a. SAR interferometry

In order to analyze the propagation component of the interferogram, the influence of the geometric component needs to be eliminated. Geometric phase differences are caused by either a change in satellite position or a coherent change in the position of the scatterers on earth, between the two acquisitions. A difference in satellite positions will measure topographic height variation in the SAR image. Using a reference elevation model, a synthetic topographic interferogram can be constructed, which can be subtracted from the observed interferogram, resulting in a so-called differential, topographic-free, interferogram (Massonnet et al. 1993). Other variations in the geometric component, for example, due to surface deformation, can be safely ignored for these short time intervals. Therefore, observed phase gradients in the differential interferogram can only be attributed to propagation delay variability and residual trends due to the inaccuracy of the satellite position during the acquisitions. Finally, the interferometric phase, which is originally “wrapped” to the interval [−π, π), is unwrapped using dedicated phase-unwrapping algorithms (see, e.g., Goldstein et al. 1988).

b. Interferometric delay analysis

After obtaining the differential interferogram, the observed phase differences can be interpreted as (i) the spatial delay variation between the radar antenna and millions of pixels in the interferogram and (ii) the difference between the two generally uncorrelated states of the atmosphere during the SAR acquisitions. Due to satellite orbit errors and the wrapped nature of the phase observations it is only possible to measure the lateral variation of the delay, rather than the total delay. The delay variation (in mm) δ_p,q = δ_p − δ_q between pixel p and q is directly related to the interferometric phase difference ϕ_p,q = ϕ_p − ϕ_q by (Hanssen 1998):

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Mapping the incidented delay variation to zenith values can be achieved by

δ^z_p,qδ_p,qθ,(4)

with look angle θ varying between 19° and 23° (see Fig. 1). Such a simple mapping function is sufficiently accurate for steep incidence angles (Bean and Dutton 1968).

The standard deviation σ_ϕ is derived from the coherence γ, that is, the amount of correlation between the two SAR images, with 0 ⩽ γ ⩽ 1 (see Just and Bamler 1994). For γ ≥ 0.8, as observed in the interferogram used in this study, we find σ_ϕ ⩽ 52°. Using Eqs. (3) and (4), and averaging five pixels to obtain 20 m × 20 m resolution cells yields a formal accuracy of the zenith delay (vertically integrated refractivity) observations of σ_δ^z ≈ 2 mm. Additional spatial averaging to a ground resolution of approximately 160 m × 160 m yields a phase standard deviation of σ_ϕ ⩽ 5° (Joughin and Winebrenner 1994) and, consequently, σ_δ^z ⩽ 0.2 mm.

Possible systematic errors in these delay observations are manifested as long wavelength gradients in the interferogram, caused by inaccuracies in the satellite’s position during image acquisition. These additional tilts are removed from the interferogram before analyzing the delay differences. This procedure also removes long wavelength delay gradients caused by pressure and temperature gradients or gradients in ionospheric electron density.

The technique now reveals incidented delay differences or integrated refractivity along every path between the antenna position, a, and the resolution cells on earth. The relation between zenith delay observed at resolution cells p and q and the refractivity distribution during SAR acquisition t_i can be written as

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where N(x, y, z, t) is the dimensionless refractivity. The zenith delay variation δ^z_p,q observed in the interferogram can be defined as

δzp,qδz,t1p,qδz,t2p,q(6)

For C-band radar and the location of the test site the refractivity can be written as (Smith and Weintraub 1953; Kursinski 1997; Hanssen 1998)

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where P is the total atmospheric pressure in hPa, T is the atmospheric temperature in kelvins, e is the partial pressure of water vapor in hPa, n_e is the electron number density per cubic meter, f is the radar frequency (5.3 GHz), and W is the liquid water content in g m⁻³. The terms k₁ = 77.6, k^′₂ = 23.3, and k₃ = 3.75 × 10⁵ are obtained from Smith and Weintraub (1953), but results from Thayer (1974) are also commonly used. The four terms are referred to as hydrostatic term, wet term, ionospheric term, and liquid term, respectively. Resch (1984) indicated that the first two parts are accurate to 0.5%.

It is obvious that the inverse problem, the retrieval of all atmospheric parameters in four dimensions from a nearly vertically integrated measurement, is ill posed. However, there are several considerations that make a reasonable interpretation of the signal delay in terms of the wet term (integrated water vapor) likely.

• The sensitivity of the delay differences to variations in the atmospheric parameters. Under standard atmospheric conditions, a change in surface pressure of 1 hPa will result in a 2.3-mm delay difference, whereas a small variation in moisture of 1 g kg⁻¹ (1.2 hPa) at 0°C already produces 6 mm per vertical kilometer. The sensitivity for a change in temperature of 1°C is 4–20 times smaller than a change in moisture of 1 hPa, depending on ambient conditions. The influence of cloud droplets is maximally a few millimeters, depending on the droplet size and cloud height. Ice crystal influence can be neglected for 5.66-cm radar wavelengths (Hanssen et al. 1999).

• The dominant spatial wavelengths of the variations. For example, the influence of the hydrostatic term depends on the spatial variation of surface pressure within a 100 km × 100 km SAR image. For most meteorological situations, this variation can be approximated by a single gradient. For the ionospheric term, similar reasoning holds in most cases for latitudes studied here. Moisture variations, however, of 1 g kg⁻¹ are common even on a 1-km spatial scale and will have a directly noticeable effect in the interferogram (Weckwerth et al. 1997).

• Assumptions on vertical stratification and topographic effects. As long as the vertical variation of atmospheric parameters is identical for every resolution cell in the interferogram, these effects will not influence the interferometric phase. Note that a different vertical layering during the two SAR acquisitions will, however, influence the interferogram if significant topography is present (Delacourt et al. 1998; Hanssen and Klees 1999). The signatures of these effects will have strong correlation with the topography. For the test site analyzed here, with height variation within a range of 100 m, no topographic induced effects are expected or observed.

c. Meteosat water vapor channel

The Meteosat WV channel, centered around 6.7 μm, is used in operational meteorology to observe the development of structures of upper-tropospheric water vapor, which carry the signatures of atmospheric conditions. Especially subsidence inversions, that is, downward vertical transport of dry air behind a frontal zone, are clearly visible in the WV images. Due to the strong absorption by water vapor at this wavelength, the observed brightness temperatures usually originate from tropospheric layers above 3 km (Weldon and Holmes 1991). Therefore, quantitative analysis is restricted to upper-tropospheric water vapor as described by Schmetz et al. (1995). Unfortunately, however, the concentration of water vapor is highest near the earth’s surface, where relatively high pressure and temperature allow the air to contain more water vapor. Therefore, in general it is not possible to make one unique parameterization of WV channel brightness temperatures into precipitable water vapor. In section 3b it is discussed whether GPS tropospheric delay time series are suitable for establishing a tailor-made parameterization to obtain Meteosat spatial precipitable water vapor values for the special case of a subsidence inversion.

a. SAR interferogram analysis

An interferogram, covering a strip of 100 km × 200 km, has been formed using SAR acquisitions on 26 and 27 March 1996, 2141:05 UTC (2241:05 local time). The orbital separation, parallel to the look direction, is approximately 32 m, which makes the configuration moderately sensitive to topographic height differences (see Fig. 1). An a priori reference elevation model is used to correct for the topographic phase in the interferogram (TDN/MD 1997). Some additional corrections for tilts in the length and width direction are applied, and water surfaces are masked. The resulting differential interferogram is shown in Fig. 2. The phase values are converted to zenith wet delays, using Eqs. (3) and (4), and consecutively to differential precipitable water vapor using Eq. (10), where we use the term “differential” to indicate the difference between the two states of the atmosphere during the SAR acquisitions.

A strong, large-scale gradient is clearly visible in the south of the interferogram, aligned approximately perpendicular to the radar flight direction. The interpretation of this phenomenon is a key topic of this study. Previous studies have shown that phase variation perpendicular to the flight direction might be caused by oscillator drift errors in the onboard reference clock (Massonnet and Vadon 1995). A possible way to examine this possibility would be the calculation of a long swath of connected SAR images, but this is outside the scope of this study. Here it is assumed that the phase variation is caused by atmospheric water vapor only. The likelihood of this assumption can be tested using the combined analysis with Meteosat and GPS.

Note that, apart from the gradient in the south of the interferogram, wave phenomena also can be observed in the interferogram (see the lower left-hand part of Fig. 2). These are possibly due to low-level moisture variations of which the structure seems to indicate the presence of boundary layer rolls.

Wet delay differences δ^z_pq can be related to integrated precipitable water (I) values, the liquid equivalent of the integrated water vapor:

δzpqΠt1TsIt1pIt1qΠt2TsIt2pIt2q(8)

Using the density of liquid water, we find that 1 mm integrated precipitable water is equal to 1 kg m−2 integrated water vapor. The conversion factor ΠtiTs is approximated using surface temperatures Ts for the image acquired at ti, defined by Askne and Nordius (1987). After first models for ΠtiTs were derived by Davis et al. (1985) and Bevis et al. (1994), a polynomial model was developed to approximate ΠtiTs for De Bilt, the reference site in the Netherlands, using 1461 radiosonde profiles (Emardson and Derks 2000):

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where T₀ is the mean annual surface temperature for the location, and t_D is the day number of the year. The used coefficients are a₀ = 6.443, a₁ = −1.33 × 10⁻², a₂ = 0.18 × 10⁻⁴, a₃ = 3.6 × 10⁻², a₄ = 3.0 × 10⁻², and T₀ = 283.80 K. Using surface temperatures provided by 18 stations in the area, we find that 1/Π^t₁ = 0.1513 ± 0.0004 and 1/Π^t₂ = 0.1507 ± 0.0005, for 26 and 27 March, respectively. Applying one factor for both days, 1/Π_m = 0.1510 ± 0.0005 is justified, since the fractional error in Π is one order of magnitude smaller than the fractional error in the delay measurement. We can rewrite Eq. (8) as

∇It1,t2pqIt1pIt1qIt2pIt2qΠ−1mδzpq(10)

where we define ∇It1,t2pq as the double difference of integrated precipitable water in space and time. Note that a single, isolated anomaly will have a different sign in the interferogram, depending on whether it appeared in the first or the second acquisition. In that case, one of the single spatial ΔItipq differences can be regarded as nearly zero, in which case we can interpret the double difference as a single spatial difference during the other acquisition.

b. Meteosat WV channel analysis

On 27 March 1996 a strong subsidence inversion passed the Netherlands from north to south, which was clearly visible in the WV channel images (Fig. 3). The subsidence inversion can be identified by the dark band in the image, corresponding to relatively high brightness temperatures. Because the descending air in the subsidence inversion is rather dry, the absorption (and emission) of radiation is low and, therefore, the air is relatively transparent. This enables radiation from lower (warmer) layers to contribute to the signal, which results in high apparent brightness temperatures. The center of the subsidence inversion moves in one day from about 53° to 48°N, as apparent from Fig. 3.

In Fig. 5 the change in I over Kootwijk is shown derived from both GPS and Meteosat. We find that the rms of 0.6 kg m⁻² is accurate enough for our analysis. The passage of the subsidence inversion from north to south is clearly visible. Values range from 1 to 7 kg m⁻².

Using the parameterization in Eq. (11), precipitable water vapor can be derived from the Meteosat WV channel image. The parameterization yields accurate values for the area near Kootwijk (location indicated in Fig. 2), but with increasing distance the accuracy will probably decrease.

For the comparison with the SAR interferogram, an elongated area is selected (indicated by the arrow in Fig. 3), which ranges from 51.38°N, 5.93°E to 52.72°N, 5.43°E. The area includes the GPS station Kootwijk Observatory for Satellite Geodesy (KOSG), to optimize the validity of our parameterization, and is parallel to the ground track of ERS. The subsidence inversion band is nearly perpendicular to the profile and moves approximately parallel to it. As a result, the analyzed precipitable water vapor signal is dominated by only one atmospheric process—the subsidence inversion. Therefore, the parameterization is assumed to be sufficiently accurate for the selected profile.

All pixels that contain high-altitude clouds (temperature below −20°C) are excluded from the analysis, to increase the reliability of the parameterization. In a cloud, the fraction of total water that is clustered in particles is generally small, but the absorption of radiation by these particles is much larger than the absorption by the water vapor in the cloud. Therefore, the correlation between the Meteosat WV channel observations and the correct amount of precipitable water vapor will deteriorate if clouds are present.

To obtain similar quantities, as in the interferogram, the extracted profiles corresponding with 2200 UTC 26 March and 2200 UTC 27 March are differenced. The resulting values are indicated by the triangles in Fig. 7, where the position of the pixels along the profile is expressed by their latitude. From the Meteosat WV images in Fig. 3 it appears that at 2200 UTC 26 March the selected area was in the front part of the subsidence inversion, which results in a decreasing amount of precipitable water vapor with latitude. At 2200 UTC 27 March, the selected area was in the back part, resulting in an increasing amount of precipitable water vapor with latitude. As a consequence of these opposite trends, the quantities in Fig. 7—which are formed by subtracting the values at 27 March from the values at 26 March—have an amplified north–south gradient. In the next section these results will be compared with the results from SAR interferometry.

c. Intercomparison

The evaluation of the water vapor observations from the Meteosat WV channel and the radar interferogram is subject to 5 degrees of freedom.

1. The interferogram shows relative delay differences. Therefore, the analyzed profile has an arbitrary bias when compared with the absolute values of the Meteosat profile.

2. Due to the oblique viewing geometry of the radar (23° from zenith) the profile will be shifted some kilometers parallel to the west (see Fig. 6). Since the atmospheric situation during the two acquisitions was nearly symmetric (i.e., perpendicular to the profile), the effect of this lateral shift is negligible.

3. The inaccuracy in the satellite orbits might lead to a small tilt in the profile. Here we assume that this tilt is sufficiently eliminated by using a number of reference points during preprocessing.

4. The Meteosat positioning accuracy is approximately 0.5 pixels, corresponding to a 4.5-km uncertainty in north–south direction.

5. Due to the geostationary position over the equator, there is an additional shift D in north–south direction when the majority of the water vapor is at a height h (see Fig. 6). In that case, the information will appear to be shifted northward in the image. For the location analyzed here, this shift is approximately D = 1.7 × h. Note that the time difference between the SAR acquisition and the Meteosat scan of the latitudes of the Netherlands (13 min) might also account for a slight additional shift.

The expected variance for the InSAR observations is assumed to be uncorrelated between adjacent values and equal for every observation. In section 2b, the delay standard deviation σ_δ^z = 0.2 mm (or kg m⁻²) was derived. Using Eq. (10), simple error propagation yields an a priori value for the integrated water vapor observations of σ_I,sar = 0.08 kg m⁻².

A goodness-of-fit parameter between the InSAR and the Meteosat observations can be optimized by adjusting the 5 degrees of freedom mentioned above, or by adjusting the a priori variances of both observations. Here, a combination of both approaches is suggested. Parameters 2–4 are assumed to be sufficiently well approximated. A northward shift of the Meteosat observations (parameter 5) of 5 km is used as a first-order approximation, derived from an average height of the dominant water vapor signal in the WV channel. The additional Meteosat profile shift, the bias of the InSAR profile (parameter 1), and the scaling factor for the variances are now estimated by deriving the minimal reduced chi-square value, χ²_ν, of the two profiles as a function of the bias of the InSAR profile (Bevington and Robinson 1992):

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The number of measurements is denoted by n, r_i are the differences between the Meteosat and the INSAR profile values, and σ_i are the a priori standard deviations of the differences, that is, σ²_i = σ²_I,sar,i + σ²_ΔI,msat,i. Here we find σ_i = 0.85 kg m⁻² for all i ∈ [1, . . . , n]. Note that the contribution of σ_ΔI,msat,i is one order of magnitude larger than σ_I,sar,i.

The χ²_ν values are expressed as a function of the InSAR profile bias and the horizontal shift of the Meteosat profile. The search window of the bias has been confined between −3 and 3 kg m⁻², whereas the horizontal shift is confined between −1.1 and 1.1 km. The minimum value, χ²_ν = 2.1, is found for a bias of 1 kg m⁻² and a horizontal shift of −1.1 km. Figure 7 shows the original position of the InSAR profile, indicated by the thin line, and the new position, indicated by the bold line. The original position of the Meteosat profile values is indicated by the triangles, while the new positions are indicated by the squares. A 95% confidence interval for the difference of the two data sources is indicated by the inner error bars. Note that the error bars are drawn at the Meteosat positions, although they also include a small component of the InSAR profile.