Retrieval of the Ocean Skin Temperature Profiles From ... - IEEE Xplore

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 54, NO. 4, APRIL 2016

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Retrieval of the Ocean Skin Temperature Profiles From Measurements of Infrared Hyperspectral Radiometers—Part II: Field Data Analysis Elizabeth W. Wong, Member, IEEE, and Peter J. Minnett, Member, IEEE

Abstract—To obtain the vertical temperature profile of the thermal skin layer on the aqueous side of the ocean–atmosphere interface from radiance measurements of hyperspectral infrared radiometers, regularization methods are used to address the nonlinearity and ill-conditioning of the inverse problem associated with the retrieval. This paper demonstrates the truncated singular value decomposition regularization technique on field data sets obtained from the Marine-Atmospheric Emitted Radiance Interferometer (M-AERI). The M-AERI takes highly accurate and spectrally resolved radiance measurements in the infrared, which allows for the sensing of temperature values within thermal skin layer depths of less than 1 mm below the ocean surface. Results showed temperature inversions found in the solution, but these are deemed unphysical, based on previous tests on synthetic data and by a scaling analysis using the Rayleigh number, which are presented here. The importance of sea surface emissivity for performing the atmospheric correction to M-AERI measurements to derive surface emissivity spectra is also discussed. The retrieved profiles exhibit a smooth exponential-like structure that agrees well with theory.

Fig. 1. Schematic of idealized temperature profiles of the upper few meters of the ocean during (a) nighttime and daytime under strong-wind conditions and (b) daytime during low-wind conditions and high solar radiation, which results in the thermal stratification of the surface layers. The vertical depth scale is nonlinear [11].

Index Terms—Infrared (IR), radiometer, skin sea surface temperature (SST), truncated singular value decomposition (TSVD).

I. I NTRODUCTION

S

ATELLITE observations of sea surface temperatures (SSTs) have been a major contributor to improving our understanding of the Earth’s climate system and have revolutionized studies in, but not limited to, air–sea interactions, energy budgets, and global climate change. The consistent long-term global maps of SSTs produced by infrared (IR) and microwave space-based radiometers are important as SST observations are required as boundary conditions in numerical weather models and climate models and for data assimilation. However, they lack the absolute accuracy of < 0.1 K [1]–[3] required for climate change projections, resulting in the need for at-sea Manuscript received October 11, 2014; revised May 10, 2015 and September 27, 2015; accepted November 10, 2015. Date of publication January 5, 2016; date of current version March 9, 2016. This work was supported by the National Aeronautics and Space Administration through the Physical Oceanography Program. E. W. Wong is with the Meteorology and Physical Oceanography Program, Rosenstiel School of Marine and Atmospheric Sciences, University of Miami, Miami, FL 33149 USA (e-mail: [email protected]). P. J. Minnett is with the Department of Ocean Sciences, Rosenstiel School of Marine and Atmospheric Sciences, University of Miami, Miami, FL 33149 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2015.2501425

measurements for validation of satellite SST measurements and to help improve its accuracy. The Advanced Along-Track Scanning Radiometer [4] has accuracy of ∼0.23 K (with continued improvements as noted in the Ph.D. thesis of O. Embury [5]). The signal measured by high spatial resolution satellite IR radiometers has its origin in the near-surface ocean layer of submillimeter thickness, and the temperature of this layer is commonly referred to as the skin SST. One main source of error causing the present inaccuracy is the variability of the cool-skin effect [6]–[10], which is a result of heat flow from the ocean to the atmosphere. This requires a thorough understanding of the physics of the skin SST layer and would not only help improve the accuracy of radiometric measurement of SST but also provide deeper insight into ocean–atmosphere interactions, which are crucial in the Earth’s climate system. Such studies can only be done through at-sea measurements, allowing for precise measurements of the skin SST layer, which is essential given that we are attempting to comprehend a temperature layer with submillimeter vertical scales. Embedded within the submillimeter skin viscous layer on the aqueous side of the ocean–atmosphere interface lies the thermal skin layer and the electromagnetic (EM) skin layer. The thermal skin layer exists within the top 10 μm of the temperature profiles and is represented in Fig. 1 (adapted from [11]) as the temperature profile between SSTskin and SSTsubskin . (Note: The vertical depth scale is nonlinear.) Below the thermal skin

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layer, we note a well-mixed layer during nighttime or daytime conditions under moderate and strong winds and thermal stratification in daytime and light wind conditions. Even when there are strong diurnal temperature gradients, a thermal skin layer exists that causes the interface temperature to be cooler than the subskin temperature. The increase in temperature with depth within the thermal skin layer is due to the occurrence of surface cooling, which occurs in nearly all conditions. Turbulent heat exchange through the thermal skin layer is suppressed by the air–sea density difference at the interface and viscosity on the aqueous side of the interface. The thermal skin layer’s properties are therefore determined by molecular conduction processes. The average thermal skin layer exhibits a smooth continuous profile and is known to be nonlinear [12], [13] with an exponentiallike curvature that is largely attributed to the absorption and emission of IR radiation within the EM skin layer being concentrated closer to the surface of the ocean [7]. The temperature difference between the skin and subskin temperature, i.e., T , can range from ∼0.1 K for wind speeds > 7 m/s to ∼0.6 K for wind speeds < 2.5 m/s [9]. Studies describing the thermal skin layer are numerous (e.g., [3], [6], [14], and [15]), with a focus on obtaining T by taking into account wind effects, wave breaking, and diurnal warming such that corrections may be made to obtain an estimate of the skin temperature from subskin temperature measurements. These theoretical studies are based on 1) pure molecular conduction (e.g., [3] and [6]), using a characteristic thermal skin layer depth, i.e., δT , and taking into account the following heat flux equation: T =

Q δT ρcp κ

(1)

where κ is the molecular diffusivity, Q is the net flux at the ocean surface, ρ is the density of seawater, and cp is the specific heat capacity, and 2) surface renewal effects (e.g., [15]–[17]), which take into account renewal time scales to parameterize T . Despite the extensive studies of T and the thermal skin layer depth, this is insufficient if one is to understand the thermal skin layer—capturing the curvature of the thermal skin layer’s vertical temperature profile through measurements is needed. Obtaining submillimeter temperature measurements in the water is difficult. Ward and Donelan [18] attempted this by using a microthermometer with a diameter of 0.005 mm to measure the viscous sublayer temperature profile and to verify the Saunders coefficient [6]. However, the resolution of such measurements is at millimeter to centimeter scales, which is substantially thicker than the scales of the thermal skin layer. Furthermore, there is the risk of direct heating or physical disruption of the layer. Passive remote sensing is therefore ideal for removing such error sources and has been demonstrated through the use of passive IR radiometers in laboratory and field experiments to obtain an approximation to the vertical temperature gradient in the thermal skin layer [19], [20]. However, both assumed a linear temperature gradient that would hold only if there were no net IR radiative exchanges nor incident solar radiation, with the result that all heat loss would occur

at the air–sea interface. Wong and Minnett [13] (hereinafter WM15) demonstrated a retrieval technique using synthetic data to derive the thermal skin layer temperature profile. This paper is a follow-up to WM15 with the objective of applying their retrieval technique to ship-board data made by the MarineAtmospheric Emitted Radiance Inteferometer (M-AERI) [21]. The M-AERIs have been deployed in a large number of cruises and are well suited for the problem to be addressed, given the high accuracy of the spectrally resolved measurements of the thermal emission from the sea surface.

II. T RUNCATED S INGULAR VALUE D ECOMPOSITION (TSVD) T ECHNIQUE The radiance emitted by the sea surface measured by spectrometers is ∞ Im (v) = −

B (v, T (z))

d(e−αz ) dz dz

(2)

0

where Im (v) is the measured radiance at wavenumber v, B(v, T (z)) represents Planck’s function with T (z) being the vertical temperature profile of interest, and d(e−αz )/dz is the weighting function given by the Beer–Lambert law, i.e., I(z) = I0 e−αz . α is the wavenumber-dependent absorption coefficient of the medium and can be found from v and the corresponding imaginary component of the refractive index of water, i.e., k, (α = 4πkv). The vertical ordinate, i.e., z, is defined positive downward, with z = 0 being the ocean surface; therefore, the intensity of the radiation decreases with depth. Equation (2) is known to be ill-conditioned [22], which means that there is a strong likelihood of amplification of the errors in the measurements occurring in the derivation of T (z) from the measurements of Im (v). This results in the need to utilize regularization techniques to constrain the solution such that the errors in the measurements would not be magnified. WM15 demonstrated a TSVD technique to obtain a retrieval of the thermal skin layer’s nonlinear temperature profile using synthetic data. The algorithm is shown in Fig. 2 and will be applied here to field data. It was noted in WM15 that the first-guess temperature profile is a very important factor in obtaining a physically reasonable retrieval. To acquire the first-guess profile, we use WM15’s direct mapping method, which utilizes the relationship between the emission scale depth with wavenumber Dp (v) = 1/α(v) (see Fig. 3). The direct mapping method is executed by deriving the brightness temperature spectrum (BT(v)), which is found by calculating the inverse of Planck’s function from the measured radiance spectrum and subsequently mapping BT(v) with Dp (v) to obtain a temperature-depth plot, i.e., BT(Dp ). The values of k used to derive α were taken from [23]. To obtain a smooth and continuous first-guess profile, a complementary error function (erfc) temperature profile, i.e., (3), is fitted to BT(Dp ). Equation (3) is based on the derivation of the temperature profiles in the thermal skin layer by Liu and Businger [24] in which the heat diffusion equation was solved

WONG AND MINNETT: OCEAN SKIN TEMPERATURE PROFILES FROM MEASUREMENTS OF IR RADIOMETERS—PART II

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Fig. 2. Flowchart of algorithm using radiance spectra averaged into 11 values and incrementing Tb by 0.01 K from [13].

assuming free convection with a constant heat flux boundary condition, i.e., 2z T − Tb = π 0.5 6i3 erfc (3) Ts − Tb 3π 0.5 δc 2 √ where 6i3 erfc(x) = ((1+x2 )e−x / x) − (1.5 + x2 )x erfc(x), erfc(x) is a complementary error function defined as 1 − erf(x), Tb is the subskin temperature, Ts is the surface temper¯ 0 ) in ature, δc is a scaling depth defined as δc = κ((Ts − Tb )/Q ¯ 0 represents the purely conductive heat flux averaged which Q over time or space, and κ is the thermal conductivity of water. The IR spectral intervals used in this study are 800–1200 cm−1 (8.33–12.5 μm) and 2640–2800 cm−1 (3.57–3.79 μm) as identified by WM15. These intervals were chosen as they have minimal atmospheric absorption along the path from the sea surface to the M-AERI, exhibit a wide Dp range, and have the additional benefit of the reflectance of water being weakly dependent on the temperature and salt content [19], [20].

III. C RUISE DATA AND I NSTRUMENTATION The field data used in this study were taken by the M-AERI [21] during the African Monsoon Multidisciplinary Analysis (AMMA) 2006 cruise. An outline of the quality controls applied to the field data and derivation of emissivity values required for the atmospheric correction of the M-AERI’s measured spectral radiances is discussed below.

A. M-AERI The M-AERI is a sea-going well-calibrated Fourier transform infrared interferometer. It was developed from the Atmospheric Emitted Radiance Interferometer (AERI) [25] in the Space Science and Engineering Center, University of Wisconsin—Madison for the Department of Energy’s Atmospheric Radiation Measurement Program [26]. The M-AERI measures radiances (units: mW m−2 Sr−1 (cm−1 )−1 ) emitted in the wavenumber range 500–3000 cm−1 (∼3–20 μm in

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Fig. 3. Plot of penetration depth, i.e., Dp , versus wavenumber, i.e., v. The values of the imaginary component of the refractive index of water, i.e., k, used to calculate Dp = 1/4πkv are obtained from [23].

from two blackbody (BB) cavities mounted with their axes at 60◦ and 120◦ to the vertical, with the upper BB cavity being maintained at a temperature of 60 ◦ C and the lower BB cavity floating at ambient air temperature. Radiance measurements are averaged over 45-s intervals consisting of 45 independent interferograms, and the accuracy of the derived BT is determined to be < 0.02 K at 20 ◦ C and < 0.04 K at 30 ◦ C with a signal-tonoise ratio (SNR) range of 135–3135 at the short wavelengths and 1135–5400 at the long wavelengths. Typically, skin SST measurements are derived at a wavelength of 7.7 μm, which corresponds to radiance values of ∼55 mW m−2 Sr−1 (cm−1 )−1 and an SNR of ∼1800 for a 45-s average. The scan mirror must be kept dry during measurements as emission from water droplets on the mirror would contaminate the measurements, and so, the M-AERI is covered with a tarpaulin during conditions of heavy rain or sea-spray, and no measurements are taken. The schematic of the M-AERI’s viewing geometry is shown in Fig. 4, and the algorithm is described in [20] and [21]. Equation (4) shows that the observed upwelling radiance, i.e., Rsea , measured by the M-AERI while viewing the sea surface, consists of the spectral radiation emitted at the sea surface, i.e., B(v, Tskin ), the downwelling atmospheric emission, i.e., Rsky , reflected at the sea surface and the component of atmospheric emission from the layer below the level of the instrument at height h, Rh (v, θ). Rh (v, θ) includes both direct and reflected emissions from the sea surface attenuated by the atmosphere between the surface and height h. v is the wavenumber, and θ is the emission angle referenced to zenith. (v, θ) is the sea surface emissivity. Thus Rsea (v, θ) = (v, θ)B(v, Tskin ) + (1−(v, θ)) Rsky (v, θ)+Rh (v, θ).

Fig. 4. Schematic of the M-AERI’s viewing geometry. The sea and sky radiances, i.e., Rsea and Rsky , are measured at height h with an instrument pointing angle of θi and an effective incidence angle of θie at the sea surface [20].

wavelength) with an effective spectral resolution of 0.5 cm−1 , which resolves many gaseous absorption and emission lines in the atmosphere. Minnett et al. [21] described the details of its operation, accuracy, and applications, and only a brief summary will be given here. The M-AERI has a scan mirror that cycles through a sequence of scene views consisting first of the upwelling radiance from the sea surface and the downwelling atmospheric radiances measured at complementary angles with typical atsea deployment measured incidence angles, i.e., θi , of 55◦ , followed by a zenith measurement of the downwelling radiation (see Fig. 4). This set of scene views is sandwiched between two calibration sequences consisting of emission measurements

(4)

Since the M-AERI is generally mounted on a ship at a height of a few meters, Rh (v, θ) is very small and will be neglected in the analysis reported here. Similarly, the atmospheric attenuation along the path lengths from the sea surface to the M-AERI can be neglected. Corrections were applied to Dp by taking into account the angle of refraction at the interface, i.e., θr , such that Dp = (1/α) cos θr , where θr = 55◦ [20]. Rewriting (4) without Rh (v, θ), we have Rwater (v, θ)= (v, θ)B(v, Tskin )+(1−(v, θ))Rsky (v, θ). (5) Solving for the skin SST, Tskin gives Tskin = B −1 {(Rwater (v, θ)−[1−(v, θ)] Rsky (v, θ)) /(v, θ)} (6) where B −1 is the inverse Planck function. Equation (6) is used to perform the atmospheric correction and to retrieve the skin SST value from M-AERI measurements of spectral radiances, i.e., Rwater and Rsky . The top panel in Fig. 5 shows a sample radiance spectrum from the M-AERI taken during nighttime and cloud-free conditions across the full wavenumber range of 500–3000 cm−1 . The bottom panel shows the radiances converted into BT using Planck’s function. The blue lines indicate the upwelling radiance from the sea surface and exhibit a nearly smooth


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Fig. 5. Sample of M-AERI radiance spectra and the equivalent brightness temperature during nighttime and cloud-free conditions. Blue line: upwelling radiance measurement at 55◦ ; red line: downwelling radiance measurement at 55◦ ; black line: downwelling radiance measurement at zenith.

Planckian shape with the exception of random spikes observed at frequencies where the signal is masked by thermal noise (e.g., 1500–1700 cm−1 ) due to the atmosphere being insufficiently transmissive for the internal BB calibration to function well. It is the small deviations from Planck’s function at constant temperature that results from the spectral dependence of the emission depth (see Fig. 3) that conveys the information on the temperature gradient in the thermal skin layer that we seek. From the bottom panel, the “drop” in the atmospheric BT values at frequencies of 700–1300 cm−1 and 2000–2900 cm−1 indicates the presence of atmospheric transmission windows, where the atmosphere is sufficiently transmissive to allow the M-AERI to measure radiances emitted from higher in the atmosphere. Away from these atmospheric spectral transmission windows, the atmosphere is less transmissive, and the measured temperature from the M-AERI is warmer because it is sensing radiation emitted from gases lower in the atmosphere. B. AMMA 2006 Cruise The M-AERIs have been deployed on many research cruises, and for our study, we chose data from a cruise of the National Oceanic and Atmospheric Administration (NOAA) ship Ronald H. Brown (RHB). The cruise took place in the tropical Atlantic Ocean from May 28, 2006 to July 14, 2006, as part of the AMMA study. AMMA was a coordinated international project conducted to improve knowledge and understanding of the West African Monsoon as well as its variability and impacts. There were two legs to the RHB cruise: Leg 1 was from May 28 to June 17, where the RHB left San Juan, Puerto Rico, heading southeast with the majority of the leg spent off West Africa. Leg 2 was from June 22 to July 14, leaving Recife, Brazil and returning to Charleston, SC. Cruise tracks with SST measurements made by the M-AERI are shown in Fig. 6. The M-AERI was installed on the starboard side of the O2 deck of the RHB, with a mirror pointing angle of 55◦ to nadir. Retrievals of the skin temperature profiles from M-AERI

Fig. 6. AMMA 2006 cruise tracks with SST measurements made by the M-AERI. Top panel shows skin SST from Leg 1; bottom panel shows skin SST from Leg 2. Dates show the position of the ship at the start of each UTC day; gaps are caused by rain or heavy seas when useful data cannot be taken.

measurements were done beginning on May 27, 2006 and ending on July 14, 2006. The gap from May 27 to June 22 corresponds to the port call in Recife, Brazil, whereas the other larger gaps in M-AERI data were due to instrument failure or periods of bad weather. The M-AERI was repaired, and data collection was restored on July 29. The skin SST temperature, measured at a wavelength of ∼7.7 μm, ranged from 295 K to 305 K. This cruise was chosen because of the availability of ceilometer data, which enables the presence of clouds to be detected and relatively low wind speeds with a mean of 7.8 m/s. The ranges of meteorological variables are: air temperatures 295–302 K, air pressure readings 1010–1030 hPa, and relative humidity 50%–80% with wind speeds 0–20 m/s. For our analysis, only nighttime data (20:00–08:00 h local time) with low winds (< 10 m/s) were used to avoid possible contamination of the measurements by reflected and scattered solar radiation and to ensure that the thermal skin layer was rarely disrupted. Data with the presence of clouds were excluded from the analysis, as will be discussed in Section III-C. C. Quality Control of Cruise Data Before applying the TSVD technique to at-sea data, we select measurements that were taken in conditions most likely to provide a good signal. Since the sea-viewing M-AERI measurements include a component of reflected sky emission, we also have to be certain that the correction term in (6) is sufficiently accurate; of particular importance is demonstrating that the spectral signal in Tskin is not contaminated by

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Fig. 8. Spectral emissivity values calculated for five averaged AMMA 2006 cruise spectra. Cyan line: 5-cm−1 bins; magneta line: 7.5-cm−1 bins; red line: 10-cm−1 bins; blue solid line: averaged and low-pass filtered emissivity values; black dotted-dashed line: emissivity values obtained from [30].

, and has been shown to successfully simulate sea surface emissivity values from spectrometers such as the M-AERI [28], [29]. The VarMinT approach reduces the uncertainties in the emissivity, which could map into errors and uncertainties in the temperature profiles by computing the lowest variance in each prescribed spectral interval. The steps are as follows. Fig. 7. Sample radiance spectrum (upper) and brightness temperatures (lower) obtained from the M-AERI during nighttime and cloud-free conditions. Red line shows the measured upwelling radiance spectrum. Black line shows the measured downwelling radiance spectrum. Blue line shows the corrected upwelling radiance spectrum.

wavenumber-dependent errors in (v, θ). Fig. 7 shows an example of M-AERI measurements in the atmospheric windows during cloud-free nighttime conditions with the atmosphericcorrected BT spectrum plotted as the blue line. Spectra that show the presence of clouds were removed as emission from clouds masks the atmospheric emission lines, which is important for the derivation of surface emissivity to be accurately performed since it shows contrast between the sea measurement (upwelling radiance spectrum, Rwater ) and the sky measurement (downwelling radiance spectrum, Rsky ). This contrast is required for the derivation of spectral emissivity using a variance-minimizing technique described in [27]. The presence of passing clouds overhead could cause temporal and spatial heterogeneity, which, because the M-AERI scan mirror measures the sea and sky radiances consecutively and not simultaneously, introduces uncertainty in the reflected sky emission correction. Clouds pose an additional problem in rough sea surfaces as it may introduce additional error due to tilted facets of roughened sea surfaces in the M-AERI’s field of view, causing specular reflection of the sky radiance away from the simple geometric path shown in Fig. 4. The piecewise-linear variance-minimizing technique, which is termed VarMinT [27], uses the variance of the BT [see (7)] in small wavenumber intervals to derive emissivity values, i.e.,

1) Choose three different wavenumber intervals (5 cm−1 , 10 cm−1 , 15 cm−1 ). 2) For each interval, calculate the BT variance at different values using (7). 3) that corresponds to the smallest BT variance is taken to be across this wavenumber interval. 4) After finding across the whole spectral range and for each wavenumber interval, average into 10-cm−1 ranges such that at least four estimates occur in each 10-cm−1 average. 5) The spectrally binned average is passed through a digital box-car low-pass filter to produce a smooth spectral (v). The assumptions for the VarMinT are that the sky radiance, i.e., Rsky , is not correlated with (v) and that (v) is a smooth function of wavenumber. Errors such as the occurrence of different spectral signatures in and Rwater due to the presence of a time lag between the sequential measurement of Rsky and Rwater may arise from uncertainties in the refractive index for seawater and the measurement procedure. This is reduced by adopting a piecewise-linear approach, which is illustrated in steps 4 and 5. Thus (v) =

Rwater − Rsky . B(v, Tskin ) − Rsky

(7)

Fig. 8 shows the derived from five averaged profiles obtained from the AMMA06 cruise data set. Moreover, plotted are values calculated by Filipiak [30] at a 55◦ view angle, 0-m/s wind speed, and at a water temperature of 300 K with a salinity of 35 PSU. The results obtained from the VarMinT agree well with Filipiak’s data but with some slight discrepancies, which


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may be due to differences in the data set’s environmental conditions (e.g., wind speed, temperature, and salinity), including the possible presence of surfactants and impurities that are likely to exist in real seawater conditions. Further details are given in [31]. The differences in the derived (v, θ) and Filipiak’s data (see Fig. 8) are less than 0.005 and are not of major concern in the calculation of the BT’s to be used in the retrievals. The values of (v) used here are those derived from the M-AERI measurements to ensure consistency. D. Emissivity Sensitivity Tests To ensure that (v) derived from the piecewise-linear VarMinT is suitable for our study of the thermal skin layer, two sensitivity analyses were done in our selected wavenumber regions of interest. The first analysis was to determine the choice of increment value in the VarMinT iteration. Four increments (δ = 0.005; 0.001; 0.0005; and 0.0001) were used in the VarMinT, and the derived (v) value for each δ is shown in the top panel of Fig. 9. δ = 0.00001 was taken as the reference, and we subtracted the values of (v) derived using δ = 0.005; 0.001; 0.0005 from this reference value to examine the significance of varying δ values (bottom panel in Fig. 9). The difference in the (v) derived using δ = 0.0005 and 0.001 compared with that derived using δ = 0.0001 is insignificant (red and blue lines in the bottom panel of Fig. 9); the difference is < 10−3 . When compared with the derived (v) using a δ of 0.005, a larger difference is observed. This indicates that in the spectral windows of interest and our current choice of δ = 0.0005, an accuracy value of < 10−3 in the derived (v) using the VarMinT algorithm can be achieved, giving us confidence in the emissivity-corrected sea surface BT from M-AERI measurements. Thus, the deviation from Filipiak’s values shown in Fig. 8 is not caused by artifacts in our algorithm. The choice of δ = 0.0005 also corresponds to about 0.05% error in the emission spectra corrected for reflected sky emission, which is calculated through (7) by assuming a mean B(v, Tskin ) of 0.5 mW Sr−1 m−2 (cm−1 )−1 , mean Rsky of 0.1 mW Sr−1 m−2 (cm−1 )−1 , and a mean of 0.95 at the higher wavenumber spectral range. The second sensitivity test relates the changes in , i.e., , to changes in the emissivity-corrected BT, i.e., (BT). For this test, we took the (v) derived using δ = 0.0005 (0.0005 ), decreased 0.0005 by 0.01, and calculated BT by considering both 0.0005 and 0.0005 − 0.01. A ratio of the changes in BT and , i.e., Se = (BT)/, as a function of wavenumber is shown in Fig. 10. Drawn in the same figure is a blue dotted reference line of Se = 20, implying that a 0.01 change in would result in a 0.2-K change in temperature. Above this line, small changes in would result in larger BT variations. From Planck’s law, the percentage change in radiance resulting from a percentage change in BT, i.e., SBT = (B(v, T )/B(v, T ))/(BT/BT), increases with increasing wavenumber in the IR region. Therefore, changes in affect the changes in radiance, and we expect Se to increase with an increasing wavenumber. This is particularly shown in the wavenumber range of 800–1000 cm−1 as a rapid radiance drop (120–50 mW Sr−1 m−2 (cm−1 )−1 ) is observed. However, Se does not continue to increase steadily

Fig. 9. Top: Spectral emissivity values derived for four different incrementation values used to calculate the surface emissivity (δ = 0.0001, 0.0005, 0.001, 0.005) in the VarMinT using a sample of AMMA 2006 cruise measurement. Bottom: Difference of spectral emissivity values (0.0005 , 0.001 , 0.005 ) with reference to 0.0001 using a sample AMMA 2006 cruise measurements. Black line: 0.0001 ; red dashed-dotted line: 0.0005 ; blue dotted line: 0.005 ; green dotted line: 0.001 .

because of a plateau in radiances from 1000 to 1200 cm−1 at ∼50 mW Sr−1 m−2 (cm−1 )−1 and 2640–2800 cm−1 at ∼0.15 mW Sr−1 m−2 (cm−1 )−1 . Hence, the variability in Se is not largely due to the magnitude of the downwelling sky radiance, but is instead dominated by the variability of . This is deduced from observing that ∼ 0.95 from 2640 to 2800 cm−1 compared with the 1000–1200 cm−1 range, which has an ∼ 0.97 (see Fig. 8), thereby implying that a given variation of would have a larger effect on BT for wavenumbers with higher [see (7)]. Plotted in Fig. 10 is the downwelling sky radiance. Note that in regions where the downwelling sky radiance “peaks,” the gradient experiences a “trough.” This is because when the downwelling sky radiance peaks, the contrast between the upwelling sea radiance and downwelling sky radiance measurement is smaller, resulting in a poorer estimate of (the denominator in (7) approaches zero). The “peaks” occur due to changes in the atmospheric transmissivity spectrum and is caused by

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Fig. 10. Plot of sensitivity of emissivity with BT values and upwelling sky radiance against wavenumber. Change in emissivity = 0.01. Change in BT = (c2v/ log(c1v3 /R1 + 1)) − (c2v/ log(c1v3 /R2 + 1)), where R1 = (Rsea − Rsky (1 − ))/ and R2 = (Rsea − Rsky (1 − ( − 0.01)))/ − 0.01.

Fig. 12. Temperature profiles; blue line: T f g (z) which is an erfc function fitted to 300 averaged M-AERI spectra represented by the green crosses; red line: iterated result at p = 3. Top panel: Tb = Tbf g ; bottom panel: Tb = Tbf g + 0.1. Fig. 11. Plot of brightness temperature with depth. Measurement values obtained from a sample AMMA 2006 cruise radiance spectrum.

the gaseous components of the cloud-free atmosphere. Fig. 11 shows a sample BT with a depth profile before and after the emissivity correction is performed. The larger oscillations seen at deeper depths can therefore be explained as the measured downwelling sky radiance spectrum is more variable at higher wavenumbers thereby implying that there is more variability in the contrast required for the VarMinT approach to produce . Thus, the oscillations are likely to be artifacts of the process, and not physical, resulting in the need to fit a smooth curve to the temperature with depth plot to attain the first-guess profile. IV. C RUISE DATA R ESULTS AND A NALYSIS The TSVD approach was first tested using both spectral windows as discussed in Section II, where there is large variability with Dp (see Fig. 3), particularly from 2640 to 2800 cm−1 ; with a 1-cm−1 resolution, this gives a total of 560 spectral values. The retrievals of the thermal skin layer temperature profile from

individual full-resolution spectra have been demonstrated in WM15 to result in noisy and highly oscillatory profiles that were clearly not physical, as the thermal skin layer is laminar and is unable to support multiple temperature inversions within the layer. The retrieved profiles also exhibit a similar behavior to noise-injected synthetic data (see [13, Sec. IV-A]) where the noise injected corresponds to the M-AERI’s SNR. In WM15, the issue of highly oscillatory profiles was overcome by averaging each radiance spectrum into 11 wavenumber intervals. However, we decided to first explore the possibility of using the entire spectral range by averaging radiance spectra to perform a retrieval. In [13, Sec. IV-B], it is shown that when a retrieval is performed on a spectrum that consists of an average of eight radiance spectra, the retrieved profile matches the temperature profile obtained by averaging the corresponding eight retrieved temperature profiles. This shows that spectral averaging can be used instead of time averaging. The result of the temperature profile retrieval obtained from the nighttime averaged spectra from the AMMA 2006 cruise is shown in Fig. 12. The top panel shows the first retrieved profile when the subskin temperature of the first-guess profile, i.e., Tbf g , has not


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Fig. 13. Logarithmic plot of the singular values of the discretized inverse problem, (2), for M-AERI data. The red dotted-dashed line indicates the noise level of 0.0001 (0.01%), whereas the red dashed line indicates the noise level of 0.01 (1%). The red solid line indicates the level of the machine epsilon (eps = 2.2204 · 10−16 ).

been incremented, and the bottom panel shows the result after Tbf g is incremented in intervals of 0.01 K until the inversion disappears. The removal of the temperature inversion occurred when the subskin temperature, Tb = Tbf g + 0.1 and a smooth profile (red) is obtained which is close to that of the first-guess profile for that iteration (blue). The result agrees well with the synthetic data tests performed in WM15. When different firstguess profiles with the same surface temperature, i.e., Ts , and Tb = Tbf g + 0.1 values but a different gradient were used, the solutions were observed to converge to the same profile, which indicates that the solution is robust. The large inversion observed in the top panel of Fig. 12 can be attributed to the resultant subskin temperature Tb tending toward Tbf g . Thus, to compensate for a deficit in the calculated surface emission caused by too cold Tb , higher temperatures near the surface are introduced. The incrementation of Tbf g therefore compensates for this artifact to ensure that a smooth profile is retrieved. The level of truncation, i.e., p, of the SVDs (see WM15), was determined by using knowledge of the M-AERI’s smallest SNR of 135 at 3000 cm−1 [21] with better SNR values of 1800 at lower wavenumbers. This translates to a fractional uncertainty of 0.0074, which was rounded up to 1% to be used to determine the level of truncation. Fig. 13 shows the logarithmic plot of the singular values obtained from the field data, and a noise level of 1%, the dashed red line, indicates that a truncation parameter of p = 3 ought to be used. It is also noted that at p = 3, the fractional uncertainty, i.e., Ψ, needs to be in the interval: 0.0067 < Ψ < 0.0256 which translates to an instrument’s SNR range of 39.0 < SNR < 149.2. This truncation level works well for the cruise spectra as oscillatory profiles are observed resulting from numerical instabilities for p ≥ 4. A noise level of 0.01% is also plotted as a reference line from WM15, which established that p = 6 is used when a retrieval from noiseless synthetic data is performed.

Fig. 14. Plots of the difference in radiances with the measured M-AERI spectral radiance, expressed as a percentage (averaged over 300 spectra). Top panel (a) and (b): radiance difference of the first-guess radiances (blue line); radiance difference of the iterated result’s radiances at p = 3 (red line). Bottom panel (c) and (d): radiance difference where no inversion exists in the iterated solution (black line); radiance difference where an inversion exists in the iterated solution (green line). Plots (a) and (c) correspond to the percentage of radiance differences from 800 to 1200 cm−1 . Plots (b) and (d) correspond to the percentage of radiance differences from 2640 to 2800 cm−1 .

The percentage difference in radiance values, i.e., ((B(v, T (z)) − Im )/Im ) · 100%, between the first-guess and iterated solution when Tb = Tbf g + 0.1 is shown in the top panel of Fig. 14. There is no significant change in this difference at the higher wavenumbers since there is not a marked change in the temperature profile at depths of z > 0.01 cm. The curvature adjustment at shallow depths of z < 0.01 cm has resulted in a reduction in the percentage error of the radiances at the lower wavenumbers. Shown also in the bottom panel of Fig. 14 is the percentage difference in radiances between the measured M-AERI spectral data and both iterated results from Fig. 12. Both lines are very close, implying that the TSVD method of obtaining a least squares fit to the spectral radiances would not help in determining whether the retrieved profile is “true.” The reasons for rejecting the iterated solutions with the inversions are, first, because of the unphysicality seen from the synthetic profile runs and, second, in theory, such a large inversion in the temperature gradient could not be supported in such a thin layer as molecular conduction would remove the heat down the

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Fig. 15. Ten sample profiles derived from spectra taken during the AMMA 2006 cruise. Blue X’s represent the mean BT of the wavenumber intervals with error bars plotted as ±1 standard error. Red solid line denotes first-guess profile; black solid line represents the retrieved profile with respect to the mean BT; gray dotted lines represent the retrieved profile with respect to ±1 standard error. A majority of the gray dotted lines are overlaid on the black solid lines, thereby indicating the similarity between the retrievals to the mean and ±1 standard error in BT.

temperature gradients in both upward and downward directions. Thus, even if such an inversion were to occur in nature, it would quickly decay to a profile similar to that in the bottom panel of Fig. 12. This will be further discussed in Section V. The temperature profiles derived from the AMMA cruise indicate that the algorithm works well. However, 300 spectra were averaged prior to the application of the TSVD method to give the profile shown in Fig. 12, but given that under normal operations, an M-AERI measurement sequence results in five spectra per hour. Thus, averaging 300 profiles means that the retrieved thermal skin layer temperature profile is an average of 60 h, which takes about five days to accumulate as only nighttime data have been used. This is not desired as the main objective is to determine a realistic temperature profile of the thermal skin layer, which is expected to vary with heat fluxes. Thus, ideally we require retrievals from individual spectra or, if necessary, an average over a smaller number of spectra.

We therefore followed the approach in WM15, which averages the individual spectra into wavenumber intervals. To obtain individual retrievals, we refer to the flowchart in Fig. 2 in which the relevant spectral ranges are averaged into 11 wavenumber intervals, and a retrieval is performed on these 11 points. The choice of 11 wavenumber intervals is driven by signal-to-noise considerations in the individual unaveraged spectrum that caused the TSVD algorithm to fail to converge. This is particularly so at higher wavenumbers that have very low radiance values (on the order of 1 mW Sr−1 m−2 (cm−1 )−1 ) as compared with the lower wavenumbers (on the order of 100 mW Sr−1 m−2 (cm−1 )−1 ), which makes the retrievals highly sensitive to noise at high wavenumbers. We follow WM15’s result of 40-cm−1 wavenumber intervals at the lower wavenumber range of 800–1200 cm−1 and because of a much higher variability in the BT observed at the higher wavenumber range of 2640–2800 cm−1 (e.g., Fig. 12), the entire range is


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analyzing the difference in radiance values between retrieved profiles with and without the iteration to adjust Tb and Ts does not indicate whether the resulting profile is physical. The difference between the solid white line and the red dashed line in Fig. 16 does not exceed ±4 · 10−3 % and ±4 · 10−4 % for the lower and higher wavenumbers, respectively. Section V discusses the benefit of implementing the iterative method to ensure we obtain monotonic physical thermal skin layer profiles. V. D ISCUSSION

Fig. 16. Plot of the difference in the retrieved profile’s radiances with the measured M-AERI spectral radiance expressed as a percentage, i.e., (B(v, T (z)) − Im )/Im · 100%. Gray lines: percentage differences for 300 profiles retrieved from AMMA 2006 cruise. White solid line: average of these percentage differences; white dotted line: ±1 standard deviation. Red dotted line: average of difference between the first-guess profile’s radiances with measured M-AERI spectral radiances expressed as a percentage, i.e., (B(v, T f g (z)) − Im )/Im · 100%. The first-guess profile’s radiance spectrum, i.e., B(v, T f g (z)), is noted to be very close to the mean spectrum, i.e., B(v, T (z)), as the retrieved profiles are such that their emission spectrum matches the measurements, even if the profile is unphysical.

averaged into one point, thereby giving 11 spectral points as the basis for the retrieval. Ten sample individual profiles are shown in Fig. 15. All the retrievals show smooth continuous profiles (black solid lines) with the number of iterations ranging from one iteration performed for yearday 148.0467 to 21 for yearday 187.928. Each iteration meant incrementing Tbf g by 0.01 K, and the criterion for the algorithm to exit is when the gradient changes are smaller than a predefined threshold of 10−5 K/cm. Moreover, plotted in Fig. 15 are the retrievals performed using ±1 standard error of the mean BT calculated for each wavenumber interval. These are denoted by the dotted gray lines which may not be visible as they coincide largely with the retrieved profile from the mean BT (black solid line). The largest deviation between the retrievals of the standard error adjusted mean to the retrievals of the true mean has an absolute BT difference of 0.018 K, which is smaller than the stated retrieval accuracy of 0.048 K in WM15. Fig. 16 shows the percentage errors of radiance values between 300 individually retrieved profiles and their measured M-AERI spectra (gray lines) with the mean and ±1 standard deviation plotted as the solid and dotted white lines, respectively. The average percentage error, i.e., ((B(v, T (z)) − Im )/Im ) · 100%, across 800–1200 cm−1 is 0.00938 ± 0.00292% and across 2640–2800 cm−1 , 0.0312 ± 0.0104%. This is equivalent to an absolute radiance difference of 0.855 ± 0.271 mW Sr−1 m−2 (cm−1 )−1 and 0.0150 ± 0.00506 mW Sr−1 m−2 (cm−1 )−1 across 800–1200 cm−1 and 2640–2800 cm−1 , respectively. The mean percentage error for the retrieved profiles using the first guess without iteration is also plotted as the red dashed line in Fig. 16. It is clear in Figs. 14(c), (d) and 16 that by only

The results obtained clearly indicate that the solution to the inverse equation is not unique and can result in the generation of different profiles with very similar error magnitudes. This is shown in the comparison between the radiances calculated from the iterated profiles and first-guess profiles in Figs. 14(c), (d) and 16. Therefore, how do we know which solution profile is closest to the true profile? The oscillatory solutions are not physical since it is not possible to support multiple large temperature gradient reversals in such a thin layer; these oscillations are due to the ill-conditioning of the inverse problem. However, the retrieval performed with M-AERI data has an inversion at depths of ∼0.05–0.1 mm for the case when Tb is not adjusted to a higher value (see Fig. 12). Although this has been deemed unphysical from the results obtained from synthetic data runs in WM15, it is still important to be certain whether such a gradient reversal occurring in the thermal skin layer is indeed possible. It is interesting to note that some previous studies performed on the thermal skin layer reported inversions whereas others did not. For example, Chu and Goldstein in [32] and Herring in [33], who studied turbulent thermal convection of water between two parallel plates, observed inversions, as did Veronis in [34] and Chang and Wagner in [35] who studied Bénard convection and changes in surface temperatures due to small amplitude surface waves. On the other hand, Katsaros et al. in [36] performed an experiment in an open tank of water and did not observe the inversions, nor did Spangenberg and Rowland in [12] who carried out a similar experiment to study natural convection in water, nor Ward et al. in [37] who used an upper decameter SST profiling instrument called SkinDeEP. Numerical simulations of free convection done by Leighton et al. in [38] did not observe inversions in their thermal skin layer profiles. Katsaros et al. [36] described the studies which observed the inversions, or “knees,” to be unrealistic in their experimental conditions and summarized that these studies were obtained at Rayleigh numbers of less than 107 . She also confirmed the findings by Chu and Goldstein [32], that for gradient reversals in the water to occur, an upper-limit Rayleigh number of ∼ 8 · 106 is required, and this requirement has some dependence on the Prandtl number. Furthermore, the presence of inversions is also noted to occur in earlier studies and at depths below the thermal skin layer, whereas the more recent studies, using improved instrumentation, did not show the presence of these inversions. Despite this, none of the studies (including the more recent ones) are focused on the upper layers of the skin layer profile of

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less than 1 mm as most of the studies are performed at centimeter scales. At depths of a few centimeters, the Rayleigh numbers are much larger, and the analysis of the thermal skin layer has, in reality, extended into the subskin layer. The submillimeter depth scales of the thermal skin layer which are of interest are not entirely applicable to the studies made by these papers and experience completely different mechanisms. The gradient reversals seen by Chang and Wagner [35] occurred at a depth of ∼1 cm with a temperature difference of 0.04 K with respect to the bulk temperature, whereas Chu and Goldstein [32] observed a reversal at 3.8 mm with a temperature difference of about 0.25 K. Both are deeper than the submillimeter thermal skin layer and inversions found in some of our temperature profile retrievals. To further assess the possibility of a temperature inversion in the thermal skin layer, we consider molecular heat conduction on small scales. If an inversion or local temperature maximum, i.e., Tmax , were to occur, this temperature would be expected to quickly decay to a value of the subskin temperature, resulting in our expected results of a smooth profile monotonic with depth. An inversion in the thermal skin layer could be conceivably caused by convection of warmer water from below. To confirm our original assumption that the thermal skin layer is embedded in a laminar viscous layer, an analysis at our depth scales of interest using the Rayleigh number was performed. The dimensionless Rayleigh number, i.e., Ra , is the ratio between the rate of heat transfer by convection to that by conduction and takes the following form: Ra =

bgL3 T αd va

TABLE I TABLE OF R AYLEIGH N UMBERS AT L ENGTH S CALES OF 1 AND 0.1 mm

TABLE II TABLE OF M INIMUM D EPTH VALUES R EQUIRED FOR C ORRECTION FOR R AYLEIGH N UMBERS OF 102, 1193, AND 657.3

(8)

where g is the gravitational constant, b is the coefficient of thermal expansion, L is the length scale, T is the temperature difference across length L, αd is the thermal diffusivity, and va is the viscosity of the medium. The effects of salinity or density variations are small and are not considered in this analysis. Rayleigh [39] proposed a finite value of the temperature difference, i.e., Tc , and a critical Rayleigh number, i.e., Rac , of 27π 4 /4 as being required for convection to occur, assuming there was free slip between the fluid and the boundaries. A value of Rac = 1709 was predicted in [40] for a fluid confined between two horizontal boundaries with no slip between the boundary and fluid, whereas Low in [41] predicted Rac = 1108 if one of the boundaries is rigid while the other is free. In an experimental tank, Spangenberg and Rowland [12] obtained Rac = 1193 for the onset of natural convection and Rac = 102 for this convection to be maintained. Importantly, for our study, they also observed that convection did not occur at water depths of less than 1 cm. Table I shows the calculations made for Ra for a temperature difference range of 0.1 K–1.0 K with length scales of 1 and 0.1 mm, i.e., characteristic of the thermal skin layer. We see that the maximum Ra number, i.e., 16.41, occurs for a temperature difference of 1 K. This Ra is very small compared with the Rac values for convection and, thus, also confirms that the mean temperature gradient in the skin layer can exist without being disrupted by convection. Rac values of 1193 and

102 as predicted by Spangenberg and Rowland in [12] were subsequently assumed to calculate the length scales required to meet the minimum Rac value (see Table II). A minimum depth of 4 mm is required for the onset of convection to occur, and a minimum of 1.83 mm is required to maintain convection. These values are larger than our skin layer depths of 0.1–1 mm, and thus, we conclude that convection is not occurring at these depth scales. Moreover, the range of length scales required for an Rac value of 27π 4 /4 = 657.5 is also shown in Table II. This simple analysis confirms that depths of ∼1 mm beneath the ocean–atmosphere interface are laminar, and thus, the thermal skin layer is indeed within the viscous sublayer at the ocean surface. VI. S UMMARY AND C ONCLUSION In conclusion, the TSVD technique has been demonstrated as a method to retrieve the temperature profile of the thermal skin layer using emitted radiance spectra obtained from a seagoing interferometer, i.e., the M-AERI. Applying the TSVD technique to 300 averaged spectra from the M-AERI from the AMMA cruise, a truncation level of p = 3 was established for the M-AERI’s SNR, and a smooth continuous profile was obtained similar to what we expected from theory. Individual profile retrievals were subsequently performed with the same requirement established in the noisy synthetic data runs from WM15 to obtain smooth monotonic temperature depth profiles:


by averaging the radiances into 11 wavenumber intervals and truncating at p = 2. The results obtained from field data agree well with the results obtained from WM15’s synthetic data. Using the removal of vertical temperature inversions as a criterion for a successful temperature profile retrieval is further justified by computing the Rayleigh number, which shows that at the depth and temperature difference scales analyzed here, the fluid is laminar, and heat transfer is dominated by molecular conduction, thereby implying that the reversal in the temperature gradient cannot be supported at these depths. The importance of the sea surface emissivity spectrum required for atmospheric correction was also discussed. A robust variance-minimizing technique was tested and shown to provide sea surface emissivity values of sufficient accuracy for the atmospheric correction with accuracy of 10−3 . Further analysis on the sensitivity of emissivity to the calculated BT showed that the variability of BT at deeper depths is unlikely to be physical. This also accounts for the need to use an average and obtain a smooth fitted erfc profile to the BT values at deeper depths for the first-guess profile. This research has shown very encouraging results for future studies of the ocean thermal skin layer. The ability to retrieve the nonlinear profile of the thermal skin layer introduces a fresh approach to the exploration of the physics of the thermal skin layer, air–sea interfacial studies, and the application of seagoing spectroradiometers. ACKNOWLEDGMENT The authors would like to thank Dr. M. Szczodrak for her contribution to the development of the TSVD algorithm and suggestions on the processing of the AMMA cruise data. The authors would also like to thank the captain, officers, and crews of the NOAA ship Ronald H. Brown for the at-sea support. R EFERENCES [1] G. Ohring, B. Wielicki, R. Spencer, B. Emery, and R. Datla, “Satellite instrument calibration for measuring global climate change: Report of a workshop,” Bull. Amer. Meteorol. Soc., vol. 86, no. 9, pp. 1303–1313, Sep. 2005. [2] “Sea surface temperature error budget: White Paper,” Interim Sea Surface Temp. Sci. Team (ISSTST). Jun. 2010. [Online]. Available: http://www. sstscienceteam.org/SST_Error_Budget_White_Paper_Final.pdf [3] C. Fairall et al., “Cool skin and warm layer effects on sea surface temperature,” J. Geophys. Res., vol. 101, no. C1, pp. 1295–1308, Jan. 1996. [4] O. Embury, C. J. Merchant, and G. K. Corlett, “A reprocessing for climate of sea surface temperature from the along-track scanning radiometers: Initial validation, accounting for skin and diurnal variability effects,” Remote Sens. Environ., vol. 116, pp. 62–68, Jan. 2012. [5] O. Embury, “Sea surface temperature for climate from the along-track scanning radiometers,” Ph.D. dissertation, School of Geosciences, The Univ. Edinburgh, Edinburgh, U.K., 2014. [Online]. Available: https:// www.era.lib.ed.ac.uk/bitstream/1842/8960/2/Embury2014.pdf [6] P. M. Saunders, “The temperature at the ocean–air interface,” J. Atmos. Sci., vol. 24, no. 3, pp. 269–274, May 1967. [7] E. D. McAlister and W. McLeish, “Heat transfer in the top millimeter of the ocean,” J. Geophys. Res., vol. 74, no. 13, pp. 3408–3414, 1969. [8] A. R. Harris, “Improved sea surface temperature measurements from space,” Geophys. Res. Lett., vol. 22, no. 16, pp. 2159–2162, Aug. 1995. [9] C. J. Donlon et al., “Toward improved validation of satellite sea surface skin temperature measurements for climate research,” J. Clim., vol. 15, no. 4, pp. 353–369, Feb. 2002. [10] P. J. Minnett, M. Smith, and B. Ward, “Measurements of the oceanic thermal skin effect,” Deep Sea Res. II, Topical Stud. Oceanogr., vol. 58, no. 6, pp. 861–868, Mar. 2011.

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[11] C. Gentemann and P. J. Minnett, “Radiometric measurements of ocean surface thermal variability,” J. Geophys. Res., vol. 113, no. C8, Aug. 2008, Art. ID C08017. [12] W. G. Spangenberg and W. R. Rowland, “Convective circulation in water induced by evaporative cooling,” Phys. Fluids, vol. 4, no. 6, pp. 743–750, 1961. [13] E. W. Wong and P. J. Minnett, “Retrieval of the ocean skin temperature profiles from measurements of infrared hyperspectral radiometers. Part I: Derivation of an algorithm,” IEEE Trans. Geosci. Remote Sens., vol. 54, no. 4, Apr. 2016. [14] J. Wu, “An estimation of oceanic thermal-sublayer thickness,” J. Phys. Oceanogr., vol. 1, no. 4, pp. 284–286, Oct. 1971. [15] A. V. Soloviev and P. Schlüssel, “Parameterization of the cool skin of the ocean and of the air–ocean gas transfer on the basis of modeling surface renewal,” J. Phys. Oceanogr., vol. 24, no. 6, pp. 1339–1346, Jun. 1994. [16] W. T. Liu, K. B. Katsaros, and J. A. Businger, “Bulk parameterization of air–sea exchange of heat and water vapor including the molecular constraints at the interface,” J. Atmos. Sci., vol. 36, no. 9, pp. 1722–1735, Sep. 1979. [17] S. L. Castro, G. A. Wick, and W. J. Emery, “Further refinements to models for the bulk-skin sea surface temperature difference,” J. Geophys. Res., vol. 108, no. C12, pp. 3377, 2003. [18] B. Ward and M. A. Donelan, “Thermometric measurements of the molecular sublayer at the air–water interface,” Geophys. Res. Lett., vol. 33, no. 7, Apr. 2006, Art. ID L07605. [19] W. McKeown, F. Bretherton, H. L. Huang, W. L. Smith, and H. L. Revercomb, “Sounding the skin of water: Sensing air–water interface temperature gradients with interferometry,” J. Atmos. Ocean. Technol., vol. 12, no. 6, pp. 1313–1327, Dec. 1995. [20] J. A. Hanafin, “On sea surface properties and characteristics in the infrared,” Ph.D. dissertation, Dept. Meteor. Phys. Oceanog., Univ. Miami, Miami, FL, USA, 2002. [Online]. Available: http://search.proquest.com/ docview/275785748 [21] P. J. Minnett et al., “The Marine-Atmosphere Emitted Radiance Interferometer (M-AERI), a high-accuracy, sea-going infrared spectroradiometer,” J. Atmos. Ocean. Technol., vol. 18, pp. 994–1013, 2001. [22] J. R. Eyre, “On systematic errors in satellite sounding products and their climatatological mean values,” Q. J. Roy. Meteor. Soc., vol. 113, no. 475, pp. 279–292, Jan. 1987. [23] J. Bertie and Z. Lan, “Infrared intensities of liquids XX: The intensity of the OH stretching band of liquid water revisited, and the best current values of the optical constants of H2,” Appl. Spectrc., vol. 50, no. 8, pp. 1047–1057, 1996. [24] W. T. Liu and J. A. Businger, “Temperature profile in molecular sublayer near the interface of a fluid in turbulent motion,” Geophys. Res. Lett., vol. 2, no. 9, pp. 403–404, Sep. 1975. [25] R. O. Knuteson et al., “Atmospheric emitted radiance interferometer. Part I: Instrument design,” J. Atmos. Ocean. Technol., vol. 21, no. 12, pp. 1763–1776, Dec. 2004. [26] G. M. Stokes and S. E. Schwartz, “The Atmospheric Radiation Measurement (ARM) Program: Programmatic background and design of the cloud and radiation test bed,” Bull. Amer. Meteorol. Soc., vol. 75, no. 7, pp. 1201–1221, Jul. 1994. [27] J. A. Hanafin and P. J. Minnett, “Measurements of the infrared emissivity of a wind-roughened sea surface,” J. Appl. Opt., vol. 44, no. 3, pp. 398–411, Jan. 2005. [28] N. R. Nalli, P. J. Minnett, and P. Delst, “Emissivity and reflection model for calculating unpolarized isotropic water surface-leaving radiance in the infrared. 1: Theoretical development and calculations,” J. Appl. Opt., vol. 47, no. 21, pp. 3701–3721, Jul. 2008. [29] N. R. Nalli, P. J. Minnett, E. Maddy, W. W. McMillan, and M. D. Goldberg, “Emissivity and reflection model for calculating unpolarized isotropic water surface-leaving radiance in the infrared. 2: Validation using Fourier transform spectrometers,” J. Appl. Opt., vol. 47, no. 25, pp. 4649–4571, Sep. 2008. [30] M. Filipiak, Refractive Indices (500–3500 cm−1 ) and Emissivity (600–3350 cm−1 ) of Pure Water and Seawater [Dataset], 2008. [Online]. Available: http://hdl.handle.net/10283/17 [31] E. Wong, “Retrieval of the skin sea surface temperature using hyperspectral measurements from the marine-atmospheric emitted radiance interferometer,” M.Sc. thesis, Dept. Meteorol. Phys. Oceanog., Univ. Miami, Miami, FL, USA, 2013. [Online]. Available: http://scholarlyrepository. miami.edu/cgi/viewcontent.cgi?article=1440&context=oa_theses [32] T. Y. Chu and R. J. Goldstein, “Turbulent convection in a horizontal layer of water,” J. Fluid Mech., vol. 60, no. 1, pp. 141–159, Aug. 1973. [33] J. R. Herring, “Investigation of problems in thermal convection: Rigid boundaries,” J. Atmos. Sci., vol. 21, no. 3, pp. 277–290, May 1964.

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[34] G. Veronis, “Large-amplitude Bénard convection,” J. Fluid Mech., vol. 26, pp. 49–68, 1966. [35] J. Chang and R. Wagner, “Laboratory measurement of surface temperature fluctuations induced by small amplitude surface waves fluctuations,” J. Geophys. Res., vol. 80, no. 18, pp. 2677–2687, 1975. [36] K. Katsaros, W. Liu, J. Businger, and J. Tillman, “Heat transport and thermal structure in the interfacial boundary layer measured in an open tank of water in turbulent free convection,” J. Fluid Mech., vol. 83, no. 2, pp. 311–335, Nov. 1977. [37] B. Ward, R. Wanninkhof, P. J. Minnett, and M. J. Head, “SkinDeEP: A profiling instrument for upper decameter sea surface measurements,” J. Atmos. Ocean. Technol., vol. 21, no. 2, pp. 207–222, Feb. 2004. [38] R. I. Leighton, G. B. Smith, and R. A. Handler, “Direct numerical simulations of free convection beneath an air–water interface at low Rayleigh numbers,” Phys. Fluids, vol. 15, no. 10, pp. 3181–3193, 2003. [39] L. Rayleigh, “On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side,” Philos. Mag., vol. 32, pp. 529–546, 1916. [40] H. Jeffreys, “Some cases of instability in fluid motion,” Proc. R. Soc. London, vol. 118, no. 779, pp. 195–208, 1928. [41] A. R. Low, “On the criterion for stability of a layer of viscous fluid heated from below,” Proc. R. Soc. London, vol. 125, no. 796, pp. 180–195, 1929.

Elizabeth W. Wong (M’14) received the B.Eng. degree in electrical engineering from the National University of Singapore, Singapore, in 2007 and the M.Sc. degree in meteorology and physical oceanography from the University of Miami, Miami, FL, USA, in 2013, where she is currently working toward the Ph.D. degree in meteorology and physical oceanography. Her research interests include ship-board remote sensing instrumentation and the physics of the air– sea boundary layer. Ms. Wong has been a student member of the American Geophysical Union since 2010.

Peter J. Minnett (M’09) received the B.A. degree in physics from the University of Oxford, Oxford, U.K., in 1978 and the M.Sc. and Ph.D. degrees in oceanography from the University of Southampton, Southampton, U.K. Since 1995, he has been a Professor with the Department of Ocean Sciences, Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, FL, USA. His research interests include deriving accurate measurements of ocean variables, primarily sea surface temperature, by remote sensing from satellites and ships. Dr. Minnett serves as the Science Team Chair of the Group for High Resolution Sea-Surface Temperature.