## Abstract

This work analyses and solves for issues encountered when retrieving surface emissivity in LWIR (750 to 1250 cm^{-1}) and MWIR (2000 to 3500 cm^{-1}) bands under outdoor conditions. The Spectral Smoothness method, which takes advantage of high spectral resolution measurements to solve for temperature emissivity separation, and which has already proven its efficiency in the LWIR domain, was applied in an experimental campaign to assess its ability to operate both in the LWIR and MWIR domains. In the MWIR band, directional behaviour of surface emissivity is shown to be a source of systematic errors in the retrieved emissivity and a new method, called SmaC (SMoothness And Continuity), corrects for this error by providing quantitative estimates on the deviation of the surface from Lambertian behavior.

© 2007 Optical Society of America

## 1. Introduction

High spectral resolution infrared (IR) radiometry, available through Fourier-Transform (FT) spectroradiometers, is developing mostly in relation to atmospheric sounding, and object or material identification. In both cases surface emissivity signature is required for either accurate inversion of atmospheric radiances, or characterization of target through specific spectral features. Emissivity extraction from radiance is a well-known under-constrained problem. In addition to providing access to detailed spectral surface information, high spectral resolution IR radiometry opens new ways for emissivity-temperature separation, as illustrated by the Spectral Smoothness Method (SpSm, [3]). Extensive results have already demonstrated the SpSm efficiency in the LWIR band (LongWave InfraRed, from 750 to 1250cm^{-1}) [3, 4, 5, 6]. The present work extends the emissivity retrieval to the MWIR (MidWave InfraRed, from 2000 to 3000cm^{-1}) band taking advantage of radiance measurements sampled at a spectral resolution of 2cm^{-1} throughout the (3–13 *µ*m) domain, identifying sources of errors and assessing their impact on the accuracy of retrieval on surfaces in outdoor conditions from measurements at ground level [1].

SpSm is thus applied to radiances acquired during a dedicated field campaign. The radiance model from which to extract emissivity is introduced in Section 2. Section 3 gives a brief description of the campaign, samples, instruments and associate measurements. In Section 4 issues related to environmental radiances are introduced and discussed. In Section 5, emissivity signatures retrieved from application of SpSm are presented and discussed. In Section 6, problems encountered in the MWIR band are pointed out, analysed and a new method, SmaC, is proposed to correct for such errors by accounting for angular dependence of the surface optical properties.

## 2. Radiance models and emissivity retrieval issues

For modelling purposes the observed target is assumed opaque and homogeneous over the radiometer’s field of view. Moreover, as we focus on ground measurements in the atmospheric windows, the absorption and upwelling radiative transfer between the target and the spectroradiometer will be omitted.

#### 2.1. Lambertian model

Under this Lambertian surface assumption, the at-sensor radiance *R*(*ν*) measured at wavenumber *ν* reads

The observed radiance is composed of an emissive term involving the surface spectral emissivity *ε* and the surface temperature *T* through the Planck function *B*(*ν,T*), and of a reflective term where *I*
^{↓} represents the downwelling irradiance incident on the surface. The first issue encountered in retrieving the surface emissivity refers to the evaluation of the irradiance *I*
^{↓} and will be treated in Section 4. Assuming the irradiance known, the second, fundamental, problem is the so-called temperature-emissivity separation [3, 2]. It amounts to solving an underdetermined system of *N* equations (the *N* radiance spectral values) with *N*+1 variables (the *N* emissivity spectral values and the temperature).

#### 2.2. Directional model

No real surface is strictly Lambertian ([9, 8]) but rather exhibits directional properties. It is thus necessary to introduce these properties into the radiance model and to analyse their impact on emissivity retrieval. Eq. (2) expresses the radiance measured by a sensor viewing a directional surface in the viewing direction (*θ _{v},φ_{v}*).

$$+{\int}_{2\pi}\mathrm{brd}f({\theta}_{i},{\phi}_{i},{\theta}_{v},{\phi}_{v},\nu ){R}_{\mathrm{env}}^{\downarrow}({\theta}_{i},{\phi}_{i},\nu )\mathrm{cos}{\theta}_{i}d{\Omega}_{i}$$

$$+\mathrm{brd}f({\theta}_{s},{\phi}_{s},{\theta}_{v},{\phi}_{v},\nu ){I}_{\mathrm{dir}}^{\downarrow}({\theta}_{s},{\phi}_{s},\nu )$$

The surface is characterized by its spectral emissivity ε, now depending on the viewing direction, and its spectral bidirectional reflectance distribution function (BRDF), *brdf*, depending on both incident (subscript “i”) and viewing directions. The irradiance received by the surface is now separated into directional radiances *R*
^{↓}
_{env}, mainly environmental thermal emission, and solar direct irradiance *I*
^{↓}
_{dir}, which reaches the surface in the (*θ _{s},φ_{s}*) direction. Ω is the solid angle, with

*d*Ω=sin

*θdθdφ*. In the most general case, retrieving emissivity thus implies in addition to Sun irradiance, determination (or knowledge) of anisotropic radiance

*R*

^{↓}

_{env}and target’s BRDF.

## 3. Field campaign and measurements

A field campaign was conducted to validate an emissivity retrieval methodology based on SpSm. It took place in June 2004, during three weeks, at the ONERA (Office National d’Etudes et de Recherches Aérospatiales) center of Fauga-Mauzac, on the PIRRENE (*Programme Inter-disciplinaire de Recherche sur la Radiométrie en Environnement Extérieur*) experiment site.

#### 3.1. Samples and materials

The measurement protocol and the emissivity retrieval method were tested on eleven samples, providing a variety of surface types, materials and emissivity spectra. These samples include soils, rocks and manmade materials. They are listed in table 1 with some information on composition or material characteristics only for indication (in this work, there is no attempt to relate emissivity spectra to sample characteristics).

A LABSPHERE infragold plate was systematically measured before each sample acquisition in order to estimate the downwelling irradiance.

#### 3.2. Instrumentation for radiance measurements

A BOMEM (MR250 Serie) Fourier transform interferometer was used to measure spectral radiances from 750 to 3000 cm^{-1}, with two detectors: MCT from 750 to 1850 cm^{-1} and InSb from 1850 to 3000 cm^{-1}. The spectral resolution was set to 4 cm^{-1} (sampling rate equal to 2 cm^{-1}) and the acquisition time to 6s (100 scans averaging). The spectroradiometer, which has a horizontal line of sight, was held at 1.2*m* above the ground and pointed to an horizontal surface via a 45° gold mirror. The total path length from the surface to the sensor was about two meters and the AMY08 lens leads to a footprint of 20 cm diameter.

In order to increase the temporal radiometric stability of the spectroradiometer in an outdoor environment, the instrument was put into an insulated chamber cooled by an air conditioning

device and shielded from wind. Moreover the interferometer was purged with dry nitrogen. Figure 1 shows the experimental setup.

The temporal stability of the radiance upwelling from the surface was monitored by a broad band LWIR radiometer, while the spatial homogeneity over the spectroradiometer’s FOV was monitored by a LWIR camera. All these three instruments were calibrated on the same M345 MIKRON black body, shielded from wind. Comparisons between interferometer acquired spectra, simulated radiances and reference radiometer measurements showed radiance differences of less than 1% over the whole spectral domain of interest and throughout the field campaign, revealing the good radiometric quality of the interferometer calibration process.

#### 3.3. Ancillary data

Jointly with surface radiance measurements, meteorological measurements were acquired to supply ancillary data for environmental terms evaluation (see Section 4). They consist in temperature and water vapor vertical profiles, provided by the ARPEGE climate model of the CNRM(*Centre National de Recherche Météorologique*). These profiles, extrapolated to ground measurements provided by local humidity and temperature probes, were used in the radiative transfer calculations.

## 4. Obtaining downwelling radiance/irradiance

#### 4.1. Simulation versus measurement approach

Accurate values of *R*
^{↓}
_{env} radiances and solar irradiance are crucial for accurate inversion of Eq. (2). Two possibilities were considered: determination based on simulation using radiative transfer codes, ancillary atmospheric data and environment characteristics, or determination based on a radiometric measurement approach. The use of simulation raises three issues: i) residual errors in instrument spectral calibration as well as atmospheric absorption line modelling induce misregistrations between measurements and simulations, jeopardizing SpSm performance; ii) inputs to modelling and modelling itself are sensitive to non-clear sky conditions; iii) the self emission of the surrounding instrumentation represents a non negligible contribution to the downwelling radiances, which is hard to accurately accounting for. With the measurement approach, the one used in this work, the downwelling radiances/irradiance are now obtained by inverting a spectral radiance measured by the spectroradiometer over a diffuse reflective panel, placed at the location of the sample, which avoids the issues mentioned above. The drawback is that the reference reflective surface is never ideal (reflectivity equal to unity and perfectly Lambertian). The reflective panel used is a LABSPHERE Infragold (S series). As it exhibits relatively strong directional effects, as shown in Fig. 2, its measured radiance *R _{gld}* is inserted into Eq. (2), which is rewritten as Eq. (3) where

*ρ*is the hemispherical - directional spectral reflectance related to the Infragold BRDF,

_{gld}*brdf*, by Eq. (4), and to emissivity by Kirchhoff’s law (

_{gld}*ρ (θ*)=1-

_{v},φ_{v},n*ε*(

*θ*)).

_{v},φ_{v},ν$$+{\int}_{2\pi}\mathrm{brd}{f}_{\mathrm{gld}}({\theta}_{i},{\phi}_{i},{\theta}_{v},{\phi}_{v},\nu ){R}_{\mathrm{env}}^{\downarrow}({\theta}_{i},{\phi}_{i},\nu )\mathrm{cos}{\theta}_{i}d{\Omega}_{i}$$

$$+\mathrm{brd}{f}_{\mathrm{gld}}({\theta}_{s},{\phi}_{s},{\theta}_{v},{\phi}_{v},\nu )\phantom{\rule{.2em}{0ex}}{I}_{\mathrm{dir}}^{\downarrow}({\theta}_{s},{\phi}_{s},\nu )$$

In the LWIR band direct solar irradiance is negligible and solely environmental thermal irradiance contributes to the total irradiance of the surface. On the contrary, in the MWIR band, both contributions exist in daytime conditions and the direct solar irradiance often exceeds the environment term beyond the *CO*
_{2} absorption band (*ν*>2200*cm*
^{-1}). As a consequence of this intrinsic difference in the physical contributions to irradiances between the LWIR and MWIR bands, a specific method is employed in each atmospheric window to retrieve the downwelling environmental terms.

#### 4.2. LWIR band

Eq. (3) reduces to (dependence on viewing angles is omitted for clarity)

It turns out that the required environment radiances *R*
^{↓}
_{env} are not yet retrievable from Eq. (5) even if the directional characteristics of the panel were accurately known. Assuming that the integrands are smoothly varying with angle, the integral of Eq. (5) can be rewritten as the product of two integrals, thus leading to a Lambertian-like expression:

Total irradiance *I*
^{↓} (equal to *I*
^{↓}
_{env} in LWIR) is then obtained from

The directional-hemispherical reflectance of the panel, required in Eq. (8), is measured in the laboratory at a viewing angle *θ*=8°, close to the actual situation (Fig. 2). *T _{gld}* is measured during the campaign using a thermocouple fitted in a hole drilled from the rear side of the reflective panel. As a rule, the high value of the Infragold reflectance makes the emission term very small; hence rather large uncertainties in

*T*may be tolerated. To estimate the error on retrieved irradiance that the Lambertian approximation of the reference panel may imply, simulations of the measured radiance were made in the framework of Eq. (3), introducing a brdf model for the reflective panel, derived from the laboratory measurements (Fig. 2) and adding a coarse estimate of the instrument emission. The error on the retrieved irradiance simulated with Eq.(8) amounts to less than 7%. Although not negligible, this error on the measured environment irradiance has eventually a small impact on the retrieval of the sample’s emissivity as long as the emission term largely exceeds the reflection term (under our field experiment clear sky conditions, the emission term exceeds at least 90% of total radiance in the LWIR band for

_{gld}*ε*>0.8).

Figure 3 presents the relative difference between the total irradiance *I*
^{↓} assessed from the radiance measurement on the reflective panel (on June the 29^{th} at 10h00 UTC) and the irradiance *I*
^{↓}[*SIMU*] simulated with MODTRAN 4.1 ([10]), using the atmospheric profiles described in section 3.3. The difference ranges from -35% to 20% and thus most of the time exceeds the error on *I*
^{↓} due to the Lambertian approximation of the reflective panel. The differences mainly arise from the modelling of the instrument’s irradiance and the spectral mismatch between measurement and simulation. This is a demonstration of the advantage of the measured over the simulated approach.

#### 4.3. MWIR band

Contrary to the LWIR band, direct solar irradiance is not negligible in the MWIR band and may even dominate. For the same reasons as in the LWIR band, the angular integration of the environment irradiance reflected toward the sensor is reduced to a Lambertian-like model used to approximate the environment term of Eq.(3). *R _{gld}* then becomes (dependence on viewing angles is omitted for clarity)

The determination of the searched direct Sun irradiance requires the knowledge of *brdf _{gld}* over 2

*π sr*. Unfortunatly, this term was not available. However an estimate of

*brdf*, noted

_{gld}*br̂d f*, at sun angles (

_{gld}*θ*) may be extracted from Eq. (9):

_{s},φ_{s}$${\u3008\frac{{R}_{\mathrm{gld}}-\left(1-{\rho}_{\mathrm{gld}}\right)B\left({T}_{\mathrm{gld}}\right)-{\rho}_{\mathrm{gld}}{I}_{\mathrm{env}}^{\downarrow}\left[\mathrm{SIMU}\right]\u2044\pi}{{I}_{\mathrm{dir}}^{\downarrow}\left[\mathrm{SIMU}\right]}\u3009}_{2735-2880}$$

〈*X*〉_{2735–2880} is the spectral averaging of the expression in brackets from 2735 to 2880 cm^{-1}. In this spectral range, atmospheric transmission is high and the irradiance level is mainly sensitive to solar angles. As a consequence the simulated direct irradiance *I*
^{↓}
_{dir}[*SIMU*] should be very close to the real one. As for *I*
^{↓}
_{env}[*SIMU*], *I*
^{↓}
_{dir}[*SIMU*] is subject to the same errors (instrumentation emission and spectral mismatching) as noted in Section 4.1. However, owing to its very small contribution to the total irradiance, these errors have a very small impact on the retrieved *br̂d f _{gld}*.

Thus, under the assumption of a spectraly independent *br̃df _{gld}* in the MWIR band, likely for this kind of reflective surface and in the narrow spectral domain [2735; 2880 cm

^{-1}], direct irradiance is obtained from Eq. (9) and (10).

In Fig. 4, the irradiance *I*
^{↓} obtained by assuming the reflective panel Lambertian (curve “Rfl lamber”), deviates strongly from the simulated irradiance *I*
^{↓}[*SIMU*]. When accounting for the directional behaviour of the reflective panel (via the assessment of its BRDF), the deviation is very close to zero in average (curve “Rfl dir”). Still, spectralmismatch between *I*
^{↓} and *I*
*↓*[*SIMU*] leads to differences higher than 5%. The measurement approach used to get the irradiance in theMWIR band both advantages the correction of the high directional behavior of the reflective panel, and the spectral coherence between radiances and irradiances, which is a key requirement for SpSm application.

## 5. Emissivity retrieval - results under Lambertian assumption

#### 5.1. Spectral Smoothness method (SpSm)

From Eq. (1), emissivity can be calculated with an *a priori* temperature by

If T deviates from the actual value, atmospheric spectral features appear in its corresponding emissivity spectrum. SpSm iteratively calculates emissivities for a set of temperature values, and searches for the smoothest emissivity among all calculated ones, by minimizing a smoothness function. Eq. (13) gives the criterion applied in this work:

Different formulations for the SpSm criterion have been tested, including second derivative criteria, high frequency thresholding in the Fourier domain, etc. They all statistically led to the same performance and we chosed to keep the simplest one. Whereas SpSm is not very sensitive to the choice of a smoothness function, SpSm’s performance depends on the spectral domain of application. It turns out that application of SpSm in the LWIR band only is most effective. This is related to the S/N, significantly higher in the LWIR band (Plank’s function maximum), thus keeping the necessary spectral coherence between the radiance terms. As a consequence, SpSm is operated in the LWIR band to give the sample’s surface temperature, that is subsequently introduced in Eq(12) to get the MWIR emissivity spectrum.

#### 5.2. Experimental results

In order to analyse the results obtained with the whole set of 11 samples, the retrieved emissivities are compared to the emissivity spectra measured in the laboratory with a BRUCKEREQUINOX spectrometer equipped with an integrating sphere. This set-up provides a spectral directional-hemispherical reflectance with a 0.03 peak-to-peak accuracy. Although there is no guarantee that laboratory spectra represent the absolute reference, especially because of the lack of spatial coherence between laboratory and outdoor measurements (2.5 and 20 *cm* diameter footprints respectively), this way gives for consistency in the analysis of field campaign results.

Two parameters are calculated for each of the eleven samples measured during the field campaign on the three spectral domains defined as:

The first parameter is the mean difference between the average of all retrieved emissivities and the emissivity measured in laboratory. This parameter is noted Δ_{M/L} and its expression is given in Eq. (14).

$$\mathrm{with}\phantom{\rule{.5em}{0ex}}M[i]=\frac{1}{{N}_{m}}\sum _{i=1}^{{N}_{m}}{\epsilon}_{j}\left[i\right]$$

In Eq. (14), *ε _{j}* [

*i*] is the emissivity retrieved by SpSm from the measurement

*mj*at the wave number

*ν*[

_{i}, ε_{L}*i*] is the laboratory emissivity at

*n*the amount of wave numbers of the spectral domain

_{i}, N_{D}*D*(

*D*∈{

*D*}) and

_{III};D_{IIa};D_{IIb}*N*the amount of measurements done on the sample of concern during the field campaign (

_{m}*N*~9 typically).

_{m}The second parameter is the standard deviation of the retrieved emissivities, defined in Eq. (15).

A synthetic view of the results is given by Fig. (5) which displays simultaneously both Δ_{M/L} and *σ _{m}*. For each sample, three deviation bars, associated to the three spectral domains, are represented. Each bar is centred on Δ

_{M/L}and its half-length is equal to

*σ*. For instance, the “DIIb” bar of the wood sample indicates that the deviation of its retrieved emissivities to the laboratory measurement is around 0.03 (location of the centre), while the spreading of the emissivities around their mean is ±0.04 (half length of the bar).

_{m}LWIR band: The small length, around 0.005, of *D _{III}* bars demonstrates a high reproducibility of the SpSm method in the LWIR band. Moreover, retrieved spectra are close to laboratory ones in this band: deviation between SpSm and lab spectra remains (but for one sample) well below 0.02, confirming the good results of the SpSm method in this band.

MWIR band: In the *D _{IIa}* spectral domain, results remain quite satisfactory for most samples. In the

*DII*domain, on the contrary, spreading is important and large deviations with respect to laboratory measurements occur.

_{b}Examples of retrieved emissivity spectra shown in Fig. (6) and Fig. (7) help explain Fig. (5):

• in the LWIR band, it can be seen that the deviation between field retrieved emissivity signatures and laboratory signatures, although very small on the average, is not spectrally homogeneous; nor is it identical from one sample to another.

• the MWIR band is characterised by the occurrence of two frequent phenomena: a discontinuity around 2400 *cm*
^{-1} and a deviation with respect to laboratory measurement. The discontinuity takes place at the wave-number where Planck’s function at the sample’s temperature equals the environment term. Indeed, when *B*(*ν,T*)=*I*
^{↓} (*ν*)/*π, ε* (*ν*) is undefined according to Eq. (12).

The deviation from laboratory measurements occurs mainly from the discontinuity wavenumber onwards. In this domain, by day-time, the reflection term is comparable to emission. Different interpretations may be put forward:

• i) errors in the estimation of the direct irradiance: because the measurements were made under clear-sky conditions and all samples measured within less than one hour, one would expect quasi systematically the same amounts of error on the retrieved emissivities of the different samples, which is not the case.

• ii) sample surface heterogeneity: no correlation is found between heterogeneity revealed by the TIR images of the surfaces and “anomalies” in emissivity signatures

• iii) directional behaviour of samples’surface. This is the explanation the present work considers the most likely, recalling the strong directional behaviour of Sun irradiance from 2400 cm^{-1} onwards. To support this assumption leads to working with a directional radiance model, dealt with in next section.

## 6. Surface emissivity retrieval - introducing directionality of surface

#### 6.1. Smoothness and Continuity method (SmaC)

### 6.1.1. Directional emissivity model

So far, only directional properties of the reference reflective panel used to get the downwelling irradiances have been introduced. The directional model of Eq. (2), in addition to the angular distribution of downwelling radiances, involves the sample’s BRDF, both quantities being unknown and difficult to assess. In the same way, as for the reference panel in Section (4), a Lambertian sample surface is assumed in the LWIR band.

According to Eq. (9), Eq. (16) gives the sample’s measured radiance, based on such an approximation.

$$+\mathrm{brd}f({\theta}_{s},{\phi}_{s},{\theta}_{v},{\phi}_{v},\nu ){I}_{\mathrm{dir}}^{\downarrow}({\theta}_{s},{\phi}_{s},\nu )$$

Hence, sample’s BRDF has a direct bearing in the MWIR band only. Next step is to assume that angular and spectral characteristics are separable (in other words: BRDF is not spectrally dependent). Again, this assumption is likely and justified by the narrow spectral range (3–5 *µm*) in which it is applied. A directional factor *f*̃ is introduced, brd *f* then verifying Eq. (17).

Using Kirchhoff’s law, Eq. (16) becomes then

$$+\frac{1}{\pi}\left(1-\epsilon ({\theta}_{v},{\phi}_{v},\nu )\right)\left[{I}_{\mathrm{env}}^{\downarrow}\left(\nu \right)+\tilde{f}({\theta}_{s},{\phi}_{s},{\theta}_{v},{\phi}_{v}){I}_{\mathrm{dir}}^{\downarrow}({\theta}_{s},{\phi}_{s},\nu )\right]$$

Finally, the emissivity is then derived from Eq. (18)

$$\mathrm{with}\phantom{\rule{.5em}{0ex}}{\tilde{I}}^{\downarrow}({\theta}_{s},{\phi}_{s},\nu )={I}_{\mathrm{env}}^{\downarrow}\left(\nu \right)+\tilde{f}({\theta}_{s},{\phi}_{s},{\theta}_{v},{\phi}_{v}){I}_{\mathrm{dir}}^{\downarrow}({\theta}_{s},{\phi}_{s},\nu )$$

Determination of emissivity from Eq. (20) thus requires the assessment of *f*̃ in addition to the temperature. Next section describes the way it is done.

### 6.1.2. Smoothness and Continuity method (SmaC)

Since discontinuities and biases observed on the retrieved spectra in the MWIR band are expected to be caused by the surface directional behaviour, assessing the form factor *f*̃ goes with correcting these discrepancies. In the same way as SpSm searches for the temperature which corresponds to the smoothest emissivity spectrum, *f*̃ will be found by looking for the most continuous spectrum. This method, complementary to SpSm in the MWIR band, is called Smoothness and Continuity (SmaC).

The first step of SmaC is to determine the wave-number of discontinuity occurrence, noted νc in the following. To achieve this, SmaC looks for the crossing point between Planck’s function and downwelling irradiance (Section 5.2).

Once *ν _{c}* is found, discontinuity is estimated using two criteria. The first one, denoted Δ, measures the difference between the emissivity value when

*ν*tends to

*ν*from upper wavenumbers and when ν tends to νc from lower wavenumbers (Eq. (20)).

_{c}The way to numerically determine Δ is illustrated on Fig. 8: Two domains are selected around *ν _{c}* (on the “left” and on the “right”), avoiding the high divergence zone (blue curves of Fig. 8). A linear regression is applied on both domains, and the resulting difference at

*ν*, Δ, is calculated. Each domain contains twenty wave numbers in order to minimize the noise effects and properly represent the trends of the emissivity spectrum on both sides of

_{c}*ν*.

_{c}A second criterion is added to the Δ criterion: it computes the standard deviation of the divergence zone around *ν _{c}*. Indeed, correctly assessed emissivity should be undetermined at

*ν*only, and should not diverge around it (Fig. 8 is an example of the contrary).

_{c}SmaC attempts therefore to look for the directional factor *f*̃ which minimises a weighted sum of both previously described criteria (a higher weight is given to the D criterion). The temperature used is the one assessed by SpSm.

#### 6.2. SmaC results

### 6.2.1. Impact of SmaC application

An example of SmaC applied to sample stone#1 is shown on Fig. 9. Both discontinuity and deviation disappear almost totally in the MWIR band. The LWIR band results are the same as SpSm original ones, because directional effects are taken into account in the MWIR band only (Eq. (16)).

A synthesis of SmaC results is presented in Fig. 10 and Fig. 11. The presentation is the same as that for SpSm results in Section 5. From Fig. (10) is can be seen that SmaC improvement over SpSm alone does exist, though only very little in the *DIIa* domain (from 1900 to 2300 cm^{-1}). This may be explained by the weak impact of *E*
^{↓}
_{dir} in comparison to *E*
^{↓}
_{env}. From Fig. (11) it is clear that SmaC improves significantly (but for one sample) the retrieval of the emisivity spectrum in the *DIIb* domain (from 2400 to 3000 cm^{-1}): the deviation of SmaC retrieved emissivities from lab values is less than 0.02 in most cases, and the dispersion is also reduced, which indicates a good robustness with respect to variations of angle of incidence of direct Sun irradiance.

### 6.2.2. SmaC and surface directional factor

In addition to improving emissivity signatures in the MWIR band, SmaC method attempts, through the directional factor *f*̃, to provide information on the directional behaviour of the observed surface. Hence, all results obtained with the set of 11 samples in the MWIR band, presented on Fig. (12) to (22), show SmaC signatures (compared to lab) and the retrieved directional factors for the three available dates and time-of-day. To illustrate, special attention may be paid to negev (Fig. (16)) and stone#1 (Fig. (17)). First, as one can expect for a sandy soil, negev looks reasonably Lambertian since the retrieved directional factor varies little with Sun angle and is close to unity. On the contrary, stone#1 looks highly directional with angle dependent retrieved directional factor. As for the large differences observed in retrieved directional factor between dates, this is tentatively assumed to reveal impact of measuring configuration on a highly directional observed surface. Indeed, to avoid spectroradiometer’s shadow projection on the sample, the viewing angle had to be changed (within +10, -10°) from morning to afternoon. As *f*̃ depends on viewing angle and Sun zenith and azimuth angles, this might explain on the one hand the dynamical range of retrieved values, on the other hand the clustering of values according to period and time of acquisition. These remarks are valid for all other samples: all three sand-type samples (negev, sand#1 and sand#2) exhibit directional factors close to unity, whereas directional factors for likely highly directional samples significantly deviate from unity. In this way, *f*̃ results, although only few, look coherent, and seem to give relevant information on directional behaviour of measured samples.

## 7. Conclusion

Emissivity assessment from a radiance measurement is an ill-conditioned problem, which can only be solved under constraint. The Spectral Smoothness (SpSm) method relies on the higher irregularity of the spectrum of atmospheric irradiance compared to the emissivity spectra of natural surfaces. SpSm is applied in this work to exploit radiance measurements acquired with a high-resolution (4 cm^{-1}) spectroradiometer on various kinds of surfaces obtained in out-doors conditions. Special attention is devoted to the accurate determination of the environmental terms (mainly atmospheric radiances and solar direct irradiance) involved in the signal, while maintaining the spectral matching necessary for applying SpSm.

The method used relies on using a gold reflective reference panel. It is shown that, although this surface is far from Lambertian, and without *a priori* knowledge of the panel’s BRDF, a proper combination of measurements and simulations of atmospheric spectra allows assessment of the environmental terms accurate enough so that residual errors have a negligible impact on the retrieved emissivity signature. SpSm is known to give good results in the LWIR (750 to 1250 *cm*
^{-1}) which work confirms. When the spectral domain of interest includes the MWIR band (2000 to 3000 cm^{-1}, 3.33 to 5 *µm*) extraction of emissivity signature from radiance faces specific problems besides emissivity-temperature separation issue, experimentally revealed through divergences and biases. Based on observations, it is proposed in this work that the origin of such biases lies in the directional (*ie*. non Lambertian) properties of the observed surface. A new method, extension of SpSm, the Smoothness and Continuity (SmaC) is introduced and tested. Elimination of divergence and biases relies on introducing a surface directional factor *f*̃, relating surface BRDF and emissivity, and searching for the *f*̃ value that best realises the constraint of spectrum continuity.

SmaC method is shown to be effective and informative: i) the retrieved signature over the whole spectral domain is improved. Assessment of performance is given by retrieved emissivity deviation to laboratory measured spectra. Statistically, SmaC emissivity deviation to laboratory remains less than 0.02 in the LWIR band (SmaC is identical to SpSm in this band), and less than 0.03 in the MWIR band, in which SmaC improves SpSm retrieval. ii) the retrieved directional factor provides information on the directional properties of the target, without a priori knowledge or independent determination of the BRDF (*ie*. directional factor close to unity for a sandy soil as expected). More experimental work and detailed analysis are needed to assess the quality of this information.

In this context ONERA is developing a means for directional measurements of optical properties in outdoor environment based on SmaC, called MISTERE (*Moyen Infrarouge Spectrom étrique de Terrain pour l’Evaluation des Réflectivités et Emissivités*). The retrieved angular emissivities and directional factors are planned to be validated with REMUSA (*REflectance MUlti-Spectrale et Angulaire*), a laboratory instrumentation for BRDF measurements at infrared wavelengths, currently under development.

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