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Abstract
This paper describes a data archaeology and rescue of temperature-dependent thermal infrared (IR) optical constants for liquid water from previously published works. The data rescue is based upon digitization of a figure published by Pinkley et al. [J. Opt. Soc. Am. 67, 494 (1977) [CrossRef] ], which showed ratios of laboratory measured spectral reflectances taken at 4 different temperatures (1, 16, 39, and 50 °C) with those measured previously at 27°C. Using our digitization of these data, we perform Kramers-Kronig analyses to derive temperature-dependent optical constants for existing standard datasets taken at room temperatures for the entire thermal IR spectrum. These datasets are of interest for the accurate modeling of thermal IR surface emissivity and reflectance over the range of global surface temperatures.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Thermal infrared (IR) environmental satellite remote sensing applications based on inversion of multispectral or hyperspectral measurements with sensitivities close to the Earth surface (i.e., window or surface channels) usually depend on a priori state specification of the spectral surface emissivity. Applications requiring a high degree of absolute accuracy, for example, estimation of global sea-surface skin temperatures (SSTs) from satellites, benefit from comparable accuracy in the a priori emissivity [1]. It is estimated that 1% uncertainty in emissivity can result in $\approx$0.2–0.6 K systematic error in longwave IR (LWIR) 11-$\mu$m window channel calculations [2].
While it has been known previously that there is a significant spectral dependence of emissivity on surface temperature [3–5], it was not until recent numerical weather prediction (NWP) model assimilation findings [6] that a significant systematic bias (on the order of $\pm 0.5$ K) was revealed on a global scale in cold ocean waters. The finding of global-scale NWP model assimilation impact [6] has brought attention back to this issue.
The temperature dependence in emissivity arises from the dependence of the complex refractive index on temperature. The complex refractive index for a dielectric medium (e.g., water), $N_\nu$, is defined by [7]
where $\nu$ is the spectral wavenumber, and the optical constants, $n_\nu = \Re (N_\nu )$ and $k_\nu = \Im (N_\nu )$, are the refractive index and extinction coefficient, respectively. Because of the dependence of water density on the bulk temperature $T$, the optical constants are implicitly a function of temperature, that is $N_\nu = N_\nu (T)$.Standard published datasets for the IR optical constants include the following in order of increasing ambient bulk water temperatures: Hale and Querry [8], 25$^{\circ }$C (298 K); Bertie and Lan [9], 25$^{\circ }$C (298 K); Downing and Williams [10], 27$^{\circ }$C (300 K); Pontier and Dechambenoy [11], 27$^{\circ }$C (300 K); Segelstein [12], 30$^{\circ }$C (303 K); and Wieliczka et al. [13], 30$^{\circ }$C (303 K). Note that all of these datasets were taken for water temperatures close to “room temperature.”
To our knowledge, the only complete set of laboratory-derived, temperature-dependent IR optical constants of liquid water obtained at more than one above-freezing surface temperature ($T > 0^{\circ }$ C) is that of Pinkley et al. [3]. More recent temperature-dependent datasets of interest are those of Rowe et al. [14] and Newman et al. [4]. However, the Rowe data were derived primarily for cloud-droplet applications, with all their measurements taken at supercooled water temperatures ($T \leq 0^{\circ }$ C). The Newman data, on the other hand, were derived only for the LWIR window region (770–1230 cm$^{-1}$) using field measurements obtained from a spectrometer mounted on a low-flying aircraft at $\approx$35 m; unlike the other datasets, their data are not laboratory-derived.
2. Methodology
Temperature-dependent optical constants for liquid water over the full thermal IR spectrum can be derived from existing published datasets (taken at room temperatures) by measuring reflectances at normal incidence, then applying Kramers-Kronig (KK) analysis. However, rather than obtaining such laboratory measurements ourselves, we have instead pursued a data archaeology and rescue [15] of the measurements published by Ref. [3] in an effort to secure datasets for the full IR spectrum.
2.1 Digitization of published data
Although Pinkley et al. [3] obtained optical constants for the electromagnetic spectrum spanning 400–5200 cm$^{-1}$, it is an unfortunate circumstance that their tabulated data are of little practical use as published. The reason for this stems from their Table I, which suffers from two fundamental limitations. First and foremost, presumably due to page constraints, they only included a small subset of the thermal IR spectrum, namely 500–820 cm$^{-1}$, 1600–1690 cm$^{-1}$, and 2650–3000 cm$^{-1}$. These omit critical atmospheric window regions in 820–1000 cm$^{-1}$, 1080–1200 cm$^{-1}$, and 2500–2650 cm$^{-1}$, which are of fundamental importance to environmental satellite remote sensing, with the latter shortwave IR (SWIR) region being the most transparent of the thermal IR spectrum. Furthermore, the data that they do include in their table are severely truncated to 3 significant figures, which are not precise enough for today’s high-precision applications (as indicated in Section 1.). The lead author (Nalli) attempted to contact surviving members of Ref. [3] (L. Pinkley, P. Sethna, and D. Williams), but subsequently found that both Pinkley and Williams have been deceased since 2004, with the whereabouts of Sethna remaining unknown.
As a work-around, we considered digitizing their results as plotted in two figures (op. cit. Figs. 3 and 4 for $n_\nu$ and $k_\nu$, respectively). However, we found it impractical to perform this with an acceptable degree of accuracy (i.e., without incurring unknown systematic errors) given that the lines for different temperatures overlap one another in the temperature/spectral ranges of interest. This was true for either the electronic reprint (PDF) or the hardbound print versions, the latter additionally suffering from axis distortion in the photocopying of pages from hardbound volumes. It should be kept in mind that it was the temperature dependence that we were interested in, so the overlap of temperature lines ultimately presented a fundamental hindrance to such an approach. These circumstances prevented us from acquiring their derived temperature-dependent optical constants reliably, which has in turn led to the present work.
Our recovery of the Pinkley data therefore involves an approach based upon a digitization of the original laboratory reflectance-ratio measurements, namely Fig. 1, op. cit., reproduced here in Fig. 1(a). This figure shows measured ratios of spectral reflectances (normal incidence), taken at 4 different water temperatures ($T$ = 1, 16, 39, and 50$^{\circ }$C), with those measured previously by Ref. [10] at room temperature $T_0$ = 27$^{\circ }$C (300 K), that is $\rho _\nu (T) / \rho _\nu (T_0)$. There are two benefits of this approach. First and foremost, there is clear delineation in the temperature curves (Fig. 1(a)), which facilitates accurate digitization of the sought-after temperature dependence. But just as important, and as will be seen, these are the original laboratory measured data from which the optical constants are derived; consequently, this is a more fundamental approach that can be applied to other laboratory datasets.
Fig. 1. Pinkley et al. [3] laboratory measured thermal IR, normal-incidence reflectance ratios of liquid water, $\rho _\nu (T) / \rho _\nu (T_0)$, where $T =$1, 16, 39, 50$^{\circ }$C and $T_0 = 27$$^{\circ }$C: (a) digital reprint of the original figure (Fig. 1, op. cit.; reprinted with permission of Optica), and (b) our digitization with slight empirical shortwave broadening of the 840–870 cm$^{-1}$ cold peaks.
The results of our digitization are shown in Fig. 1(b), which closely duplicate those in the original figure (Fig. 1(a)). The peak cold sensitivity found in the $\approx$840–870 cm$^{-1}$ range was extended slightly on the shortwave side, empirically based on computed differences between spectral observations and radiative transfer calculations (obs $-$ calc) from the NOAA National Centers for Environmental Prediction (NCEP) Gridpoint Statistical Interpolation (GSI) Global Data Assimilation System (GDAS) [16] (briefly discussed more in Section 3). These digitized data allow for a direct derivation of temperature-dependent optical constants for each of the existing standard datasets as described below.
2.2 Deriving optical constants from measured reflectances
The complex reflectances (or reflectivities) for the parallel and perpendicular polarizations (with respect to the plane of incidence or surface) are given by [17]
At normal incidence (i.e., at local zenith observing angle $\theta _0 = 0^{\circ }$), the polarized amplitude coefficients differ only in sign, thus $|R_\parallel | \equiv |R_\perp$| and the degree of polarization becomes zero. In this case, the complex Fresnel equation reflectances reduce to [17]
where we have introduced the shorthand $\nu$ subscript to denote the spectral dependence. From Eqs. (2) and (3) we thus have for normal incidence [18] with the reflectance for radiant intensities (or radiances, typically in units of mW$\,$m$^{-2}$sr$^{-1}$cm) then given by [3,7,17]To obtain the optical constants $n_\nu$ and $k_\nu$ as a function of $r_\nu$ and $\phi _\nu$, we first solve Eq. (4) for $N_\nu$
Then, substituting $e^{i\phi _\nu } \equiv \cos \phi _\nu + i\sin \phi _\nu$, we can break Eq. (6) into the real and imaginary parts and solve for $n_\nu$ and $k_\nu$. Note that this operation implicitly requires rationalizing the denominator by multiplying the top and bottom by the complex conjugate factor, $(r_\nu \cos \phi - 1) - r_\nu \,i \sin \phi$. Then, simplifying the result by applying the trigonometric identity $\sin ^{2}\phi + \cos ^{2}\phi = 1$, substituting the squared amplitude reflectances with intensity reflectances, $r_\nu ^{2} \equiv \rho _\nu$, and making the temperature dependence explicit, we arrive at the following expressions [10,18–20]
Thus, given measured spectral nadir-viewing ($\theta _0 \approx 0^{\circ }$) reflectances, $\rho _\nu (0,T)$, and phase shift, $\phi _\nu (T)$, one may experimentally derive the optical constants at different laboratory temperatures, $T$.
2.3 Kramers-Kronig (KK) analysis
Following Ref. [3], we can estimate the phase shift at an arbitrary water temperature via the Kramers-Kronig (KK) phase-shift dispersion relation for each channel $\nu$ as [3,7,10,19,20]
However, rather than use Eq. (9) for direct estimation of the phase shift for all temperatures $T$, a more accurate approach [3] is to estimate instead the deviation, $\Delta \phi _\nu (T)$, from the phase shift at a reference laboratory temperature, $T_0$, that is
where both $\hat {\phi }_\nu (T)$ and $\hat {\phi }_\nu (T_0)$ are estimated from the KK integral Eq. (9). The phase shift at temperature $T$ may then be computed as where the exact phase shift at reference temperature $T_0$, $\phi _\nu (T_0)$, can be derived from Eq. (7) asThe above KK methodology enables the retrieval of optical constants at temperature, $T$, from those measured at a laboratory reference temperature, $T_0$, along with spectral reflectances, $\rho _\nu (0,T)$, measured at $T$. Thus, it is not necessary to re-derive the optical constants explicitly in the laboratory at varying temperatures to arrive at a temperature-dependent set.
Following Ref. [21], we numerically performed the KK spectral integration of Eq. (9) using simple trapezoidal quadrature. This is performed for each of the published datasets discussed above in Section 1. A reasonable approximation can be attained if the integration extends well beyond the spectral region of interest [3,7,19], which in our case is roughly 600–3000 cm$^{-1}$. Given that the data of Segelstein [12] span the widest spectral range (from 0.001 to $10^{6}$ cm$^{-1}$), our approach was to append the Segelstein data to each of the datasets outside of their spectral limits, followed by linear interpolation to a common wavenumber coordinate from 0 to $5 \times 10^{6}$ cm$^{-1}$ (i.e., $\nu = 0,\,20,\, 40,\, \dots,\, 5 \times 10^{6}$ cm$^{-1}$). To get around the singularity at $\nu ' = \nu$, we simply used alternate wavenumber indices in a manner that avoids the singularity; thus, the point spacing used in the quadrature was $\Delta \nu = 40$ cm$^{-1}$. This spectral interval is the same order of magnitude as the original datasets in the spectral region of interest. As reported in Ref. [21], sensitivity of the quadrature to the number of points flattens out, and we found this to be the case for spectral intervals $\Delta \nu < 100$ cm$^{-1}$, with no notable improvement in accuracy < 40 cm$^{-1}$.
3. Results and discussion
Figure 2 shows an example of the spectral phase shifts estimated from the KK analysis methodology described in Section 2 based on the Ref. [9] optical constants (which are the most recent of the sets considered here). The top plot shows the comparison of the estimate for the reference temperature, $\hat {\phi }_\nu (T_0)$, obtained from the KK integral Eq. (9), along with the exact value calculated from Eq. (12). Here the KK-estimates from our simple trapezoidal quadrature scheme are in very good agreement with the theoretical values; we also found very good agreement using datasets [8,10,13], with slightly less agreement found using datasets [11,12]. However, given that we are ultimately using the approach for estimating the deviation $\Delta \phi _\nu (T)$ in Eq. (10), the KK estimates for all cited datasets are of more than sufficient accuracy. The bottom plot (Fig. 2(b)) shows the resulting temperature-dependent phase shifts, $\phi _\nu (T)$ derived from Eq. (11).
Fig. 2. Phase shifts, $\phi _\nu$, based on laboratory measured optical constants by Ref. [9] at 27$^{\circ }$C: (top) $\phi _\nu$ calculated directly from the optical constants using Eq. (12) (blue line) and $\hat {\phi }_\nu$ estimated from KK-analysis using Eq. (9) (red line); (bottom) temperature-dependent phase shifts derived from Eq. (11) (colored lines) along with the original phase shift from Eq. (12) (black dotted line).
Based on these phase shifts, the temperature-dependent optical constants, $n_\nu (T)$ and $k_\nu (T)$, can be calculated from Eqs. (7) and (8), the results of which are plotted in Fig. 3 for the same set of laboratory data [9]. Note that similar results (not shown here) were obtained for all the published optical constant datasets considered in this paper [8,10–13], and are available as Dataset 1 (Ref. [23]).
Fig. 3. Derived temperature-dependent thermal-IR optical constants for pure liquid water (Dataset 1, Ref. [23]), here based on lab measurements of Ref. [9] taken at 25$^{\circ }$C: (a) refractive index, $n_\nu (T)$, and (b) extinction coefficient, $k_\nu (T)$. Similar results were obtained for all of the remaining published datasets considered herein [8,10–13].
From these, temperature-dependent thermal IR emissivities may be calculated for any zenith observing angle, $\theta _0$, as $\epsilon _\nu [\theta _0,N_\nu (T)] = 1 - \rho _\nu [\theta _0,N_\nu (T)]$, where $\rho _\nu$ is the Fresnel intensity reflectance. Plotted in Fig. 4 is a comparison of flat-surface spectral emissivities at 4 different zenith observing angles within the LWIR atmospheric window region (750–1250 cm$^{-1}$). Figure 4(a) shows the results based on our data rescue (using Ref. [9]), and Fig. 4(b) shows the same but based on the Newman et al. dataset [4]. Reasonable agreement is found between the two sets of calculations, especially in the region below 900 cm$^{-1}$. However, there are some subtle, but significant, differences above 900 cm$^{-1}$, where the Newman data generally predict systematically lower emissivities (especially at larger $\theta _0$, where they are $\approx$0.005 less), along with considerably more temperature dependence.
Fig. 4. Computed water (flat surface) spectral emissivities, $\epsilon _\nu [\theta _0,N_\nu (T)]$, spanning the LWIR atmospheric window for emission angles, $\theta _0 =$ 30$^{\circ }$, 50$^{\circ }$, 60$^{\circ }$, and 65$^{\circ }$: (a) based on temperature-dependent optical constants from our data-recovery (Dataset 1, Ref. [9,23]), and (b) based on Ref. [4].
The critical region around 910–940 cm$^{-1}$ is of interest, as the Newman data [4] predict greater temperature dependence here. As alluded to above, this dependence was also suggested in global NCEP GSI assimilation obs $-$ calc analyses (not shown here). According to the Pinkley [3] measurements (Fig. 1(a)) there is an extremely steep slope in this region from maximum temperature dependence to zero; in our digitization (Section 2.1, Fig. 1(b)) we very slightly extended the cold temperature peaks on the shortwave side to 880 cm$^{-1}$, but in the interest of not deviating too far from the original measurements, we did not make any other alterations. Notably, the region of negligible cold dependence, 920–1100 cm$^{-1}$, was left unchanged, although this region may very well be the source of the discrepancy. We also compared our results against those of Zelsmann [22] for the far-IR within the overlap region 400–600 cm$^{-1}$ (not shown here) and found the same temperature dependence in terms of sign, although there were some differences in magnitude not unexpected given this region is normally considered outside of the thermal-IR.
As discussed above, the discrepancies in temperature dependence identified in the 910–940 cm$^{-1}$ region are not altogether surprising given that it is a region of transition between maximum and minimum temperature dependence (Fig. 1), but these thermal IR surface channels are otherwise important for NWP data assimilation over oceans. Thus, minimization of remaining global biases may ultimately require an updated set of laboratory-measured water reflectances obtained at different temperatures. This will potentially be the subject of future work; results from application to IR sea-surface emissivity models will be reported in a separate paper.
Funding
National Environmental Satellite, Data, and Information Service; Joint Polar Satellite System STAR Cal/Val Program and Proving Ground and Risk Reduction Initiatives; Joint Center for Satellite Data Assimilation 2019 Annual Operating Plan.
Acknowledgments
We wish to acknowledge the support of M. D. Goldberg (JPSS) and K. Garrett (NOAA/STAR); the STAR IR Soundings Team (K. Pryor, S. Kalluri, et al.); and S. English, S. Newman, and the ISSI Reference Ocean Surface Emissivity Model Team. The scientific results and conclusions, as well as any views or opinions expressed herein, are those of the author(s) and do not necessarily reflect those of NOAA or the U.S. Department of Commerce.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available in Refs. [3,4,8–13]. Data generated from the results presented in this paper (Dataset 1) based on Ref. [3] for previously published datasets [8–13] are available online via figshare (Ref. [23]).
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