLA), leaf thickness and leaf water content, may directly or indirectly contribute to the spectral signatures of leaf radiation through modulating leaf bulk dielectric property. Normally,εleafis not a constant scalar, but a frequency-dependent complex variable, inline image, which is subject to the constraint of Kramers-Kronig (KK) relations [Lucarini et al., 2005]. The KK relations state that

  • display math
  • display math

where the symbol P denotes Cauchy integration in the complex variable space; ωis the angular frequency. It is worth noting that the only mathematical precondition in deriving the KK relations is the non-singularity (analytic and bounded) of inline image in the upper half plane of the complex space, which is physically guaranteed by the causality property of dielectrics in the real time space [Nussenzveig, 1972]. Similar KK relation pair holds for the complex refractive index inline image in that inline image. The imaginary part k(ω) is the extinction coefficient of dielectrics in physics, which characterizes the absorption and scattering property of dielectric materials. n(ω) is the ordinary refractive index. Unless the term “complex refractive index” is used, refractive index is referred to as n(ω) hereafter.

[3] The KK relations imply that the real and imaginary parts of inline image or inline image are analytically related. Therefore they cannot change in an arbitrary manner individually in the complex space. This constraint allows one to construct the spectra of one part from the spectra measurement of the other. If the measurements of both parts of inline image and inline imageare available, one may analyze the data quality and self-consistency using the KK relations. Nevertheless, a practical problem of applying KK relations is that measurement data is usually available over limited spectrum range, while KK relations require the information over whole spectral space. The past decades have seen extensive leaf optical studies in the range of 0.4 μm to 2.5 μm largely because the spectrophotometric instrumentation is only readily available in this region. Fortunately, solar incident radiation also predominantly occurs in this range. And according to equation (1), the spectral details of high-frequency end and local spectral “window” have the very crucial weight for successful KK analysis. The spectrum too far from this range will have very little impact on the KK integration of this range. It is expected that a simple extrapolation would suffice the KK analysis in the range of 0.4 to 2.5 μm.

[4] In this study, we explore the application of the KK relations for estimating of leaf refractive index n(ω) from well-calibrated leaf absorption measurements in the range of 0.4 μm to 2.5 μm. Although not being specific for the improvement of the widely used PROSPECT leaf optical properties model [Jacquemoud and Baret, 1990; Jacquemoud et al., 2009], this study is stimulated by the fact that the specification of n(ω) is not as straightforward as that of k(ω) [Allen et al., 1969]; and a static n(ω) spectrum has been simply used in the PROSPECT model. Physically, n(ω) should not be a constant spectrum since plant leaves are not pure dielectric media. Leaf mass composition may vary significantly among different plant types. In the PROSPECT model, k(ω) is the sum of the specific absorption coefficients of all absorbing constituents (e.g., leaf water, dry matter, pigments) weighed by their respective mass contents. Thus, k(ω) varies implicitly with respect to leaf types. The KK relations between n(ω) and k(ω) should be one essential mechanism for the PROSPECT model to properly account for the interspecies changes of leaf optical property.

[5] In the following sections, the KK relation pair of complex refractive index is first introduced, which is followed by the numerical implementation using the variational function fitting (VFF) method [Kuzmenko, 2005]. The validity of the KK numerical implementation is demonstrated with the measurements of pure liquid water, where both refractive index and absorption data are available from the literature. The VFF-KK algorithm is then coupled with the PROSPECT model to estimate the leaf refraction index n(ω) from the model predefined leaf absorption coefficient data set. Performance of the KK-constrained PROSPECT model is analyzed with the Leaf Optical Properties EXperiment 93 (LOPEX) data.

2. KK Relations of Complex Refractive Index

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. KK Relations of Complex Refractive Index
  5. 3. A Testing With Pure Liquid Water
  6. 4. VFF-KK Analysis of Leaf Refractive Index
  7. 5. Analysis Results and Discussion
  8. 6. Summary and Concluding Remarks
  9. Acknowledgments
  10. References
  11. Supporting Information

2.1. KK Relations in Continuous Spectrum Space

[6] In dielectric material physics, complex dielectric constant (relative permittivity) is a linear function of the susceptibility of the material, which in turn is the Fourier transformation of the response function of the dielectric material in the real time space. Therefore, inline image must satisfy inline image, where inline image is the conjugate function of inline image. Since inline image, inline image. Thus, the real and imaginary parts of inline image are even and odd functions, respectively, that is, n(ω) = n(−ω) and κ(ω) = − k(−ω). This particular property leads to further simplification of the KK relations of equations (1) and (2). By multiplying the integrands of equations (1) and (2) with inline image, and replacing the real and imaginary parts by n(ω) and k(ω), one may have the following KK relations of complex refractive index:

  • display math
  • display math

Equation (3) and (4) are based on angular frequency space. One may easily have the equivalent equation system on wavelength space with ω = 2πc/λ, where c is the speed of light in vacuum.

2.2. KK Analysis by Variational Function Fitting (VFF) Method

[7] In practical calculations, the data range is unavoidably discrete and finite. The divergence at ω′ = ωwill cause trouble in numerical integration. In order to reduce the errors caused by the finite spectrum, variants of the KK relations were introduced, e.g., FFT-based KK analysis [Bortz and French, 1989], singly subtractive KK analysis [Ahrenkiel, 1971; Bachrach and Brown, 1970] and multiply subtractive KK [Palmer et al., 1998]. The basic idea of subtractive KK analysis is to remove the n(ω) values at one or more anchor (reference) frequencies from the KK integration. The n(ω) values at anchor frequency points come from independent measurements or other previous studies. Although the subtractive form of a KK integral converges more rapidly than a conventional KK relation, the requirement to have anchor-frequency n(ω) values is not practicable for the study of leaf optics. As argued above, leaf refractive index varies with respect to leaf types and is generally hard to accurately measure. It is thus desirable to have a numerical KK analysis that allows the calculation of n(ω) just with k(ω) data, meanwhile with good convergence property. The variational function fitting method [Kuzmenko, 2005] provides efficient solutions on this aspect. The central idea of VFF method is that the analytical form of k(ω) is parameterized independently at each anchor frequency, while n(ω) is dynamically coupled to k(ω) by the KK transformation.

[8] The choice of functional form of k(ω) is somewhat arbitrary, but must sufficiently account for the ‘local’ spectral weight near ω.It was demonstrated that a simple triangular line-shape function which is nonzero only inside a small region adjacent toω works well if the anchor points are dense enough [Kuzmenko, 2005]. The triangular line-shape function was defined as

  • display math

where ωi is the ith anchor frequency, Substituting equation (5) into equation (1) and making finite integration in the rang [ωi−1, ωi+1], one may have

  • display math

where,

  • display math

Although not explicitly mentioned by Kuzmenko [2005], this method is virtually a Rayleigh–Ritz method in direct variational analysis, where a function is approximated by a linear combination of elemental (base) functions of the same linear space. This method may yield solutions even if an analytical form for the true solution is intractable.

[9] Equation (6) is the contribution of the element δk which is centered on the anchor frequency ωi. To calculate the contribution of all the elements δk to the refractive index at ω, one needs sum up over all the anchor frequencies. In a discrete case, it follows that

  • display math

In the case that both n(ω) and k(ω) are unknown, equations (5)(8) may be coupled with radiative models to inverse n(ω) and k(ω) from reflectance or transmittance measurements. If k(ω) is already known, equation (8) may be used to calculate n(ω) directly from k(ω).

[10] The drawback of the above analysis system is that it does not take into account the spectral weight outside the frequency range [ω1ωN]. But the same problem of extrapolations exists in all the conventional KK analysis, which actually does not have any universal solution. In Kuzmenko [2005], two-step fitting was suggested to address the extrapolation issue. The first-step fitting uses formula-defined functionnmod(ω), e.g., Drude-Lorentz model, to match the big-picture of experimental spectra. The second-step applies the VFF methodnvar(ω) to fit the fine spectral details. The general solution is

  • display math

As shown in this study, if the considered range is not large and nvar(ω) is able to describe the local spectral details properly, a simple nmod(ω) form, e.g., a linear function or constant value may suffice to account for the contribution from outside the data range.

3. A Testing With Pure Liquid Water

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. KK Relations of Complex Refractive Index
  5. 3. A Testing With Pure Liquid Water
  6. 4. VFF-KK Analysis of Leaf Refractive Index
  7. 5. Analysis Results and Discussion
  8. 6. Summary and Concluding Remarks
  9. Acknowledgments
  10. References
  11. Supporting Information

[11] The optical properties of water have been under extensive studies because of the water involvement in various radiative transfer processes. Liquid water is also one of the major solar radiation absorbers of fresh plant leaves. The high absorptions of leaf in the infrared beyond 1.3 μm are mainly by leaf water [Allen and Richardson, 1968]. As shown in numerous studies, the absorption spectrum structure of leaf water is not much different from that of pure liquid water. For fresh leaves, the radiative absorption by leaf water may actually serve as the “backbone” of the general absorption spectrum of a leaf. It is therefore natural for us to start the VFF-KK analysis with pure liquid water data.

[12] The optical absorption properties of water have been well documented in the literature. Except for the visible range, there are no significant discrepancies among different sets. The data set by Hale and Querry [1973] covered the optical constants of water from visible to far infrared range (0.2 μm–200 μm). Both n(ω) and k(ω) are available from Hale and Querry [1973], which enable us to test the efficiency of the VFF method in a fairly wide spectral range. Since the range of our interest is visible to middle-wave infrared (0.4 μm–2.5 μm) in this analysis, the extra measurements, i.e., the data of 2.5 μm–200 μm, may be used to reduce the uncertainty of the considered range. Although the extended range is also limited, equation (3) indicates that the k(ω) spectrum contributes little to the calculation of n(ω) if anchor frequency ω′ is far from ω. n(ω) and k(ω) values are digitized directly from Table 1 of Hale and Querry [1973], and used in this testing analysis.

[13] The calculation of nvar(ω) from water k(ω) data is straightforward following equations (5) through (8). To specify nmod(ω), one needs to assume the functional form and determines the function parameter(s) from other measurements, e.g., the reflectance and/or transmittance. Since n(ω) is already available in the testing data set, the n(ω) data can be used as the true value to determine the functional parameter(s). Without a prior knowledge, we started with the assumption nmod(ω) = constant, and inversed the constant through the Shuffled Complex Evolution optimization [Duan et al., 1988, 1993, 1994]. The Shuffled Complex Evolution (SCE) algorithm is a general purpose global optimization method widely used in hydrometeorology community. The objective (or merit) function of the optimization was the mean square root of the difference between nKK(ω) and the true values. Although the considered range is limited to 0.4 μm–2.5 μm in this study, different cutting ranges were tested to evaluate the impact of limited data range on the performance of VFF method in this testing analysis.

[14] It turned out that nmod is around 1.34, which is the mean water refractive index. As shown in Figure 1, larger deviation may be seen at longer-wave (low frequency) end. Yet it may be avoided by using some sophisticated function ofnmod(ω) or by using extended k(ω) data. For instance, Curve 1 in Figure 1deviates very much from the true-value curve in the range of 0.4 μm–2.5 μm, which just catches the mean water refractive index value; yet as the sub-range of Curve 2,nKK(ω) matches the true-value curve very well in the range of 0.4 μm–2.5 μm. The same feature bears for Curve 2 when it is taken as the sub-range of Curve 3. In general, the extended measurement beyond the considered range should be more than 2.0 μm to ensure the accuracy of the considered range.

image

Figure 1. The specific absorption coefficient (cm−1) and the refractive index of pure liquid water. The two solid lines are the data by Hale and Querry [1973]. The refractive index by Hale and Querry serves as the true value. Dotted lines are the refractive index by VFF-KK analysis with different cutting ranges: (1) 0.4–2.5 μm, (2) 0.4–4.5 μm, and (3) 0.4–7.5 μm.

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4. VFF-KK Analysis of Leaf Refractive Index

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. KK Relations of Complex Refractive Index
  5. 3. A Testing With Pure Liquid Water
  6. 4. VFF-KK Analysis of Leaf Refractive Index
  7. 5. Analysis Results and Discussion
  8. 6. Summary and Concluding Remarks
  9. Acknowledgments
  10. References
  11. Supporting Information

4.1. Leaf Optical Property and Leaf-Level Radiative Transfer Modeling

[15] In comparison with leaf refractive index, n(ω), leaf absorption properties have been well documented in the literature. In the visible range (0.4 μm–0.7 μm), leaf pigment components, primarily the chlorophylls, dominate leaf absorption. The high absorptions of leaves in the infrared beyond 1.3 μm are mainly by leaf water [Allen and Richardson, 1968]. There are three strong water absorption bands in the visible to mid-wave infrared range, with the peaks around 1.45 μm, 1.95 μm and 2.95 μm, respectively. The dry matter of leaf cells (e.g., cellulose, lignin, waxes) also contribute to leaf absorption, particularly at longer wavelengths, but generally secondary to the influence of leaf water. At wavelengths beyond 1.3 μm, the absorption features of leaf dry matter may be overshadowed by water absorption [Kumar et al., 2001; Kokaly and Clark, 1999]. The dry matter absorption is usually measured using dry foliage in laboratory studies, where both the system error and instrumental precision may be easily controlled.

[16] Although the total absorption of a leaf may differ among individuals, the basic absorption features of all leaves appear similar. By contrast, the specification of leaf refractive index n(ω) is not so straightforward. Both leaf absorption and scattering impact the computation of n(ω) [Allen et al., 1969]. The multiple scattering within a leaf is related to the leaf intercellular air spaces, which depends on the internal structures of the leaf, and varies with the leaf type. To account for the scattering mechanism at leaf level, one may resort to a leaf radiative transfer model (RTM) with a proper conceptual model of leaf structures. Provided that the leaf-level RTM handles the internal scattering properly,n(ω) may be readily calculated from the leaf effective absorption.

[17] The PROSPECT model is an open-source leaf radiative transfer model [Jacquemoud and Baret, 1990; Jacquemoud et al., 2009]. It has been widely used over the past two decades to simulate the leaf-level directional-hemispherical reflectance and transmittance, and to retrieve leaf bio-parameters from leaf radiance measurements. PROSPECT is based on Allen's Plate model [Allen et al., 1969, 1970] where the leaf is represented as N staked layers with the mass absorption coefficients of leaf individual constituents predefined. The layered structure is used to account for the intercellular scattering. In the forward mode, the model needs four input variables [Jacquemoud and Baret, 1990]: the structure parameter N, which specifies the average number of air-cell interfaces; the chlorophyll a + b content, the equivalent water thickness and leaf dry mass content. The model is usually used in the visible and near infrared range of 0.4 μm to 2.5 μm. A recent effort by Gerber et al. [2011] has extended the model's application to 5.7 μm. When coupled with an optimization program, e.g., Shuffled Complex Evolution, PROSPECT may be used to inverse model parameters from leaf reflectance and transmittance measurements.

4.2. Mixture Model of Leaf Effective Refractive Index

[18] In PROSPECT, the mass absorption coefficients of leaf individual constituents have been defined and validated from various data sources. The application of the model has been limited to the range of 0.4 μm to 2.5 μm due to data availability and the enhanced complexity of scattering in the thermal infrared range. In a recent effort, Gerber et al. [2011] extended the model's application to 5.7 μm with the construction of dry mass absorption coefficient and the leaf refractive index in the extended range. It is not the purpose of this study to construct both the absorption coefficients of leaf individual absorbers and the refractive index with VFF-KK analysis although this may be the general application of VFF-KK analysis. Instead, we use model predefined absorption coefficients to reevaluate the refractive index, which was static in the model but should be dynamically determined with respect to leaf mass compositions. In PROSPECT, the bulk or effective leaf absorption coefficient is the summation of the absorption coefficients of all the leaf component absorbers weighted by their respective mass content:

  • display math

where Ci is the mass content per unit leaf area of component absorber i, αi is the corresponding specific absorption coefficient. Note that α(ω) is mass or Lambert absorption coefficient. Since α(ω) = 2ωk(ω)/c, it follows that

  • display math

where c is the speed of light in vacuum. For a specific leaf, the mass content Ci is a constant. It is easy to prove with the substitution of equation (10) into equation (3) that

  • display math
  • display math

where nvar,i(ω) and nmod,i(ω) are the VFF-KK analysis of leaf componenti.Consequently, one may perform VFF-KK analysis for each leaf component separately, and use theequations (12) and (13) to calculate the effective leaf refractive index.

[19] In general, the mass contents of leaf pigments are much smaller than leaf water content and leaf cell material mass. n(ω) mainly depends on leaf water and leaf cell materials. The VFF-KK analysis of leaf cell materials is similar to that of pure liquid water inSection 3. nvar(ω) of leaf cell materials is calculated from equations (5)(8), and nmod(ω) of leaf cell material is assumed to be a constant value. We inverse the constant value nmod with PROSPECT + SCE from leaf reflectance and transmittance measurements.

4.3. LOPEX Data

[20] The Leaf Optical Properties EXperiment 93 (LOPEX) database was established by the European Commission Joint Research Centre (JRC) in 1993. Besides the leaf reflectance and transmittance, a variety of biochemical constituents are available from the database, including lignin, proteins (nitrogen), cellulose and starch, as well as chlorophyll and foliar water. Leaf mass contents may be directly derived from these biochemical data. About 260 leaf samples representative of 60 species were selected from the database for the VFF-KK analysis. Each species has about 4 samples with variable leaf thickness and mass contents. The leaf structure parameterNand the functional parameters of VFF-KK analysis are inversed from the leaf reflectance and transmittance of LOPEX.

5. Analysis Results and Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. KK Relations of Complex Refractive Index
  5. 3. A Testing With Pure Liquid Water
  6. 4. VFF-KK Analysis of Leaf Refractive Index
  7. 5. Analysis Results and Discussion
  8. 6. Summary and Concluding Remarks
  9. Acknowledgments
  10. References
  11. Supporting Information

5.1. Leaf Refractive Index Constant and Structure Parameter

[21] There are two functional parameters in our VFF-KK analysis, which are thenmod of liquid water, nmodw, and the nmod of leaf dry matter, nmodd, respectively. Since the absorption property of leaf liquid water is not much different from pure liquid water, nmodw is set to 1.34 in performing the inversion of other parameters. The constant nmoddneeds to be inverted from leaf radiometric measurements. With the internal scattering handled by PROSPECT model, the leaf-level optic property, e.g., reflectance and transmittance, relies on the setting of bothnmodd and the internal scattering parameter N. Therefore, N and nmodd must be inversed simultaneously from leaf reflectance and transmittance measurements.

[22] The inversion was performed by optimizing the PROSPECT model simulations against the LOPEX reflectance and transmittance measurements. Summarized in Table 1 are the N and nmodd values inversed from the selected leaf species. Both N and nmodd are found to vary among leaf species. The variation range of nmodd is between 1.4 and 1.7, with the mean value around 1.64. The range of N is between 1.4 and 3.0. In general, the N values of thicker leaves are larger than those of thinner leaves, but large N may be found with thinner leaves, too.

Table 1. LOPEX Leaf Samples Used in VFF-KK Analysisa
Leaf SpeciesLeaf Z(μm)Nnmod
  • a

    There are totally 258 samples of 60 leaf species. Averaged leaf thickness (μm) of each species is shown together with the optimized PROSPECT structure parameter N and the refractive index constant of leaf dry matter.

Quercus pubes.1381.321.61
Robinia pseudoac.941.251.59
Castanea sativa1441.381.63
Corylus avellana.1121.391.64
Prunus laurocer.3342.141.69
Prunus laurocer.3712.241.70
Laurus nobilis2271.781.69
Zea mays1961.521.69
Helianthus annuus3301.751.69
Soja hispida1291.571.69
Populus canadens1611.331.57
Oryza sativa1961.751.66
Phleum pratense1241.351.51
Zea mays2041.371.67
Lactuca sativa3322.131.64
Trifolium pratens1321.921.70
Acer pseudoplata961.261.51
Fraxinus excelsior1711.551.66
Tilia platyphyllos1231.261.66
Fagus sylvatica861.271.50
Solanum tuberosu2221.761.70
Urtica dioica1181.451.67
Beta vulgaris4371.891.70
Morus nigra1511.621.44
Vitis silvestris1621.451.67
Zea mays1991.401.57
Armeniaca vulgaris1761.411.65
Salvia officinalis5832.151.65
Prunus serotina1601.621.57
Quercus rubra1341.411.57
Quercus pubescens1971.401.67
Tilia platyphyllos1361.361.67
Betula alba1501.591.67
Castanea sativa1361.371.61
Alnus glutinosa1521.701.69
Salix alba1751.601.68
Corylus avellana1281.521.65
Populus tremula1461.331.64
Fraxinus excelsior961.811.64
Robinia pseudoaca1171.281.64
Zea mays1571.491.60
Phragmites com.1661.551.52
Musa ensete2121.651.70
Ulmus glabra1141.381.59
Iris germanica7402.811.68
Vitis vinifera1281.521.65
Ficus carica3481.591.65
Bambusa acundina1711.561.66
Armeniaca vulgar2011.501.66
Acer pseudoplata.1081.231.64
Hedera helix2511.891.70
Morus alba1521.511.69
Bambusa acundina1191.361.59
Lycopersicum2241.651.69
Vitis vinifera1681.391.68
Brassica oleracea1281.521.65
Zea mays1641.591.59
Oryza foliis siccis2091.931.69
Medicago sativa1321.551.69
Soja hispida1281.521.65

[23] In PROSPECT, an identical specific absorption coefficient is used for the leaf dry matter of all leaf types. So, the variable nmodd is somewhat unexpected because it should be constant if it is analytically associated only with the absorption property of leaf dry matter. There are two major reasons leading to variable nmodd values. First, it was assumed that the internal scattering will be perfectly handled by PROSPECT model, and no scattering but absorption contributes to the KK analysis of the refractive index. Note that the imaginary part k(ω) is the extinction coefficient in KK relations, which characterizes both the absorption and scattering property of dielectric materials. In practice, no model would be perfect. If a RTM cannot account for all the leaf internal scattering details, nmodd will take up the missed mechanism. In other words, the inversed nmodd value may include the contributions from scattering. Second, both nmoddand N are physically related with the redistribution between reflectance energy and transmittance energy. For inversion algorithm, this is an ill-posed problem, where multiple solutions may exist.

[24] Although a variable nmodd is physically and analytically possible, it is required that a specific nmodd be determined for each individual leaf type. Since nmodd and N are physically complementary, an ad hoc approach for qualitative analysis is to fix nmodd with the mean value of Table 1.

5.2. Leaf Refractive Index as the Function of Leaf Mass Compositions

[25] As mentioned above, leaf refractive index is physically a function of leaf mass compositions. Once the parameters of VFF-KK analysis are determined, thenKK(ω) of each leaf component may be calculated with equations (5)(9), and the leaf refractive index may be determined with the mixture equations (12)(13). With the mean value of nmodd in Table 1, i.e., nmodd = 1.64, a generalized refractive index of dry matter may be constructed. Figure 2 shows the refractive index of pure (100%) dry leaf and the specific absorption coefficients of the dry matter and the chlorophyll a + b over the range 0.4 μm–5.7 μm. The absorption data of dry matter and chlorophyll a + b over the range 0.4 μm–2.5 μm comes directly from the PROSPECT model. The dry matter data over the range 2.5 μm–5.7 μm was digitized from Gerber et al. [2011]. Due to the “edging effect” of VFF-KK analysis, the refractive index of dry matter near 5.7 μm may not be reliable. This extended dry matter absorption data is mainly used to ensure the analysis quality of the range 0.4 μm–2.5 μm. For leaf water, we have the data in sufficiently long wavelength range from pure liquid water measurements. The analysis quality of leaf water over the range of 0.4 μm–2.5 μm is ensured by using the water absorption data till 5.7 μm.

image

Figure 2. (top) The specific absorption coefficients (cm2μg−1) of leaf dry matter and chlorophyll a + b from the PROSPECT and Gerber et al. [2011]and (bottom) the refractive index of dry leaf by VFF-KK analysis.

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[26] As indicated in Figure 2, the refractive index of dry matter has much weaker local “oscillations” than the refractive index of leaf water in Figure 1. The local spectral details of leaf water refractive index dominate those of dry matter around 2.95 μm. This feature results from the fact that the absorption of leaf water dominates that of leaf dry matter in this region. And only the absorption property is analytically coupled with refractive index in the current VFF-KK analysis. In case that the model is not able to account for the scattering properly, the refractive index of dry matter must include the “missed” scattering mechanism. This is particularly true when thermal infrared range is considered.

[27] The variation of the leaf refractive index with respect to mass compositions may be investigated with different setting of leaf mass contents. Shown in Figure 3 is the ensemble of all the samples in Table 1. The static leaf refractive index of PROSPECT is plotted for the purpose of comparison. It may be found that leaf refractive index is almost a monotonic decreasing function of wavelength in the range of 0.4 μm–2.5 μm. But the magnitude at each specific wavelength varies with respect to leaf mass composition. The more the dry matter percentage is, the larger the leaf refractive index is. Interestingly, the PROSPECT static data characterizes very well the general trend of leaf refractive index in the considered range. Over the visible range, there exists a significant difference between the VFF-KK analysis and the static data in local spectral details. It is not sure that those “oscillations” come from real physics. In the early version of PROSPECT, a monotonic decreasing refractive index was used over the same range, which conforms to the VFF-KK analysis.

image

Figure 3. Leaf refractive indexes with different leaf mass compositions. The mass contents of leaf liquid water, dry matter and pigments are from LOPEX measurements. The one-for-all static refractive index of PROPSECT is shown for comparison.

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5.3. Verification Analysis

[28] Two parallel PROSPECT simulations are conducted for the verification analysis, with one using the static refractive index and the other using the coupled VFF-KK analysis. Compared inFigure 4 are the simulations for two samples of Beta Vulgaris species. These two samples are very different in leaf thickness and mass compositions. In general, both PROSPECT simulations match the observations very well. Although the static refractive index has more “oscillations” over the visible range (0.4 μm–0.7 μm), there is no significant reflectance and transmittance difference found in the two parallel runs. In comparison with the LOPEX measurements of 0.4 μm–0.7 μm, both PROSPECT runs match the measurement much better in the case of thicker leaf (Figure 4b). In the visible range, leaf pigments are the major absorbers. Since the pigment mass contents are much smaller than other cell matters, it is generally hard to have accurate measurements. The thinner a leaf is, the harder the measurements might be. The larger bias in Figure 4a over the visible range may mainly due to the mass content measurements.

image

Figure 4. PROSPECT simulations of two LOPEX samples of Beta Vulgaris species. These two samples are very different in leaf thickness and mass compositions. The leaf thickness of the sample is (a) 302 μm and (b) 630 μm. In Figures 4a and 4b the upper part is transmittance and the lower part is reflectance. Dashed lines are from the run with VFF-KK analysis, dotted lines are from the run with static refractive index, and solid lines are the corresponding LOPEX measurements.

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[29] In the near-infrared (0.7 μm–1.3 μm), the two parallel runs also show some distinct features in Figure 4. The simulation with VFF-KK analysis favors the matching with reflectance measurements while the simulation with static reflective index favors the matching with transmittance measurements. The transmittance with VFF-KK analysis tends to have positive bias in comparison with the measurements. By contrast, the reflectance with static refractive index tends to have positive bias over the same range (0.7 μm–1.3 μm). Such difference results from the energy redistribution between reflectance and transmittance, and the refractive index is the factor causing the difference in this case. Nonetheless, as shown in Figure 3, the refractive index by VFF-KK analysis is a function of leaf mass compositions, which may be larger than the static one in some cases and be smaller in other cases. To check if the coupling of PROSPECT with VFF-KK analysis has certain systematic features, ensemble analysis was performed with the simulations of all the samples ofTable 1. Let Oω,qi and Mω,qi denote the observation value and model simulation value at frequency/wavelength ω in the i-th sample for the radiative quantity q, respectively, where q designates reflectance or transmittance. It is generally meaningless to perform ensemble analysis over ω space for the radiative quantity q (reflectance or transmittance) because q has strong variation in the range of 0.4 μm–2.5 μm. In other words, q is not a “stationary” quantity in terms of ω. To perform ensemble analysis, the overall scattering property is first analyzed in the space consisting of all the (Oω,qi, Mω,qi) pair points, i.e., Pq = {(Oω,qiMω,qi)|0.4 μm ≤ ω ≤ 2.5 μmi = 1 …, 258}, which is shown in Figure 5. To further quantify the difference between (Oω,qi, Mω,qi), the root mean square error (RMSE) at each individual wavelength ω is calculated by inline image. The results are shown in Figure 6.

image

Figure 5. Scatterplots of the simulated reflectance and transmittance versus LOPEX measurements. Different colors are used for different wavelength ranges: dark yellow for visible range (0.4–0.7 μm), light green for near infrared (0.7–1.3 μm), and purple for mid-wave infrared (1.3–2.5 μm).

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image

Figure 6. Root mean squared errors of (a) simulated reflectance, (b) simulated transmittance, and (c) leaf-level emissivity. For each LOPEX sample, two parallel simulations were conducted, one with static refractive index (solid) and the other with VFF-KK analysis (dotted); 258 samples were simulated. RMSE at each wavelength is based on the ensemble of the 258 samples.

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[30] As shown in Figure 5a and Figure 5b, both two parallel runs have relative larger bias in the visible range. And as argued above, the major reason may be due to the specification of pigment mass contents. The simulations with VFF-KK analysis generally have better consistency with the measurements than those with static refractive index, especially over the near-infrared range of 0.7 μm–1.3 μm. In Figure 5c, a systematically negative bias may be found in the transmittance of the range of 1.5 μm–2.5 μm, Such negative transmittance bias may be rooted in the refractive index bias of this range. As shown in Figure 3, the refractive index by VFF-KK analysis is generally larger than that by the static one in this range. The negative transmittance bias indicates that the overall level of the refractive index may be a little bit higher than the true value of this range. Nevertheless, the simulations with VFF-KK analysis have much smaller scattering than those with static refractive index. A quantitative comparison of the simulation errors was shown inFigure 6.

[31] Figures 6a through 6care the RMSE of leaf-level reflectance, transmittance and emissivity, respectively. The leaf-level emissivity is estimated from reflectance and transmittance with Kirchhoff's law. The RMSE at each wavelength is calculated with all the 258 samples listed inTable 1. It may be seen that both parallel runs have relatively large but commeasurable RMSE values over visible range in Figure 6a, which conforms to the features in Figure 5a and Figure 5b. In the near infrared range of 0.7 μm–1.3 μm, the PROSPECT run is systematically improved with VFF-KK analysis. And the transmittance with VFF-KK analysis is also much better than that with static refractive index from 1.3 μm to 2.0 μm. Over the range of 1.5 μm–2.5 μm, the reflectance RMSE with VFF-KK analysis is slightly larger than that with static refractive index. Yet, the emissivity RMSE, which takes the errors from both reflectance and transmittance, is smaller with VFF-KK analysis than with static refractive index over the range of 1.5 μm–2.5 μm.

6. Summary and Concluding Remarks

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. KK Relations of Complex Refractive Index
  5. 3. A Testing With Pure Liquid Water
  6. 4. VFF-KK Analysis of Leaf Refractive Index
  7. 5. Analysis Results and Discussion
  8. 6. Summary and Concluding Remarks
  9. Acknowledgments
  10. References
  11. Supporting Information

[32] In comparison with leaf absorption properties, leaf refractive index is relatively hard to measure in practice. Both absorptive and scattering processes contribute to the definition of the refractive index. KK dispersion relations enable one to analytically derive leaf refractive index from leaf extinction property or check the consistency between leaf refractive index data with leaf extinction measurements. In this study, a mixing model is introduced to calculate leaf refractive index from leaf component absorption properties. The model is based on the variational function fitting method of KK analysis. It provides a very flexible and efficient framework to separately account for the contributions from the absorption and the scattering. With the refractive index by Kramers-Kronig analysis, the performance of the widely used PROSPECT model is generally improved, especially in the near infrared range where reflectance and transmittance are much stronger. In the mid-wave infrared range, a systematically positive bias may be found in the reflectance simulation with KK analysis. This reflectance bias results from the positive bias in the analyzed refractive index. Instead of using a constant, an advanced functional form may be needed for the leaf dry matter in the two-step variational fitting method. And this is particularly true when we extend this method to the thermal infrared range, where leaf internal scattering and the specular reflection by leaf cuticle layer become even more complicated.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. KK Relations of Complex Refractive Index
  5. 3. A Testing With Pure Liquid Water
  6. 4. VFF-KK Analysis of Leaf Refractive Index
  7. 5. Analysis Results and Discussion
  8. 6. Summary and Concluding Remarks
  9. Acknowledgments
  10. References
  11. Supporting Information

[33] This work was jointly supported by Chinese Ministry of Science and Technology project 2010CB951600 and the Joint Center for Satellite Data Assimilation (JCSDA) program of the National Oceanic and Atmospheric Administration (NOAA), National Environmental Satellite, Data and Information Service (NESDIS). The views, opinions, and findings contained in this publication are those of the authors and should not be considered an official NOAA or U.S. Government position, policy, or decision. The authors would like to thank the two anonymous reviewers for their suggestions to refine this paper.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. KK Relations of Complex Refractive Index
  5. 3. A Testing With Pure Liquid Water
  6. 4. VFF-KK Analysis of Leaf Refractive Index
  7. 5. Analysis Results and Discussion
  8. 6. Summary and Concluding Remarks
  9. Acknowledgments
  10. References
  11. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. KK Relations of Complex Refractive Index
  5. 3. A Testing With Pure Liquid Water
  6. 4. VFF-KK Analysis of Leaf Refractive Index
  7. 5. Analysis Results and Discussion
  8. 6. Summary and Concluding Remarks
  9. Acknowledgments
  10. References
  11. Supporting Information
FilenameFormatSizeDescription
jgrd18051-sup-0001-t01.txtplain text document2KTab-delimited Table 1.

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