The GMI is a conically scanning radiometer whose antenna beam maintains an approximately constant incidence angle with respect to the Earth's surface as it rotates about the vertical axis that connects the satellite with the satellite's subpoint (Fig. ). The parameters of importance include (a) the relative speed of the satellite subpoint across the Earth's surface, which is determined by the orbital period and, to a far lesser degree, by the Earth's own rotation; (b) the rotation rate of the antenna; (c) the incidence angle of the antenna beam and thus the angular radius of the scan; (d) the integration time Δt, which determines both the along-scan separation between pixels and the smearing effect that expands the EFOV relative to the IFOV; and (e) the total number of sampled pixels along one scan. The latter is in turn tied to the fraction of one complete circular scan that is actually sampled as well as to the total swath width.

Note that there are two different sets of feedhorns associated with the 10.65–89 GHz channels (1–9) and with the 166 and 183.3 GHz channels (10–13). The latter channels view the Earth at a slightly steeper angle; consequently, their data swath is narrower and their scan pattern is spatially misregistered with that of the lower-frequency channels, as shown in Fig. . Finally, because of the oblateness of the Earth, the relative registration in the along-track direction fluctuates by up to several tenths of the spacing between scan lines.

The geometry of the GMI scans must be accurately modeled to compute both the actual EFOV sizes and shapes and the overlap between adjacent EFOVs. Both are required in order to be able to determine the correct weights for constructing synthetic (resolution-matched) fields of view (FOVs) for each channel.

Here we model the orbit of the GMI as circular with fixed altitude above the Earth's surface and fixed period, and we ignore the oblateness of the Earth. By carefully examining actual post-launch GMI data, a set of geometrically self-consistent values for all major parameters of the scan geometry was either directly measured or inferred. These values are reported in Table .

Note that we ignore the variable correction due to the Earth's own rotation, but we introduce a small constant correction to the subtrack velocity relative to that predicted from the orbital velocity at the given altitude. Thus, our model can be thought of as approximating the mean scan geometry of the GMI while being subject to minor fluctuating errors that are negligible for nearly contiguous pixels but larger for widely separated pixels.

The EFOV of each GMI channel results from convolving the IFOV, or the antenna gain pattern projected onto the Earth's surface, with the spatial displacement due to scanning and satellite motion during the integration time. The IFOV is currently modeled as Gaussian, as measured sidelobe gains are at least 30 dB below that of the main beam for all channels and are negligible for the present purpose. The beamwidths were determined from the half-power points of field-measured antenna patterns supplied to us by C. Kummerow (personal communication, 2014) and consistent with those employed by . The computed IFOV and EFOV dimensions are reported in Table , and a schematic depiction of the change in EFOV resolution relative to the IFOV is shown in Fig. . Because the smearing effect of the time integration is almost entirely in the along-scan direction, only that dimension is measurably changed for the EFOV relative to the IFOV. It is most pronounced for the highest-resolution IFOVs.

Note that the interscan distance of 13.15 km is significantly larger than the cross-scan EFOV dimension of 7.2 km for the 89.00 GHz channels. In other words, these channels provide noncontiguous coverage. That in turn implies that no spatial average of 89 GHz EFOVs can closely approximate the EFOV of any lower-frequency channel.

To address the large mismatch in EFOV sizes between lower- and higher-frequency channels, we have two choices. We can

spatially average (convolve) higher-resolution channels to approximately match the coarser resolution of a lower-frequency channel or

It must be emphasized that it is generally not possible to achieve a perfect match. One can only aim to achieve the best possible match and then examine the empirical quality of the outcome. This is especially true in the case of deconvolution, as weighting coefficients must be determined so as to achieve reasonable improvements in resolution without unwanted artifacts such as excessive noise amplification and/or “ringing”. Also, deconvolution is only possible when the pixel spacing is significantly smaller than the size of the EFOV whose resolution one is seeking to improve. As a practical matter, this limits the use of deconvolution for the GMI to the 10.65 GHz channels. Finally, the ability to match FOVs is degraded at the edge of the swath, where there is an incomplete set of overlapping or contiguous pixels.

Our efforts here are similar to those reported for earlier microwave imagers, such as the Special Sensor Microwave Imager (SSM/I) , the TRMM Microwave Imager , the Advanced Microwave Sounding Unit , and the Advanced Microwave Scanning Radiometer for the Earth Observing System . Apart from and , most of these do not examine the actual properties of the resulting synthetic EFOVs.

In the classic method of , which was in turn adapted to satellite passive microwave imagers by , a cost function is defined that incorporates both a measure of noise amplification and a quadratic measure of resolution or “spread”. A tuning parameter γ allows the relative emphasis on each of the two terms to be varied. The method employed here is essentially the Backus–Gilbert method, but we replace the second term (“spread”) with one representing spatial correlation between the synthetic FOV (constructed from a linear sum of overlapping real EFOVs) and the target EFOV, in this case the native EFOV of the 18.7 GHz channels. Thus, for the 10.65 GHz channels, our procedure attempts to sharpen the resolution within the limits of spatial sampling and noise amplification considerations. For the higher-frequency channels, the procedure leads to a spatial averaging.

For convenient reference, the full mathematical derivation is provided in Appendix A. take a similar approach to ours in their deconvolution of SSM/I brightness temperatures, except that their target EFOV was an idealized uniform disk with sharp edges rather than a real (and therefore smooth) EFOV; otherwise the mathematics is the same.

Fundamentally, the method entails taking a linear sum of multiple FOVs overlapping the target FOV. That is, if the target FOV is denoted F0(x,y), then our goal is to create a synthetic FOV F′(x,y) for another channel such that F0(x,y)≈F′(x,y)=∑iwifi(x,y), where wi is the appropriately chosen linear weight applied to each of the original FOVs (or pixels) fi in the neighborhood of F0. Note that to conserve brightness temperature, the weights must sum to unity: ∑iwi=1.

The quality of F′ as an approximation to F0 can be defined in various ways. Here, we choose the squared deviation integrated over area: χ2=∫∫F′(x,y)-F0(x,y)2dxdy. If the channel being operated on has a higher frequency than the reference channel, then its native resolution is generally higher than that of the reference channel. FOV matching then reduces to a spatial averaging or blurring procedure, and most or all of the coefficients in Eq. () are positive. If, however, coarser-resolution FOVs are being combined in an effort to match a finer-resolution target FOV, then this amounts to a deconvolution, or sharpening procedure, and the weights will necessarily be both positive and negative as needed to cancel the response outside the target FOV.

A well-known problem with deconvolution, when not done carefully, is that the individual magnitudes of wi can become quite large (while still satisfying Eq. ), leading to severe noise amplification as well as “ringing” in the deconvolved image in the presence of sharp brightness temperature gradients. The measure of the noise variance amplification associated with a linear filter is N2=∑iwi2, since the effective noise variance in the processed image is then σpost2=N2σpre2, where σpre2 is the “native” noise present in the original image, including possibly geophysical noise and/or uncertainties in the precise FOV shape in addition to instrument noise.

Given Eq. (), N2 is absolutely minimized when the wi are all positive and equal, corresponding to a pure averaging or blurring procedure. In contrast, N2 can become arbitrarily large when pushing the limits of a deconvolution or sharpening procedure. In any case, whether sharpening or blurring the image, it is important to consider the inevitable tradeoff between achieving the best possible fit to the target FOV and controlling noise amplification and ringing.

Even apart from noise amplification considerations, it is generally impossible to exactly match an arbitrary target FOV via a sum over the discrete set of neighboring FOVs of different size and shape. This is especially true when the pixel density (spatial sampling) is poor relative to the resolution of the target FOV. The target FOV is therefore indeed only a target and is never actually achieved in the footprint matching procedure. Rather, one must examine the resulting synthetic FOV F′ to determine how good the fit actually is and whether the procedure is of sufficient utility to be worth the effort.

Figure  depicts the tradeoffs between noise factor and fit to the target EFOV. For 10.65 GHz, we did not want to exceed a noise amplification factor of about 2, which limited the quality of the fit to the target EFOV defined by the 18.70 GHz channels. Even without this constraint, the fit could only be improved by a few percent. For 23.80 and 36.64 GHz, an excellent fit approaching 100 % is achievable for all but the edge pixels without any noise amplification. For 89.00, a relatively poor fit is achieved due to significant undersampling by the native FOVs in the cross-scan direction. Overall, we find that a constant tuning value of γ=6×10-6 yields a reasonable compromise between fit and noise amplification for all channels. The resulting coefficients are available for download from .

Figure  depicts the shapes of the final synthetic EFOVs for pixel 110 (center of the swath). Of particular note are the following points. While there is slight improvement in the 10.65 GHz fit to the target 18.70 GHz EFOV, it is not possible to actually match that resolution. As found previously by for the TRMM Microwave Imager, the improvement is somewhat better in the along-scan direction due to more oversampling in that direction. There are significant negative sidelobes in the synthetic EFOVs for 10.65 GHz. This appears to be unavoidable given the available sampling for these channels. The fit for 23.80 and 36.64 GHz, however, is excellent.

Because the 89.00 GHz channels are badly undersampled in the cross-scan direction, the synthetic FOV fit to the target EFOV is poor in that direction. It is quite good in the along-scan direction. All of the above results are the worst cases for the entire interior of the data swath, as the sampling density improves toward the edges. At the edges, however, the fit deteriorate again. Table 3 gives the half-power beamwidths of the synthetic EFOVs compared to the native resolution for each channel. (Note that the half-power value does not give a useful measure of the improvement in the fit for 89.00 GHz due to the multimodal shape.)

It is not possible to precisely characterize the impact of resolution matching on retrieval errors, as any improvements will be highly context and algorithm specific. Nevertheless, we can, as just one example, examine the efficiency with which coastal signatures can potentially be separated from other geophysical signatures.

From the data depicted in Fig. , we computed the 9×9 channel brightness temperatures covariances for both the unmatched and the matched data. We find that the brightness temperature variances for the 10.65 GHz channels increase following the resolution matching procedure, while those for 23.80 GHz and higher frequencies decrease as a result of the same procedure (Fig. ). These outcomes are consistent with moderate spatial sharpening (and associated noise amplification) for the 10.65 GHz channels and spatial averaging for the higher-frequency channels.

The algorithm of utilizes only the seven channels from 18.7 to 89 GHz in over-land precipitation retrievals. We therefore extracted the covariance matrices (pre- and post-matched) for just those channels and calculated the eigenvectors (principle components) and eigenvalues. By far, the dominant eigenvector for this data set describes the brightness temperature contrast between land and ocean. In the pre-matched data, this eigenvector explains 99.1 % of the total channel variance, leaving 0.9 % unexplained. This unexplained residue necessarily includes all exploitable geophysical signatures (e.g., precipitation) was well as the nonlinear component of the correlations between channels due to the disparate resolutions (see for example Fig. g). In the post-match covariances, only 0.4 % of the variance is unexplained by the first eigenvector, implying at least a factor of 2 improvement in the signal-to-noise ratio, and possibly much more.

Figure  shows how the residual unexplained variance decreases with the inclusion of each successive principle component. We see that a significant reduction is achieved for all of the first four principle components. For reference, the algorithm of relies on the first three principle components after variability associated with the background (including coastlines) has been accounted for. We therefore conclude that the resolution matching procedure is likely to improve the ability of the algorithm to distinguish precipitation signatures in the presence of coastlines and other strong gradients in emissivity.