1. Introduction

Passive remote sensors of airborne chemicals and particulates using long wave IR (LWIR, also known as the “thermal infrared” region) spectroscopy are being used extensively [1–7] for environmental enforcement, atmospheric science, and domestic security applications. In such applications, LWIR radiation from naturally occurring targets in the background (e.g., buildings or the sky) is detected and absorption (or emission) features of chemicals, such as contaminants, interferants, or atmospheric species, are measured and analyzed spectrally. In the LWIR, radiation measurements can be interpreted as a brightness temperature T, which represents the actual temperature of the target when its emissivity ε = 1. For passive detection of chemicals, measurement sensitivity increases as the brightness temperature difference, ΔT, between the target chemical and the radiative background source increases [8–9]. When the atmosphere and the radiating background are near thermodynamic equilibrium, the brightness temperature difference is small (e.g., ΔT < 1 K), and slight variations in ΔT, which may be induced by atmospheric fluctuations, such as wind and turbulence, can significantly affect the measurement. Such variations limit the measurement accuracy, even when other parameters (e.g., detector noise) are well controlled throughout the measurement period. Furthermore, when measurements require long data collection cycles (e.g., 10 – 20 s), atmospheric fluctuations, which characteristically occur at a long time scale, may introduce noise that significantly exceeds other noise components, thereby becoming the sensitivity limiting parameter. Such measurements may also require subtraction of previous measurement results from current data (background subtraction). Since background measurements may be stored for extended periods of time, the temporal coherence of such radiative data may also become a sensitivity-limiting parameter. Additionally, time dependent fluctuations may be spectrally incoherent (i.e., wavelength dependent). Some of the most common passive remote sensors, e.g., Fourier transform spectrometers (FTS) or differential absorption radiometers (DAR) depend on the spectral analysis of the radiometric measurements for the detection and identification of chemical vapors. Therefore, lack of spectral coherence, or band-to band-variations in the atmospheric brightness temperature, can seriously limit the chemical detection sensitivity and specificity of such techniques.

The effect of atmospheric fluctuations on remote sensing was addressed by Menyuk et al. in a series of papers [10–13] where the fluctuations in the backscattering of a CO₂ lidar beam from a hard target were measured. The source of the fluctuations in the lidar measurements was primarily due to variations in the atmospheric refractive index, resulting from eddies causing beam wander and speckle pattern movement. The time scale of these atmospheric fluctuations was determined to be on the order of 1 – 5 ms. Residual temporal fluctuations were noticed for longer time scales (on the order of a few seconds) and were attributed to atmospheric humidity variations. The time scale of atmospheric aerosol movements on wind velocity measurements using a coherent CO₂ Doppler lidar was also investigated [14]. The time scale of such atmospheric aerosol movements was determined to be 2 – 2.5 μs (or shorter) and was attributed to atmospheric turbulence.

Little attention has been given to brightness temperature fluctuations (δT) in the LWIR and their effect on passive remote sensing. The objective of the present work was to measure the magnitude and time-dependence of atmospheric brightness fluctuations and to evaluate the spectral coherence of such fluctuations in number of settings, atmospheric conditions, and along various lines of sight (LOS). Further objectives were to compare the measured brightness temperature variations to the inherent detector noise, expressed by the noise-equivalent-temperature-difference (NETD), and to demonstrate that under certain conditions, the sensitivity (and possibly specificity) of passive remote sensing of airborne chemicals may be seriously limited by such fluctuations. The results presented in this paper include outdoor measurements obtained on two different days, each having different atmospheric conditions (a clear day and a cloudy day), with the sensor pointed towards a building (0.5 km), a mountain (6 km), the horizon (∞ km) and with the sensor pointing towards blackbody source in the laboratory are presented.

2. System description

Atmospheric brightness temperatures were measured using a DAR consisting of two 1 mm ², liquid-nitrogen cooled photoconductive mercury cadmium telluride (MCT) detectors fitted with bandpass filters. The DAR is based on an earlier design [7, 17] and was selected for these measurements because it allows simultaneous detection in two spectral bands. Such simultaneous detection is necessary to determine if fluctuations in brightness temperature are spectrally correlated. Although a DAR was used for these tests, the results are applicable to other remote sensors, e.g., FTS, which rely on multi-band infrared measurements. One detector was fitted with a bandpass filter having a transmission band centered at 9.82 μm, a full-width at half-maximum (FWHM) of 0.19 μm and a peak transmission of 0.85 (Table 1).

Table 1. System specifications for the dual MCT detectors

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The second detector was fitted with a bandpass filter centered at 10.52 μm, having a FWHM of 0.23 μm and a peak transmission of 0.56. Radiation was collected and focused on the detectors with a 6.86 cm diameter f/3 polyethylene Fresnel lens. To remove the 1/f noise that is characteristic to MCT detectors, the incoming radiation was modulated at 1 KHz with a spinning-wheel chopper having a 50% duty cycle. The chopper wheel was coated with a spectrally uniform paint having an emissivity of ε_C = 0.92. To ensure the same LOS for the two detectors (spaced 1.5 mm apart), and to avoid projecting different images onto each of the detectors, the polyethylene lens was cut along its diameter and the two semicircular segments were separated by an opaque spacer with a width matching the detector centerline spacing. The FOV of each of the detectors was determined geometrically to be 0.28° whereas the overlap of their LOS was confirmed experimentally to be within 1°. Using this configuration, the amplitude of the modulated voltage output of each MCT detector was proportional to the difference between the scene irradiance, as collected by the lens, and the chopper wheel irradiance whose emissivity was known and whose temperature T_C was measured. The time varying voltage output of each detector was digitally sampled and demodulated using a fast Fourier transform routine every ten chopper wheel modulation cycles, thereby providing a detection bandwidth of 100 Hz. The magnitude of the 10^th bin of the FFT power spectrum, corresponding to the 1000 Hz modulation rate, was proportional to the radiative contrast between the scene and the chopper blade. The difference S_k between the scene and chopper blade irradiances for the k^th channel can be expressed empirically by:

where A_k is an offset, B_k is the response of the k^th channel to scene irradiance (e.g., atmosphere or other targets within the FOV), W_s is the irradiance of the scene at temperature T and at the center wavelength λ_k of the transmission curve of the k^th channel, C_k is the response of that channel to the chopper wheel’s irradiance (C_k also includes the chopper wheel emissivity ε_C ) and W_C is the irradiance of that wheel at temperature T_C and a center wavelength λ_k . The irradiances W_s and W_C are given by Planck’s function. The response coefficients A_k , B_k , and C_k (Eqn. 1) were calibrated by measuring the irradiance from a thermally uniform hot source at various temperatures. The source that filled the entire FOV of both detectors was coated with a paint having a spectrally uniform emissivity of 0.92. The source temperature could be adjusted and controlled within 1°C and, once equilibrated with the environment, was found to be stable to within 0.1°C. A multiple regression analysis was performed on the resulting calibration data to obtain the response coefficients.

The scene brightness temperature T was calculated from S_k using these coefficients together with the measured chopper temperature T_C , by solving Eq. (1) for W_s (T,λ_k ). T was computed by inverting the Planck blackbody radiation function while assuming a scene emissivity of unity. The brightness temperature T may be significantly different from the actual temperature when the scene emissivity is not unity.

3. Measurement procedure

Time-dependent brightness temperature measurements T(t) were conducted at Charlottesville, Virginia on 10/25/2004 (clear day, afternoon) and 10/29/2004 (cloudy day, morning) with the detector LOS pointing toward (i) a building at 0.5 km (ii) a mountain at 6 km (iii) the horizon and (iv) a laboratory calibration-blackbody at room temperature ~ 50 cm away from the sensor. The air temperature during the clear day tests was ~ 16°C, the wind velocity was ~ 1.5 m/s and the relative humidity was ~ 80%. For the cloudy day, the air temperature was ~ 13°C, the relative humidity was ~ 98% and there was no measurable wind. The two detectors LOS measurements (building, mountain and horizon) were simultaneous for each line of sight but the measurements for the three lines of sight were not simultaneous. Outdoor brightness temperature measurements for the three different LOS were acquired within a 2 hour test period. The time-dependent variation of the brightness temperature as detected by each detector of the sensor is shown in Fig. 1 (for the clear day tests) and Fig. 2 (for the cloudy day). Additionally, the results of laboratory measurements, conducted after the outdoor measurements on each test date, are shown. The brightness temperatures for the horizon and mountain LOS in both tests are lower than the measured temperatures partly because the actual emissivities for these LOS are < 1 (while they were assumed to be unity in the computations). As expected, in the laboratory, while pointing at the temperature-controlled source, both channels recorded the same brightness temperature. However, when the sensor was outdoors, each channel recorded a different brightness temperature and both channels showed pronounced low-frequency, high-amplitude brightness temperature fluctuations δT of a few degrees. While both detectors did show a similar T(t) pattern the brightness temperature (and to a lesser extent the phases) of these fluctuations did not coincide. Owing to the LOS offset of 1° between the two detectors, the spatial displacement between their FOVs is < 9 m when pointing at the building and ~ 100 m when pointing to the mountain. Although some of the differences between the brightness temperatures recorded by the two detectors may be attributed to non-uniform temperature distributions at the targets (e.g., the building), variations of 3–4 K over 10 – 100 m spans are difficult to justify as due solely to the temperature distribution of the solid targets within the FOV. Furthermore, when pointing over the horizon, both detectors view nominally uniform sky and thus the slight misalignment cannot justify these measured differences. Therefore, the measured brightness temperature and the phase differences in the brightness temperatures fluctuations suggest that the irradiance recorded in each of the spectral bands are not spectrally coherent. Such large-scale fluctuations can overwhelm any passive measurement where most sensitivity requirements can be translated into variations of < 1 K in the brightness temperature. By contrast, the measurements in the lab, with the exception of high frequency noise, appear relatively stable over the duration of this test thereby demonstrating the stability of both sensor channels.

 

Fig. 1. Variation of the brightness temperature with time as measured in a clear day (afternoon) by detector 1 (thick lines) and detector 2 (thin lines) for line of sight pointing toward (1) a building at 0.5 km (blue), (2) a mountain at 6 km (red), (3) the horizon (green), and (4) a blackbody at room temperature in the lab (black). Note: the two detectors measurements are simultaneous for each line of sight but the measurements for the three lines of sight are not simultaneous and were taken within ~ 1 hour.

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As in Fig. 1 the laboratory measurements show nearly same brightness temperature at both detectors whereas the outdoor brightness temperatures were different for each detector. Unlike the results obtained during the clear day, the outdoor brightness fluctuations during the cloudy day (Fig. 2) are smaller and are nearly in phase; however, the measured brightness temperatures differ.

4. Brightness temperature fluctuations

In most measurements, where noise is described by i.i.d. (identically and independently distributed), e.g., a normal probability distribution function, the effects of a low signal to noise ratio (SNR) are compensated by increasing the integration time interval Δt. The requirement of independent samples can be interpreted as a process with a short coherence time which is a characteristic of white noise. The requirement of identical statistics implies that the samples originate from the same probability density function. For example, if a total group of n samples includes n₁ samples from a normal (Gaussian) distribution with standard deviation std₁ and n₂ = n - n₁ samples from normal distribution with standard deviation std₂ , the probability density function for the total group of n samples is given by the convolution of the two normal densities, which in this example is a normal distribution with a shifted mean and a wider standard deviation. The standard deviation of the mean of the n samples will be proportional to n ^1/2 only if std₁ = std₂ . If atmospheric brightness temperature fluctuations δT cannot be corrected, e.g., by band-by-band subtraction, the present results suggest that, contrary to common intuition, long integration times may be detrimental to many remote sensing applications. To estimate the effect of the integration time interval Δt on the atmospheric brightness temperature fluctuations δT, the m data points gathered by each detector in each of the tests were segmented into time intervals Δt, each containing N data points.

 

Fig. 2. Same as Fig. 1 but for a cloudy day (morning). The measurements for the three lines of sight are not simultaneous and were taken within ~ 2 hours.

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The variance (${\sigma }_N^2$), i.e., the temperature fluctuations δT²(Δt), in each such time interval Δt (or segment) is:

where T_j is the brightness temperature of the j^th point within the i^th segment and

is the segmental mean brightness temperature. The expected value (given by the expectation operator E[∙]) of the variance ${\sigma }_N^2$ is given by averaging all the variances within the n = m/N time intervals:

The average of the variances of the n segments provides a better estimate of σ_N for a process that is not wide-sense stationary (i.e., a process where the mean and autocovariance of the measurements are not stationary and do vary in time). Note that although Δt is varied throughout this analysis, the actual detection bandwidth remains at 100 Hz independently of the magnitude of Δt. The detection bandwidth is affected by Δt only when the segmental mean temperature μ_i , is calculated, whereas σ_N (Eq. 2) is the measure of fluctuations around a mean of data points collected at 100 Hz. Of course for a white-noise-like process the standard deviation of the mean decreases with segment size as σ_μi = (σ_N )_i/√N.

The variations of the brightness temperature fluctuations, $\delta T\left(\Delta t\right)=\sqrt{E\left[{\sigma }_N^2\right]}$, with Δt for detector 1 as measured during the clear day (Fig. 1) are shown in Fig. 3 (similar results were obtained for detector 2 but are not shown here). Brightness temperature fluctuations that were measured in the lab (black line) remained nearly constant (~30 mK) independently of Δt. This constant level of fluctuations was attributed to inherent sensor noise that may include both the detector (MCT) and sensor electronics noise. This is consistent with the sensor data (Table 1) where a 10 mK noise was projected for the MCT detectors. Similarly, δT(Δt) remains constant at ~30 mK when Δt < ~3–5 s. This too suggests that all measurements, even those performed in the outdoor environment, remained limited by detector or overall sensor noise for sufficiently short detection windows. Since the detection bandwidth was maintained at 100 Hz for all of the points in Fig. 3, the detector (or sensor) noise can be specified as $\frac{3\mathit{mk}}{\sqrt{\mathit{Hz}}}$. Note that the presentation of the detectors noise in units of temperature assumes that the electrical signal - and its fluctuations - varies linearly with temperature over a small range of temperatures (< 1 K). This assumption can be readily justified by expanding Planck’s function and was also verified experimentally.

As Δt increases beyond 3–5 s, δT(Δt) increases dramatically until at Δt = 100 s, δT(Δt) → hundreds mK. Of course as Fig. 1 suggests, these fluctuations are introduced by high amplitude (~ 5 K) low frequency (~1/14 min^-1) oscillations. Clearly, such large fluctuations exceed significantly any detector or sensor noise and can seriously limit any radiometric passive remote sensing application. To further illustrate the consequences of these results consider a detection window of Δt = 20 s, which is often the duration of integration intervals for high sensitivity measurements. As Fig. 3 illustrates, at Δt > 20 s, δT(Δt) > 100 mK for the mountain and horizon LOS. This is significantly higher than the measured detector limit of ~30 mK and thus suggests that the advantage offered by cooled MCT detectors is no longer realized. Furthermore, extending Δt just increases δT(Δt) and further reduces the measurement sensitivity. At the conditions of clear day tests, passive remote sensing of airborne chemicals, either with a DAR or FTS would have reached its maximum sensitivity with an integration interval Δt = 3–5 s with sensitivity diminishing fast with longer detection windows. This counter-intuitive observation impacts numerous remote sensing applications.

 

Fig. 3. Variation of brightness temperature fluctuations δT with detection time window Δt as seen by channel 1 for a clear day using brightness temperature measurements shown in Fig. 1.

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Fig. 4. Variation of brightness temperature fluctuations δT with detection time window Δt as seen by detector 1 for a cloudy day using brightness temperature measurements shown in Fig. 2.

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Figure 4 shows the brightness temperature fluctuations δT(Δt) of the channel 1 data obtained on the cloudy day, corresponding to Fig. 2 (similar results, not shown here, were obtained for channel 2). As in Fig. 3, the detector (or sensor) noise limit of $\frac{3\mathit{mk}}{\sqrt{\mathit{Hz}}}$ was reached in the lab and in all outdoors tests with detection window of Δt < 10 s. Consistent with the results of Fig. 2, the brightness temperature fluctuations are smaller than those for the clear day (Fig. 3). Here, the benefit of longer integration time (and thus larger total signal) can be slightly extended, relative to the conditions that prevailed in the clear day, without increasing the noise level. However, as Δt increases beyond 10 s, the noise increases rapidly and eliminates the advantage of longer integration times, even at relatively uniform atmospheric conditions.

5. Semi-empirical modeling of the brightness temperature fluctuations

Although the exact cause for the brightness temperature fluctuations δT(Δt) was not identified here, the results are sufficient to develop a semi-empirical model that describes the relationship between δT(Δt) and Δt. Such a model can be useful to project the performance of several common passive remote sensors in realistic conditions. With the exception of minor modifications, the derivation follows an earlier and well-accepted approach by Menyuk [11]. When a data set consisting of m data points is divided into n segments, each consisting of N data points, the expression for the expected value of ${\sigma }_N^2$ Eq. (4) of the i ^th segment can be written as:

where $E\left(T_i^2\right)=\frac{1}{N}\sum _{j=\left(i-1\right)N+1}^{\left(i-1\right)N+N}{T}_j^2$ is the second moment of the temperature within that segment, μ_i Eq. (3) is the mean brightness temperature and j is the running index within that segment. It can be shown (see Eqs. (B1) and (B5) of Ref [11]) that:

where μ is the mean of the entire data set (i.e., the global mean) and ${\sigma }^2=\frac{1}{m}\sum _{k=1}^{m}E{\left[T_k-\mu \right]}^2$ is the global variance relative to the global mean of the entire data set. Note that σ can also be stated as: ${\sigma }^2=\frac{1}{n}\sum _{k=1}^{n}E\left(T_k^2\right)-{\mu }^2$ With these results, Eq. (5) can be rewritten as:

where

is the time-lag autocorrelation function for the fluctuations relative to the global-mean μ and N is the time-lag index.

When the correlation coefficient is constant, ρ = ρ ₀ the brightness temperature fluctuations reduce to:

Thus, ${\sigma }_N^2$ approaches zero as ρ_o → 1 (i.e., when the signal is well auto-correlated and the absolute value of its coherence function approaches unity). At the other extreme, when ρ ₀ = 0 (e.g., a white noise), ${\sigma }_N^2$ approach the global variance σ ² with increasing segment size N.

Eq. (7) can be used for a recursive solution for the time-lag autocorrelation function ρ_N as a function of the time lag Δt_N = N×dt

Eq. (10) can be solved using the measured brightness temperature fluctuations δT(Δt) (Figs. 3–4). However, this approach caused propagation of large errors. Instead, the curves for δT(Δt) were smoothed using a 4^th order polynomial:

The coefficients of this 4^th order polynomial (Table 2) provided an excellent fit for δT(Δt) for all tests.

Table 2. Coefficients of the polynomial fit of δT(Δt > 1 s) (Eq. 11) for the six outdoor tests.

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The variation of the time-lag autocorrelation function ρ(Δt) with Δt = N×dt (Eqs. 9–10) was calculated for the results of the outdoors experiments during the clear day (Fig.5) and the results of the outdoors experiments in the cloudy day (Fig. 6). Both figures show strong correlation when Δt < 20–30 s and falling rapidly afterwards to zero at Δt > 100 s. These results are intuitively correct and represent the expectation of high correlation between closely spaced data points and a small correlation as these points are drawn apart. Although not surprising, when the results of Figs. (5 and 6) are introduced into Eq. (5), the general trend of the increase of δT(Δt) with Δt as presented by Figs. (3–4) is reproduced well.

 

Fig. 5. Variation of the time-lag autocorrelation function ρ with Δt for the results of detector 1 during the clear day tests.

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Identification of the exact mechanism that induces the brightness temperature fluctuations δT(Δt) was not the objective of this paper. However, analysis of the time-lag autocorrelation function is likely to be a critical tool in the identification of the controlling process. At Δt < 3 s the measurements (Figs. 5–6) appear to be highly correlated, i.e., ρ → 1. Of course the high-frequency noise at that range is highly uncorrelated but the relatively high SNR (i.e., the small δT that was attributed to sensor noise in Figs. 3–4) at that range still provides for the high correlation at that range. On the other hand, the low correlation at Δt > 30 s (Fig. 5) and Δt > 100 s (Fig. 6) may suggest that atmospheric eddies with periods varying from 30–100 s could be the primary source of these variations. Thus, as long as the sensor’s FOV overlaps a single eddy the time lag autocorrelation is high. But as Δt increases and as eddies are swept by the sensor the autocorrelation diminishes until it fully disappears.

This is consistent with the high amplitude, low frequency oscillation that appeared in the clear day tests (Fig. 1) when viewing both solid targets (building or mountain) and when viewing the atmosphere alone (LOS over the horizon). Even when the sensor was pointed toward solid targets, there was a significant layer of atmospheric air between that target and the sensor. Since all three outdoors tests (including the horizon) viewed a significant layer of the atmosphere and since the oscillation patterns are similar, these high amplitude brightness temperature fluctuations can be attributed mostly to atmospheric effects. Fig. 3 suggests that more than half of the observed fluctuations, δT(Δt), during the clear day were induced by the nearest 0.5 km layer along the LOS to the building. The atmospheric layers to the mountain (6 km) or the horizon only increased the fluctuations by less than twice while the path length along those LOS increased by at least ten fold. The results of the cloudy day tests (Fig. 4) further support these observations. There δT(Δt) when the LOS is towards the building (0.5 km) is even larger then when pointing towards the mountain or horizon.

Several mechanisms can contribute to the fluctuations seen in Figs. 1–2. Clearly the atmosphere consists of cells, or turbulence eddies, of various sizes. The temperature and composition within each such cell may be reasonably uniform but may vary from cell to cell. If the atmosphere could be viewed as an ideal blackbody then only cell-to-cell temperature variations need to be considered to explain the observed brightness temperature fluctuations. However, in that case, both channels will have to detect these fluctuations at nearly the same brightness temperature and synchronously (i.e., in phase). As simple observation of Fig. 1 and as results of the spectral coherence analysis below suggest, these fluctuations do not have the same brightness temperature and are not exactly in phase and thus explanation of the large magnitude of δT(Δt) must also include considerations of optical interaction with atmospheric species along the LOS.

 

Fig. 6. Variation of the correlation coefficient ρ with Δt for the results of detector 1 during the cloudy day tests.

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Figure 7 illustrates the optical transmission along a 5 km horizontal path by water vapor (H₂O and H₂O continuum), CO₂ and O₃ at as computed with MODTRAN atmospheric radiative transfer code [18], for 1976 standard atmosphere at 1 cm ^-1 resolution and standard rural visibility of 23 km. These are the three strongest, naturally occurring absorbers in the atmosphere. Channel 1, with a bandpass filter centered at 9.82 μm, is strongly affected by absorption by O₃ and water vapor but only slightly by CO₂. By contrast, channel 2 with a bandpass filter centered at 10.52 μm has no overlap with O₃ absorption but has a strong overlap with CO₂ and water vapor. If the fluctuations were to be exclusively dominated by O₃ or CO₂ absorptions (or emission) characteristics, either channel 1 or channel 2 would experience strong fluctuations whereas the other would remain nearly unaffected. But as the results of Figs. 1 and 2 suggest, both channels experience strong fluctuations, although not at the same amplitude or phase. A more likely explanation is that the observed fluctuations are interplay between varying concentrations and temperatures of these three dominant absorbing (and emitting) species as various cells, mostly in the near field (0.5 km), are swept through the sensor FOV by the prevailing winds. A likely explanation for the distinct difference between the clear and cloudy days results (Figs. 1–6) is that the atmosphere of the cloudy day was more uniform thermally (i.e., less thermal fluctuations and longer correlation times). The cloud layer is a radiating blackbody source (the emissivity of a thick cloud is nearly unity) and the deep cold sky, which is the main factor for the angular elevation angle dependence of infrared brightness temperature in the atmosphere, is shielded by the presence of clouds. Thus, cloudy skies tend to originate thermodynamic equilibrium (for the LOS) and thus reduce differences among brightness temperatures observed at different wavelengths while enhancing their temporal correlation.

 

Fig. 7. Transmission spectra of atmospheric water vapor (H₂O and H₂O continuum), O₃, and CO₂ along horizontal 5 km path as computed using MODTRAN for 1976 standard atmosphere, with 1 cm^-1 resolution and 23 km visibility.

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6. Spectral coherence analysis

Most remote sensing techniques depend on spectroscopic evaluation of the absorption or emission characteristics of the analytes within the FOV. In many techniques, the spectrum is scanned sequentially (e.g., dispersive scanning spectrometers) whereas in others all spectral bands can be detected simultaneously (e.g., FTS and DAR [17]). When spectra are scanned sequentially it is expected that the brightness temperature of the background source remains unchanged until the last spectral band is detected. Random variations in T(t) between bands may either be inadvertently interpreted as significant or may mask relevant information on the sought after analyte. Techniques that depend on simultaneous measurements of all spectral bands may provide an opportunity to reject δT fluctuations if they are spectrally coherent within the time scale of the measurements. Thus, the spectral coherence aspect and the time scale of the fluctuations may be of importance to the utility and sensitivity of passive remote sensing. The present experiment was designed to provide an initial evaluation of the spectral coherence of the fluctuations in the brightness temperature by measuring simultaneously T(t) and evaluating δT(Δt) at two separate spectral bands. Although ideally, more bands may be needed to fully evaluate the spectral coherence characteristics and to ascertain that overall spectral coherence does exist - detection in just two bands is sufficient to demonstrate lack of coherence.

To evaluate the coherence between the two spectral bands of channels 1 and 2, their power spectra were computed by the Welch method [19] where the estimated power spectrum is computed from overlapping (75% overlap) of 100 s data segments, appodized with a Chebyshev window (relative attenuation of -100 dB for side lobs), and the linear trend was removed (detrended) from the signals. The power spectrum of channel 1 during the clear day experiment is shown in Fig. 8. The high frequency component at f > ~ 0.3 Hz which is attributed to noise carries relatively little power. This is the component at Δt < 3–5 s that was identified in Fig.3 as the detector noise having a value of $\frac{3\mathit{mK}}{\sqrt{\mathit{Hz}}}$. The line of 10 dB above the noise (i.e., SNR=10) shows that the majority of the signal power is contained at frequencies f < ~ 0.1 Hz. This is consistent with visual comparison with Fig. 1 where slow, high-amplitude oscillations of several minutes are modulated by much lower-amplitude fluctuations having cycle times of < 1 min.

The coherence function (cross power spectra analysis), which describes the correlation in the frequency domain between the power spectra of the two channels, was computed by the Welch method [19]. The frequency dependent coherence function between the two detectors for the clear day tests is shown in Fig. 9 and for the cloudy day in Fig. 10. A significant coherence between the outputs of the two channels (Fig. 9) is evident for frequency components f < ~ 0.03 Hz. The coherence approaches zero when F ~ 0.2–0.3 Hz (Δt < 3–5 s). Clearly the lack of spectral coherence at the higher frequencies is consistent with the observation that the fluctuations in brightness temperatures at the short detection windows are dominated by random detector or sensor noise. On the other hand, the relatively long shared atmospheric path length (0.5 km, 6 km, and ∞ km), which is the likely source for most of the brightness temperature fluctuations, may also explain the relatively high coherence between the two signals at low frequencies. Also note that the coherence between the two bands is higher (> 0.8) when the LOS is pointing at the mountain (6 km) and the horizon and is the lowest when the LOS is pointing at the building (0.5 km) thereby reconfirming the earlier assertion that a large component of the fluctuations is introduced in the near field.

A high correlation (large value of the coherence function) between the two channels may allow for correction of part of the brightness temperature fluctuations. However, even when the coherence is high, e.g., 0.7, as much as 50% of the spectral power (0.5 ≈ 1 - 0.7²) is contained in the incoherent portion of the signal. That component cannot be corrected by a linear regression between the two detectors. Therefore, even after band-by-band subtraction, the fluctuations in brightness temperature still introduce significant measurement error.

 

Fig. 8. Power spectrum density of the brightness temperature of Fig. 1 (clear day, channel 1).

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Two sources may contribute to the incoherent component of these measurements. The most obvious is the lack of spatial overlap between the FOVs of the two detectors. Although the FOV of both detectors is 0.28° (Table 1) the LOS offset between the detectors LOS is ~1°. Thus in the near field (up to a distance of 0.5 km) where a large part of the fluctuations where detected (Fig. 3 and 4), any structure with size < 9 m cannot be resolved by the DAR (i.e., the two detectors will not simultaneously view the same structure) and temperature fluctuations within that spatial scale must contribute to the incoherent component. Alternatively, the incoherence can be attributed to variations in the temperatures and concentrations of the three dominant absorbers (Fig. 7) in the standard atmosphere (water, O₃ and CO₂). There was no attempt here to identify the source of this incoherence. However, it should be noted, that if indeed the entire, or most, of the incoherent power is introduced by the LOS offset between the two channels and is attributable to fluctuations in the temperature or concentration of absorbing species between cells that are < 9 m, any remote sensing measurement that requires an integration time longer than 3– 6 s will still suffer from random fluctuations by small cells drifting through the FOV, even if the LOS were overlapped. For example, even the modest 1.5 m/s cross wind that prevailed during the clear day test will carry at least one such cell across the FOV during a 6 s detection window.

Figure 10 shows the variation of the coherence function with detection frequency for the cloudy day measurements. As observed in Fig. 2, the correlation between the two detectors at the low frequencies (f < 0.03 Hz) is much higher than that for the clear day. A likely explanation is that the atmosphere of the cloudy day is thermally more uniform. With a higher humidity, the concentration of the primary absorber (water vapor) is also more uniform. Also, at the high frequencies (f > 0.3 Hz), where random detector or sensor noise is dominant, the coherence between the two detectors approaches zero.

 

Fig. 9. Variation with detection frequency of the coherence function between channels 1 and 2 for the clear day tests (Fig. 1).

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Fig. 10. Same as Fig. 10 but for the cloudy day tests (Fig. 2).

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7. Effects of time integration of brightness

To this point, the analysis focused on the brightness temperature fluctuations within varying time windows Δt. However, in most remote sensing experiments, the measurement consists of evaluating the local, or segmental mean μ_i (Eq. (3)), of the N data points within a preset time window Δt (i.e., integrated measurements). For a fully i.i.d. data set such extended averaging should result in reduction in the standard deviation of the mean as σ_μi = (σ_N )_i/√N. Note that σ_μi is the standard deviation of the means of the various windows and is different from (σ_N )_i which is the standard deviation relative to the local mean of the N measured brightness temperatures of the i^th window. The well known behavior of σ_μ ∝ 1/√N is usually observed when a running mean (moving average) consisting of N points is used to smooth a series of T(t) measurements and to reduce the standard deviation of μ_i or when computing the standard deviation of the mean of N samples within consecutive, non-overlapping segments.

We want to emphasize the difference between estimating the value of standard deviation (δT) of discrete measurements and estimating the standard deviation (σ_μi) of a computed mean temperature within a time window i of N data points. As N increases σ_μi decreases and the estimated (computed) μ_i approaches the true value. We may think of the sampled T(t) values as numbers sampled from several “random number generators”, with different probability density functions that represent; detector noise, electronic noise, atmospheric drift, turbulence, scintillations, etc. In the context of signal processing we view these physical processes as “random number generators” from which we draw samples. Thus each of the T(t) values is sampled from a random number generator characterized by a mean value μ_i and a standard deviation σ_i and δT is our estimate of the standard deviation of the entire process.

The theory of stochastic processes and probability [20] states that the relationship σ_μi ∝ 1/√N, and thus the ability to increase the SNR of measurements by time-integration, depend on the circumstances of the stochastic process. If the stochastic process is mean-ergodic, i.e., if its autocovariance approaches zero with increasing time window Δt, then σ_μi ∝ 1/√N. The autocovariance is an un-normalized correlation coefficient and is given by the numerator of Eq. (8). For example [20, pp. 524–525], a stochastic process for measurements T(t) with exponential correlation coefficient ρ(Δt) ∝ e ^-c|Δt| will be mean-ergodic for a timescale ≫1/c. Therefore, when the correlation coefficient is a slow decaying function (small c) the timescale required for mean-ergodic process is very long and σ_μi will not improve (decrease) as ∝ 1/√N. In the present measurements σ_μi did not decrease significantly with Δt (∝ N) because the correlation coefficient ρ did not decrease with Δt sufficiently fast to meet the mean-ergodic process criterion.

To illustrate the consequence of such Δt integration, the data of the clear day tests were divided into 1 minute sub-tests (i.e., from 0 to 1 min, 1 to 2 min, etc.). The data set within each of such sub-test was segmented into individual windows with 10ms <Δt<15s and the mean μ_i for each Δt segment was calculated. Surprisingly, the standard deviation did not diminish with increasing integration time Δt (or with N), but remained at an almost constant value. Similar qualitative results were obtained for sub-tests of longer durations (e.g., 3 min) for which the allowable (computationally) range for Δt was much longer, e.g., 10ms < Δt < 45s. In Fig. 11 the standard deviation of the mean temperature σ_μi for all the 1 min sub-tests is shown for the clear day measurements of Fig. 1. The error bars indicating the range of variation in σ_μi for the various time windows Δt within each of these tests are too small (< 5 mK) to be shown in Fig. 11. Comparing these strong fluctuations with Fig. 1 shows that indeed between 0 and 1 minute, the brightness temperature of the horizon remained relatively steady and consequently σ_μi for the horizon in Fig. 11 is low. Then from 1–2 min the brightness temperature shown in Fig. 1 declined rapidly thereby translating into a sharply increased σ_μi > 0.8 K in Fig. 11 (second data point). Even more surprising is the large magnitude of the standard deviation, which significantly exceeds the magnitude of δT(Δt) = 30 mK that was recorded in Fig. 3 when Δt < 3 s. However, note that in Fig. 3, the fluctuations were evaluated for individual windows whereas in Fig. 11 they represent the fluctuations throughout the entire duration of the sub-test and thus the high-power, low-frequency fluctuations (Fig. 8) that can be observed even during the first 1 minute sub-test affect directly the standard deviation of the various segmental averages. Since the energy content of these low frequency fluctuations is significantly larger than the power carried by the high frequency components (Fig. 8), they dominate the standard deviation σ_μi calculated over that test window irrespective of the duration of the integration window.

In many remote sensing experiments, chemicals are detected and identified by recording a background spectrum that may contain the effects of pollutants and subtracting it from subsequent spectra. As Fig. 11 suggests, when such background spectra are “aged”, even by as short of time as 1 minute, the uncertainty associated with such measurements increase dramatically. Similar calculations, using sub-test durations of two and three minutes show further increase in the uncertainty of up to 1.7 K.

The marked deviation from the expected behavior of σ_μi = (σ_N )_i/√N warrants a further discussion in the context of the atmospheric correlation coefficient. As was shown by Menyuk [11] the variance σ_μi ² for segmental averaging (which is a fast practical approximation to a moving average) for a global mean-subtracted measurements is given by

where ρ is the correlation coefficient for integrated measurements. The rate of the optimal improvement of ${\sigma }_{\mu }^2$ with 1/N is conditional to the rate the correlation coefficient ρ decreases. For the standard deviation σ_μ to decrease by a factor of Q one must increase the number of measurements in the moving average from N → Q ² N, and at the same time ρ must decrease within the N^th measurement to the (Q ² N)^th measurement such that $1\gg 2\sum _{j=N+1}^{Q^{2}N-1}\left(1-\frac{j}{N}\right){\rho }_j.$

We note that $2\sum _{j=N+1}^{Q^{2}N-1}\left(1-\frac{j}{N}\right){\rho }_j\le \frac{\left(Q^{2}N-N\right)\left(Q^{2}N-N-1\right)}{Q^{2}N}{\rho }_{max}\cong N{Q}^2{\rho }_{max}$ for a maximum constant value ρ _max within the N^th measurement and the (Q ² N)^th measurement. Therefore, to ensure an improvement of the standard deviation σ_μN by factor Q we must obtain a drastic decease of ρ within the N^th measurement to the (Q ² N)^th measurement such that 1≫ NQ ² ρ _max and

For example, assume that initially for N=10 we obtain ${\sigma }_{\mu }^2$ = 1 and we increase the number of measurements to 100 in hopes of improving the standard deviation σ_μN by a factor Q = 10, such that ${\sigma }_{\mu }\to \frac{1}{\sqrt{Q^{2}N}}=0.1$. In order for this reduction to be realized we must ensure that, for the next 90 measurements (the 11^th measurement to the 100^th measurement), the maximum correlation coefficient is ${\rho }_{max}\ll \frac{1}{N{Q}^2}=0.001$. Solving Eq. (12) for the correlation coefficient ρ of the integrated measurements in a recursive manner, similar to the solution for [Eq. (10)], we obtain:

Implementing this solution on the clear day measurements, we obtained the correlation coefficient ρ for the integrated measurements. The correlation coefficient exhibited an almost constant value for an extremely long time period and did not comply with the constraint (Eq. 13) imposed on rapid reduction of ρ with time. Therefore, a significant reduction of σ proportional to 1/√N was not apparent.

 

Fig. 11. The standard deviation for the mean temperature σ_μi for the various one-minute sub-tests of the clear day experiment.

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8. Summary and conclusions

The effect of atmospheric brightness temperature fluctuations δT(Δt) in the LWIR region on passive remote sensing measurements was measured and analyzed by simultaneously measuring the time dependent T(t) by two low-noise (δT ~ 30 mK) MCT detectors equipped with band-pass filters (centered at 9.82 μm and 10.52 μm) at a temporal resolution of 0.01 s. Measurements were repeated on a clear day and a cloudy day and along three nearly horizontal lines of sights. The sensor was pointed toward a building at 0.5 km, a mountain at 6 km and the horizon. The non-stationary measurements (wide-sense stationary) were analyzed for atmospheric brightness temperature fluctuations δT(Δt) by segmenting the data into various time intervals Δt and computing the standard deviation within each segment. Results of all measurements (i.e., both days and along all three LOS) showed that the detector noise limit of δT ~ 30 mK was achieved in all measurements in which Δt < ~ 3 s. However, on the clear day, as the time interval increased beyond 3–5 s, the brightness temperature fluctuations increased rapidly to a few tenths of a degree. On the cloudy day as Δt increased beyond 10 s, δT(Δt) increased to 0.1 K. By analyzing the autocorrelation function of these measurements it was demonstrated that the data is highly correlated up to ~30 s in the clear day tests and ~100 s in the cloudy day tests. This strong correlation, even at frequencies as slow as 0.01 Hz, suggests that atmospheric turbulence eddies may be the source for these brightness temperature fluctuations. Since the spectral band of one of the detectors overlaps the absorption features of O₃ and water vapor, whereas the other band overlaps the absorption by CO₂ and water vapor, temperature variations among eddies must affect the spectral features detected by one channel differently from the other. Thus, as eddies are transported within the FOV, temperature variations among these eddies result in variations in the brightness temperature between the two channels. This was demonstrated by analyzing the spectral coherence of the two channels. At high frequencies, where most of the random detector noise is perceptible, the coherence between the two channels approaches zero. At frequencies < 0.1 Hz, the coherence between the two channels increases rapidly, as would be expected when individual eddies are observed by both channels. However, owing to the high power content at low frequencies, the slight incoherence between the two channels may leave a significant level of uncorrected signal, even after band-by-band subtraction.

These results suggest that the sensitivity and specificity of various passive remote sensing techniques may be seriously limited by atmospheric fluctuations. When the detection time window exceeds 3– 5 s, depending on the atmospheric conditions at the time of measurement, random fluctuations as high as a few tenths of degrees may introduce measurement uncertainty that exceeds the detector noise. The lack of full spectral coherence may limit the potential of correcting these fluctuations using a band-by-band subtraction.

A possible explanation for the difference between the clear day and the cloudy day measurements of δT(Δt) coherence and correlation is that the atmosphere of the cloudy day was more thermally uniform. The cloud layer is a radiating blackbody source (the emissivity of a thick cloud is nearly unity) and the deep cold sky, which is the main factor for the angular elevation angle dependence of infrared brightness temperature in the atmosphere, is shielded by the presence of clouds. Thus, a cloudy sky tends to originate thermodynamic equilibrium (for the LOS) and reduces differences among brightness temperatures observed at different wavelengths, enhancing their temporal correlation.

Finally, the effect of integrating measurements over extended periods for reduction of measurement uncertainty was also analyzed. Results showed that even with integration of as much as tens of seconds, the measurement uncertainty, as defined by the standard deviation of these averages, is dominated by the low frequency fluctuations in the atmosphere and may introduce uncertainties in the brightness temperature as high as 1.7 K. The correlation coefficient ρ for the integrated measurements exhibited an almost constant value for extremely long time periods. Thus the constraint (Eq. 13) imposed on rapid reduction of ρ with time was not met and a significant reduction in the standard deviation of the integrated measurements was not realized. Thus, the potential of improving the signal-to-noise ratio of temperature brightness measurements T(t) by signal averaging (integration) is limited due to their non mean-ergodic property.

To the best of our knowledge, this is the first attempt to quantify the effects of brightness temperature fluctuations on passive remote sensing applications. The conclusions of this work clearly demonstrate that in many realistic conditions, environmental effects are likely to dominate and seriously limit the measurement sensitivity and specificity. Attempts to correct for such fluctuations by long-term integration, band-by-band corrections or background subtraction may provide only limited improvement.