Introduction

Spatial filtering techniques are well known through their wide usage in reducing noise effects of different observing systems. The nominal objective is to determine a function (filter) over a spatially varying measurement that provides an optimal (in some sense) measurement estimate. Currently, in the routine processing of the Geostationary Operational Environmental Satellite (GOES) sounder spectral measurements, linear averaging over 3 × 3 elements is used as the spatial filter for all spectral channels. This uniform spatial filter inadequately reduces the noise effects for some channels and overly smoothes the spatial variations for other channels. Our goal is to find an optimal size for the averaging area, also called the field of regard (FOR) in this paper, for each GOES-8 sounder spectral channel.

In section 2, we consider the physical aspects of the noise filtering problem in atmospheric remote sensing. In section 3, a traditional mathematical model of noise filtering is introduced; then in section 4 we consider the results of a statistical spatial analysis of the spectral measurements and estimate the appropriate size of the spatial filter for each GOES-8 sounder spectral channel. In section 5, results of temperature profile retrievals using spatially filtered spectral measurements are presented. Conclusions are offered in section 6.

Physical considerations

GOES sounder measurements contain noise that varies substantially from one spectral channel to another. The measured radiances are a composite function of the optical properties of the GOES sounder instrument and the thermodynamical properties of the “earth surface and atmosphere” system. The latter are described by the surface emissivity, surface temperature, and atmospheric vertical profiles of moisture and temperature. These parameters have different spatial distributions (that can also be thought of as spectra). Surface properties of land can vary drastically and are found in the shortwave part of the spatial spectrum. Atmospheric moisture fields are smoother and correspondingly they are attributed to the median-wave part of the spatial spectrum; and atmospheric temperature fields are attributed in the first approximation to the longwave part of the spatial spectrum. Temperature and moisture fields become smoother with increasing height, and thus their spatial spectral properties shift in the direction of the longer spatial waves. We observe these spatial properties in the GOES sounder spectral measurements of outgoing thermal radiances of the “earth surface–atmosphere” system; a spectral measurement describes a specific atmospheric layer (sometimes in combination with the surface) and will have the corresponding spatial properties. Spectral bands sensitive to upper tropospheric and stratospheric layers, where radiances are not disturbed by clouds or the surface, exhibit large spatial uniformity and, therefore, shortwave spatial variation of the radiances describes the noise component of these measurements. The differences between spatial spectrums of measurement noise and meteorological parameters can be effectively used to reduce the influence of noise on the solution of the atmospheric profile retrieval;the objective is to estimate the spatial distribution of a meteorological parameter on the basis of measurements in the radiance spectrum.

The physical model of a spectral measurement is based upon the radiative transfer equation (RTE), which is Fredholm’s equation of the first kind. The corresponding inverse problem is ill posed. There is no one-to-one relation between spectral measurements and meteorological parameters. The solution of the problem is unstables; a small variation in a measurement and/or a physical model of a measurement can cause significant variation in the solution. It must be noted that the RTE is nonlinear with respect to all parameters: surface emissivity and temperature as well as atmospheric temperature and moisture profiles. For such an ill-posed problem, the mean solution of individual measurements (after spatial averaging of solutions within the field of regard) could substantially differ from the solution for the mean measurement (after spatial averaging of measurements within the field of regard). Thus, noise filtering will depend on the noise spectral distribution, the signal spectral and spatial distribution, the physical properties of the desired parameters and model parameters, and the numerical properties of the corresponding inverse problem.

Noise filtering

We consider a formalism for noise filtering based on linear estimation. The measurements are described by the model

fx, yfx, yξx, y(1)

where f(x, y) is a precise value of a function at a coordinate point (x, y), and f(x, y) is a measurement contaminated by a random noise ξ(x, y). Our objective is to construct a function F[f], which provides a measurement estimate f̂ = F[f] with the desired characteristics. When constructing F[f], we assume that ξ(x, y) is not correlated with f(x,  y), and has a zero mean and spatial structure

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where σ²_ξ is noise variance and δ(x − x′) is the delta function defined by f(x′) = ∫ f(x)δ(x − x′) dx, the symbol (. . .) designates the operation of averaging. This is the traditional “white” noise model. Furthermore, we consider f(x, y) to be differentiable. We define spatial averaging, F[f], over the rectangular domain by the equation

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where the unknown parameter Δ defines the dimension of the spatial averaging. Using a first-order Taylor expansion of f(x′, y′) at (x, y)

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where R(x′, y′) is the remainder term, we obtain

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The variance of the measurement estimate f̂ is defined by

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where δf = f − f̂ and the symbol δ is now used from here on to designate parameter variation. For the white noise model expressed in (2), the linear filter F[f] will be optimal when σ²(δf) is minimized for

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(here Δ is dimensionless and β is the corresponding conversion parameter with dimension km⁻¹). Supposing that

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over the averaging area and discarding the cross derivative ∂²f/∂x ∂y, which is filtered out by the integration in (6), the term R(x, y) can be approximated by

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Last, we obtain the relation

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which defines the optimal dimension Δ̂ for the spatial averaging

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Equation (11) can be used for verifying a priori information about the statistical properties of the noise and for estimating the spatial properties of the measured spectral fields and the corresponding meteorological parameter fields. For that purpose, we have to determine σ²[δf(Δ)] for different values of Δ.

In practice we cannot measure σ²[δf(Δ)]. Only the measurement of f(x, y) is available, and the corresponding function σ²[δf(Δ)] for f(x, y) is defined by (7) to be

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We can assess the applicability of the white noise model by inspecting whether or not (13) correctly describes σ²[δf(Δ)] as a function of Δ. If we know from a priori meteorological information that the spatial variations of a parameter in a given atmospheric layer are small, so that

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then we must obtain good correspondence between two measurements of σ²[δf(Δ)] for close values Δ₁ and Δ₂. If σ²[δf(Δ₁)] and σ²[δf(Δ₂)] satisfy (13) under condition (14), then the signal variation is negligible and the dimension of averaging can be increased to Δ₃ > Δ₂ > Δ₁. In addition, (13) can be used for estimating the spatial variability of a field, if we know that σ²_ξ ≪ 1.

In principle, we can use this approach {measuring σ²[δf(Δ)] for different Δ} to estimate model components, σ_ξ, M̈ from which we can estimate the optimal Δ̂ from (12). However, in practice, the measurement f(x, y) contains noise that is substantially different from the white noise model used. Thus all the relations introduced in the previous paragraphs would only represent approximations and can be used only as a reference in the data analysis.

We need to solve (13) for the two unknowns σ_ξ and M̈ to establish a robust spatial analysis procedure; but (13) describes the signal properties and we need an estimate for describing the properties of the meteorological parameter field in (3). To construct the required estimate, we suppose that the mean field f(x, y) from (1) is sufficiently smooth, so that it is three times differentiable with small variations of second derivatives and the negligibly small third derivatives

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Then we can apply the Laplace operator

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to the estimate f̂(x, y) = F[f(x, y)]. The operator L is widely used in image processing (Hall 1979; Rosenfeld and Kak 1982). Using (3) and (5), the differential of the resulting field can be expressed in the form

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In (16) Σ(ξ) is a function of noise, which can be written:

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For the noise model in (2), Σ(ξ) is characterized by Σ(ξ) = 0, Σ(ξ)² = 4βΔσ²_ξ.

The last two terms of (16), in accordance with (15), can be discarded. Considering the spatial variations (δx)² ≈ (δy)² ≈ (β⁴Δ²)⁻¹ for large Δ, we obtain

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Thus the differential (16) has a mean

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and a variance

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It follows that δL[f̂(Δ)] describes the spatial properties of the signal model f(x, y), and σ²{δL[f̂(Δ)]} describes properties of the noise model ξ(x, y). Of course, (18) and (19), like (13), will only be approximations; but the order of magnitude of their dependencies on Δ will be important for us in analyzing (1) and (2).

Last, we define the corresponding relations for discrete rather than continuous measurements, f_i,j. Given Δ = (2n + 1)/2, where 2n + 1 is number of elements in one dimension of the square averaging domain, (13) is written

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and (16) has form

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Analysis of measurements

GOES-8 sounder spectral channels and their intended purposes are presented in Table 1. The temperature-weighting functions, moisture sensitivities, and atmospheric transmittances are shown in Fig. 1. It follows from Fig. 1 that there are many similarities in the vertical distribution of the contributions to the measurement among the different longwave and shortwave spectral bands: channels 3 and 15 (in Fig. 1a), channels 4, 5, and 14 (in Fig. 1a), and channels 6, 7, 13, and 16 (in Fig. 1b) have very similar weighting functions. From Figs. 1b,c, it follows that the measurements in channels 10 and 11 will have similar spatial characteristics with respect to both temperature and moisture. Then the horizontal spatial structure of measurement variations in channels 10 and 11 with respect to the temperature should correspond to the horizontal spatial structure of measurements in spectral channels 3–5 and 13–15.

Instrument noise values (converted into a temperature value for a target temperature appropriate to each spectral channel observing a typical meteorological scene) are presented in Fig. 2. Significant variations from one spectral channel to another are evident. The statistical functions (13–21) have been applied to measurements from 1000 UTC on 13 August 1999; this nighttime scene was selected because the conditions on the surface were somewhat homogenous and solar reflection in the shortwave bands was avoided. All 18 channels were analyzed. We are especially interested in channels 1, 2, 3, 4, 12, 14, and 15 because they are substantially affected by noise. They are not disturbed by surface effects (see Fig. 1) and are sensitive to layers of the atmosphere where the temperature fields are sufficiently smooth. We expect the spatial smoothing to be most effective for noise reduction in these channels.

The spatial characteristics of the measurements in channel 1 are presented in Table 2. Here, N is the number of samples (in the picture fragment); N was a variable because only those locations (i, j) that satisfied the following spatial condition were included in the sample:

f̂_i+k,j+1f̂_{i,jk,1=−1,0,1}(22)

Equation (22) was used to filter out obvious measurement errors and measurements in inhomogeneous scenes (e.g., partial cloudiness in the field of view). In constructing f̂ and calculating the associated statistics, we used measurements within the range 180 K < f < 340 K. Analogous to (21), we introduced

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to describe the spatial variations in the original signal. From (2), we have the noise model:

σ²δLf5σ²_ξ4M²₂η²(24)

where η is a space increment in a measurement matrix. Table 2 contains results for the median and the linear estimate of the noise. For the median f̂_1/2, the error estimate ξ̂_1/2 relates to the linear estimate error ξ̂ by

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Next we analyze the noise model from (2) using the appropriate value of σ²[δf(Δ)] from Table 2. The corresponding graphics are presented in Fig. 3. The measurements demonstrate substantially different properties than those predicted by the white noise model. This could be explained by the fact that the observed noise is as a mixture of two stochastic components,

ξζϑ,(27)

where component ζ is the white noise, and component ϑ is described by another stochastic model differing from (2). If ζ and ϑ are not correlated, and ϑ does not correlate with the signal f(x, y), then we obtain for (27) by analogy with (13) and (20),

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where ρ[ξ̂(Δ), ϑ] designates some correlation function for the noise ξ̂(Δ) in estimate (5) and the noise component ϑ in (27). In order for (28) to describe Fig. 3, ρ[ξ̂(Δ), ϑ] must be a decreasing function of Δ and ϑ² > ζ². Both conditions are physically realizable and mathematically reasonable with respect to the basic model expressed in (1)–(3). Then from (28) it follows that the variance of observed noise (27) is described by the relation ζ² + ϑ² ≥ σ²(δf̃) for large Δ and ω² ≈ Δ⁴M̈² ≪ 1. From Table 2 it follows that in channel 1 the noise variance has the level

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We compare σ²(ξ) from (29) with σ²δL(f) (last column in Table 2), which can be expressed by analogy with (24) for noise model (27) as

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where ψ(ϑ) is a positive function 0 ⩽ ψ(ϑ) ⩽ 1, describing the spatial correlative properties of the noise component ϑ. We find (29) and (30) are not compatible for the estimates given in Table 1, if ψ(ϑ) ≡ 0. It indicates that the noise component ϑ is spatially correlated. Inequalities (29) and (30) confirm this. The estimate δL[f̂(Δ)] from Table 2 along with (29) and (30) indicates that the noise model and corresponding estimate (18) do not describe properties of the observed noise; the model predicts an exponential decrease of δL[f̂(Δ)] with increasing Δ; in our experiment δL[f̂(Δ)] is close to a constant. The noise estimates from (29) and (30) are in good agreement with the data presented in Fig. 2; estimates of σ²(ξ) are less than the predicted value (≈4 K²) after filtering the noise by (22). The filtering (22) also explains the number of samples N(Δ) associated with the increasing dimension of the field of regard: the larger the Δ, the smoother the estimated field and correspondingly the fewer of estimates discarded.

To summarize the results of the spatial analysis of the spectral measurements in GOES sounder channel 1, see the following:

1. The measurements are very contaminated by noise, so that substantially different properties are observed than those described by a white noise model. The noise is spatially correlated.

2. Variations in the thermal field are sufficiently small that a noise spatial filtering with an averaging domain of 11 × 11 elements can be used. The comparison of median and linear estimates shows that the latter is preferable. The linear estimate indicates that the approximation accuracy variance is better than 0.06 K² for a field of regard with 11 × 11 elements.

Estimates of spatial roughness σ²[δL(f̂)] of averaged measurements for median and linear estimates in spectral channels of the GOES-8 sounder for different averaging domains (or fields of regard) are presented in Table 3. The spectral field is modeled using the temperature and moisture fields derived from the Eta Model forecast (Black 1994; Rogers et al. 1996) up to 100 hPa and extrapolated into the upper pressure levels 100–0.1 hPa. The spatial roughness is less than 0.057 K² in all spectral channels under a constant surface temperature and emissivity. The minimum spatial roughness of the averaged measurements should be found in the stratospheric channels 1 and 2 and upper-tropospheric channels 3, 12, and 15 (see Fig. 1), starting from 0.057 K². The spatial roughness should increase in the midtropospheric channels and reach a maximum of about 0.44–0.60 K² in channels 7, 8, 17, and 18 (the “atmospheric window”). Using (25) to compare the spatial roughness for the median and linear estimations, we find that the linear estimate is optimal in channels 1, 2, 3, and 12. Figure 4 shows the spatial roughness σ²[δL(f̂)] with spatial averaging of 3 × 3 elements for all the spectral channels of the GOES-8 sounder (see Table 3, median estimate n = 1). The spectral distribution of spatial roughness does not correspond to the associated horizontal thermal structure of the atmosphere; consequently, the solution to the inverse problem will not reproduce them either. Comparing Figs. 2 and 4 shows that spatial roughness from Fig. 4 reproduces spectral distribution of the noise maximums from Fig. 2. This means that a uniform 3 × 3 spatial averaging is not effective.

Figure 5 contains the corresponding characteristics for measurements in channels 1–8. Statistics for channel 2 are analogous to the statistics for channel 1. The quantity σ²₂(δf) slowly increases from 1.3641|_n=1 to 1.596|_n=5 K². This implies that the spatial variation of the associated temperature field is small in comparison with the spatial averaging under consideration. We can use spatial averaging with a domain of 11 × 11 elements (Δ₂ = 5). The linear estimate suggests that

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Repeating the steps (27–30) for channel 2, we obtain

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To satisfy both inequalities, we again must consider that the observed noise in channel 2 is spatially correlated. The noise estimates are in good agreement with data presented in Fig. 2. Here σ²(ξ₂) is slightly less than the predicted value (≈2 K²) after the preliminary noise filtering by (22).

Statistics for channels 3 and 4 demonstrate different properties than the statistics in channels 1 and 2. We observe a linear increase of σ²(δf) with increasing Δ that can be attributed to the signal spatial variation. Channel 8 atmospheric window measurements, which have a maximum surface contribution and therefore significant spatial variation, demonstrate a linear increase as well. Thus channels 3 and 4 cannot be processed in the same way as channels 1 and 2. Signal spatial variations σ²[δL(f)] for channels 3 and 4 equal 4.0 and 4.1 K², respectively, which is 3 orders of magnitude larger than the spatial variation obtained for channel 2 (Fig. 5c). Measurements in channels 3 and 4 are not contaminated by surface effects or atmospheric moisture. Channel 3 must be spatially smoother than channel 4. In Figs. 5a,c, we find that the appropriate dimension for the spatial averaging is n₃ = 2 for channel 3 and n₄ = 1 for channel 4. Figure 2 suggests an approximation accuracy of better than 0.2 K for channels 3 and 4. Measurements in channel 4 can be affected by clouds, thus a median estimate is preferable.

Channel 5 measurements are noticeably noisier (by a factor of at least 3) than measurements in channel 4 (after averaging) and channel 6 (Fig. 2). Channel 5 measurements are affected by the surface and midtropospheric moisture (see Figs. 1c,d). Figures 5b,d statistics for channel 5, σ²(δf₅) and σ²[δL(f̂₅)], show features associated with window channels, in which measurement variations are mostly affected by spatial variations in surface temperature/emissivity and lower-tropospheric moisture. Channel 5 should have substantially different spatial properties than an atmospheric window, but Fig. 5d shows that the atmospheric signal in channel 5 has features dominated by the large surface signal in channel 8 (with respect to the spatial averaging). This contradiction suggests that the statistics in channels sensitive to the mid- and lower troposphere are affected by another factor. Another indication is the linear increase of σ²[δL(f̂_6–8)] with respect to Δ (Fig. 5b). According to (13) and (20), σ²[δL(f̂_6–8)] should increase with Δ⁴; the discrepancy between measurement and prediction could be explained by an interaction of nonhomogeneity effects in the data sample between “cloudy–clear” nonhomogeneity of atmospheric conditions, and “land surface–sea surface” nonhomogeneity of surface conditions. In different conditions, the estimate of the second derivative M̈ in (13) and (20) will be substantially different. For example, in overcast conditions window channel measurements can be spatially smoother than atmospheric channel measurements. The same can be true for measurements in cloudless atmospheres over sea surfaces. Last, in channel 5 we note that spatial averaging n₅ = 1 avoids the spatial oscillation of moisture estimates in the midtroposphere (see Fig. 1c) and provides the same spatial smoothness as channel 4: σ²[δL(f̂₅)] = 0.4 K². In comparison, the natural signal spatial variation is σ²[δL(f₅)] = 4.7 K².

Channels 6–8 have significant signal spatial variation. The statistics σ²[δL(f_6–8)] have values 6.0, 6.2, and 6.0 K², respectively. There is no physical reason to apply spatial averaging with these signal properties.

Statistics for channels 9–16 are shown in Fig. 6. Measurements in channel 9 depend on surface temperature and emissivity, upper-tropospheric temperature, and ozone. Channel 9 statistics are similar to those in channels 4 and 5 in Figs. 5a,c, from which we conclude that n₉ = 1 should be applied. Measurements in channels 10–12 depend on mid- and upper-tropospheric temperature and moisture. The surface slightly affects measurements in channel 10. The statistics σ²[δL(f_10–12)] have values 3.1, 2.4, and 9.1 K², respectively. For comparison, in channels 7 and 8 the corresponding statistics have values 6.2 and 6.0 K², respectively. Measurements in channels 10 and 11 are smooth, in comparison with channels 3–8. Channel 12 is noisy. Figure 6a shows that channels 10 and 11 statistics have spatial variations. Figure 6c shows that spatial filtering in channels 10 and 11 can cause a loss of information regarding moisture spatial variations. In Figure 6a, σ²(δf₁₂) is slowly increasing with increasing n (or Δ). Channel 12 reveals the spatial variations of midtropospheric temperature and upper-tropospheric moisture; both fields are sufficiently smooth. Corresponding thermal fields are described by measurements in channels 3 and 4 (see Fig. 5a), for which we estimated the spatial variation to be 0.18–0.38 K². From Fig. 6c, we see that the filter dimension n₁₂ = 2 provides the required spatial smoothness in channel 12; a linear estimate is preferable. In Figs. 6a,c, the observed noise in channel 12 is noticeably larger, by a factor of at least 4–5, than the noise in Fig. 2. As with channels 1 and 2, we conclude that the noise in channel 12 has a spatially correlated component. When filtering channel 12, the scale of spatial variations in channels 11 and 12 must be similar (see Figs. 1b,c). To make them comparable in this sense, we use the filter with the dimension n₁₁ = 1 in channel 11 (see Fig. 6c). Last, we conclude from the statistics that channel 10 n₁₀ = 1 should be used to provide vertical homogeneous property to the inverse problem solution.

Considering the statistical properties of measurements in channels 13–16 presented in Figs. 6b,d, we note that they complement the measurements in channels 1–12. Thus, the spatial features and accuracy approximations in channels 13–16 must have common scales with those in channels 1–12. The measurements in channels 14 and 15 describe the mid- and upper-troposphere, respectively. They are analogous to channels 3 and 4 in sounding the atmosphere (“3 → 15,”“4 → 14”). Channels 13 and 16 are analogous to channels 5 and 6. Measurements in channels 13 and 16 are affected by the surface (see Fig. 1d). Channels 13–16 are not affected by atmospheric moisture, unlike measurements in channels 5–12. Thus the measurement estimate in channels 13–16 should be spatially smoother than those in channels 5–12. Considering Figs. 5 and 6, we notice the substantial differences between measurements in channels 13–16 and channels 4–12. The shape and amplitude of σ²(δf) and σ²[δL(f̂)] in channels 13–15 demonstrate that the measurements are very noisy (see also Fig. 4). Comparing channel 15 in Figs. 6b,d with channels 1–3 in Figs 5a,c, we find that channel 15 noise has an analogous spatial structure. Filter estimate n₃ = 2 in channel 3 (see Fig. 5c) provides the spatial smoothness σ²[δL(f̂₃)] = 0.18 K². Figure 6d shows that we must use the filter dimensions n₁₄ = 2 and n₁₅ = 6 in channels 14 and 15 to provide the required smoothness, defined by estimates in channels 2–5 and 10–12. Channels 13 and 16 have σ²[δL(f_13,16)] of 9.5 and 7.5 K², respectively; these values are noticeably larger than σ²[δL(f_5,6)] in the longwave channels 5 and 6 (5.0 and 6.0 K², respectively). We expect that σ²[δL(f_13,16)] will have to satisfy conditions (see Fig. 1d and Fig. 2):

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that is, measurement properties in channels 13 and 16 substantially differ from the model.

Channels 17 and 18 have σ²[δL(f_17,18)] of 9.3 and 7.8 K², respectively; these values are noticeably larger than σ²[δL(f_7,8)] in the longwave window channels 7 and 8 (6.2 and 6.0 K², respectively). Here, σ²(δf) and σ²[δL(f̂)] in channels 17 and 18 are presented in Fig. 7 (along with channel 8 for comparison). Figure 7a shows that σ²(δf) in channels 8, 17, and 18 is similar, indicating that there are some similarities in the measurement properties of the short- and longwave window channels. However, Fig. 7b shows that the measurements in the short- and longwave window channels have noticeably different spatial properties (see also Fig. 4). Those differences cannot be attributed to differences in noise. Here, σ²(δf) for channels 17 and 18 decreases faster than in channel 8; it becomes even less in channels 17 and 18 than in channel 8 with increasing n (or Δ). At larger n (or Δ) the difference seems to become constant. In contrast, channels 5–8 in Fig. 5d do not demonstrate such properties. Observed differences in σ²[δL(f_17,18)], σ²[δL(f_7,8)], σ²[δL(f̂_17,18)], and σ²[δL(f̂₈)] could be explained by variations in surface optical properties within the spectrum. Thus properties (spatial and spectral) of the measurements in channels 13, 16–18 noticeably differ from the properties of the measurements in channels 5–8. The longwave spectral measurements play a basic role in the solution of inverse problem. The spatial smoothing is not applied to the measurements in channels 6–8, because the signal has a substantial spatial variation and the noise influence is comparatively small. However, the spatial filtering in channels 13, 16–18 is mostly responding to the properties of the inverse problem and the spectral model used, and therefore the consideration of the signal filtering is appropriate. Figures 6d and 7b (as compared with Fig. 5d) show that we can reduce observed differences between short- and longwave spectral measurements using the spatial filter with dimensions n₁₃ = 2, n₁₆ = 2, n₁₇ = 2, and n₁₈ = 2 in channels 13 and 16–18.

Table 4 presents a summary of the spatial averaging;it presents the dimensions of the spatial filter Δ, an estimate of the spatial roughness σ²[δL(f̂)] (K²), and the approximation accuracy σ²(ξ̂) (K²) for the 18 spectral channels of the GOES-8 sounder.

Comparing temperature retrievals for two spatial averaging strategies

Measurements from the GOES-8 sounder at 1000 UTC on 13 August 1999 were processed using two spatial averaging strategies: the variable spatial averaging suggested in Table 3 and the uniform 3 × 3 spatial averaging currently used routinely (Menzel et al. 1998). Temperature profiles for the two datasets were compared with radiosonde observations (raob) from 1200 UTC on 13 August 1999. The collocation distance is within 0.5° latitude and 0.5° longitude. Twenty-seven raobs and 831 retrievals (in cloud-free conditions) were matched. The geographical location of the collocated soundings is shown in Fig. 8a. The average absolute differences of GOES sounder temperature profiles versus raob temperature measurements for 20–400 hPa are shown in Fig. 8b. The results of processing radiances with the variable spatial averaging (plot 2) are noticeably better than the results of processing radiances with the uniform 3 × 3 spatial averaging (plot 1).

Conclusions

Noise reduction in the GOES sounder measurements was investigated in order to estimate the appropriate size of a simple square filter for each spectral channel as a function of the instrument noise and spatial variability of the detected radiation. A data analysis technique was developed using a traditional statistical approach to spatial filtering of noise. The physical basis for the technique is that estimates of the radiance fields (resulting from the spatial filtering) should be sufficiently smooth spatially and spectrally so that the retrieved meteorological fields exhibit appropriate spatial smoothness. In this sense, the technique can be considered as a simplification of 3D to 2D filtering, with some restrictions imposed on the spatial and spectral roughness of the spectral measurements.

Analysis of GOES-8 sounder data indicates that the measurements are noisier than the instrument noise specification. In addition, the noise exhibits spatial correlation. Visual inspection of the spectral images reveals line to line spatially correlated noise attributed to detector recovery or smearing (after a “hot scene” the detector observes a “cold scene”) or to calibration instability.

For the GOES-8 sounder dataset from 1000 UTC 13 August 1999, a spatial filter was estimated for each spectral channel. Profiles were retrieved from the radiative transfer equation. Solving the inverse problem involved estimating the surface emissivity and temperature (attributed to the shortwave spatial domain) as well as the atmospheric temperature and moisture profiles (where the spatial domain varies strongly with height from the shortwave in the atmospheric boundary layer to the longwave in the upper troposphere and stratosphere). The linear dimension of the spatial averaging filter ranges from 1 element in the longwave spectral window (channels 6–8) to 13 elements in the shortwave sounding channel 15. Retrievals from the variable filter are improved over those from a constant 3-element square filter.