For the comparison of validation and reference measurements and the GPROF retrievals, all data have to be co-located. To this end, ground- and satellite-based results are remapped to a common, approximately equidistant 5 km × 5 km grid that follows the satellite swath. An equidistant grid is required for the analysis of the effective resolution. The grid resolution of 5 km × 5 km was chosen finer than the expected resolution of the PMW-based GPROF retrievals in order to avoid interference of the spatial sampling with the analysis of the effective resolution of the retrievals.

An example of the co-locations used in this validation study is provided in Fig. . The scene displays an overpass of the GPM core observatory over the midlatitude cyclone associated with the 2020 Easter tornado outbreak . The various panels show surface precipitation estimates from the validation and reference measurements and the different GPROF retrieval results.

Four water years (October through September) from 2019 until 2022 are used for the validation of the GPROF retrievals from GMI. This time period was chosen because it includes the water year 2019, which is used to derive the a priori database for the GPROF retrievals. The analysis is extended to include several years in order to allow assessment of the temporal variability of the retrieval accuracy.

For the other sensors of the GPM constellation, the retrieval accuracy was assessed using a single year of co-locations. The co-location periods were chosen according to the availability of observations and ground validation data. The periods chosen for the different sensors are listed in Table .

The distance between the centers of neighboring pixels of the GMI swath is approximately 13.5 km in the along-track direction and 6 km in the across-track direction. Since GPROF traditionally retrieves precipitation at the same grid as the input observations, the resolution of the retrieved precipitation fields cannot exceed the spatial sampling of the input observations. In general, however, not all variability at scales larger than the spatial sampling of the output data is truthfully represented in the retrieval results. This motivates the definition of an effective resolution as a numerical measure of the retrieval's ability to resolve small-scale spatial variability.

The consistency between retrieved and reference precipitation fields across different spatial scales can be quantified using the relative spatial Fourier spectral coherence between reference and retrieved fields. The spectral coherence can be interpreted as the complex correlation between the Fourier coefficients of two signals within a narrow frequency (or wavenumber) band. In two dimensions, the Fourier spectral coherence between two fields g(x,y) and h(x,y) is defined as follows: 1Cg,h(kx,ky)=Pg,h(kx,ky)Pg(kx,ky)×Ph(kx,ky), with kx and ky being the wavenumbers in the two orthogonal directions x and y, respectively. Pg(kx,ky) is the Fourier power spectral density (PSD) of g(x,y), defined as follows: 2Pg(kx,ky)=g^(kx,ky)×g^∗(kx,ky), with g^∗ denoting the complex conjugate of the Fourier coefficient field g^. Pg,h(kx,ky) is the Fourier cross power spectral density (CPSD) between g(x,y) and h(x,y) defined as follows: 3Pg,h(kx,ky)=g^(kx,ky)h^∗(kx,ky), The computation of the PSD and CPSD is performed through a discrete cosine transform (DCT) and averaged over the evaluated retrieval scenes.

Passing into the polar coordinate system through the change in variables k=kx2+ky2 and θ=atankxky, the spectral quantities can be analyzed as a function of the spatial wavenumber k and the azimuthal direction θ. Since we did not find any significant dependency on the azimuthal direction, we integrated the spectral quantities along θ to analyze the coherence as a function of k only.

The effective resolution of the retrieved fields can then be defined from the spectral coherence between retrieved and reference field as 4ER=12argmaxkCg,h(k)≥12, i.e., half of the shortest wavelength for which the spectral coherence is larger than 12. Defined in this way, the effective resolution corresponds to the smallest spatial scale at which the signal-to-noise ratio of the retrieved precipitation fields exceeds 1.

The validation of the GPROF GMI retrievals uses co-locations from October 2018 until October 2022. The first year of this period coincides with the a priori database. For the comparison against MRMS, only data points with an RQI of at least 0.8 were used. The validation is performed at the 5 km resolution of the co-locations. We choose to evaluate the retrieval at this resolution instead of the footprint size of the PMW observations because it makes accuracy metrics between different sensors easier to compare.

Retrievals over snow-covered and mountain surfaces are excluded from the validation due to the uncertainties in both the satellite estimates as well as the validation data. In addition to this, the GPROF a priori database for snow-covered surfaces is derived from co-locations with MRMS, while precipitation over mountains is obtained by scaling the GPM CMB precipitation to account for the orographic enhancement of precipitation. These two modifications aim to counteract known weaknesses of the GPM CMB retrievals but would skew the comparison between GPM, GPM CMB, and MRMS.

Similarly, precipitation that is identified as frozen by MRMS is excluded from the validation. The retrieval of frozen precipitation from both PMW and radar is particularly challenging due to its uncertain radiometric properties. Because of these increased uncertainties and the small contribution of frozen precipitation to the total precipitation in the validation data, the retrieval accuracy for frozen precipitation should be assessed in a separate, dedicated study.

For the analysis of the effective resolution of the retrievals, the discrete cosine transform of the retrieved and the MRMS precipitation fields were calculated over non-overlapping windows of 160 km × 160 km in which neither any of the GPROF retrievals nor the ground radar measurements had more than 2 % of missing values. For this analysis, retrievals over snow-covered and mountains surfaces as well as of frozen precipitation were included in the analysis because omitting these values would have significantly reduced the number of windows with valid precipitation estimates. Of all extracted windows, only those with a minimum radar quality index of 0.5 were considered. Only co-locations from the water years 2020 and 2021 were used in order to exclude scenes which are also present in the GPROF a priori database. The coherence was computed once using retrievals from the full GMI swath, and once only over the central part of the swath in order to allow comparison to the GPM CMB product.

The spectral coherence between the GPROF retrievals and the reference radar data as a function of the wavelength is displayed in Fig. . Over the CONUS, GPROF V7 has slightly higher spectral coherence than GPROF V5. The GPROF-NN retrievals achieve considerably higher coherence than the conventional GPROF retrievals across all wavelengths. The coherence curves of the GPROF-NN retrievals are similar up to spatial scales of about 15 km, above which GPROF-NN 3D achieves slightly higher coherence than GPROF-NN 1D. The coherence curves of the conventional GPROF retrievals and the GPROF-NN 1D and 3D retrievals are not significantly affected by the swath coverage.

The GPROF-NN HR retrieval achieves the highest coherence of the PMW retrievals, even matching that of the GPM CMB retrieval, when it is evaluated only over the swath center. However, when evaluated over the full swath, the coherence decreases sharply and falls below that of the GPROF-NN 1D retrieval. This indicates that the retrieval overfits on the viewing geometry of the central part of the GMI swath, which is the only part of the swath for which reference measurements of precipitation are available. Although the training of the GPROF-NN HR retrieval applies data augmentation that simulates the effect of perspective distortion across the swath, it does not account for the parallax effect, i.e., the displacement between observed precipitation signal and the actual precipitation caused by the slant viewing angle of the sensor. Since the parallax effect changes direction towards the edges of the swath, it can potentially explain the decrease in accuracy observed for the GPROF-NN HR retrieval.

Over the Kwajalein site, all GPROF retrievals yield significantly improved coherence curves due to the direct relationship between microwave signals and the liquid water content in the clouds. Also here, the GPROF-NN 1D and 3D retrievals yield higher coherence than the conventional GPROF retrievals albeit by a smaller margin. The GPROF-NN HR retrieval exhibits similar overfitting as over the CONUS. Evaluated over only the central part of the swath, the coherence curve of GPROF-NN HR is higher than that of every other retrieval. Evaluated over the full swath, however, the accuracy decreases below that of the GPROF-NN 1D retrieval. Interestingly, the coherence curve of the GPM CMB retrieval remains at the level of the GPROF-NN HR retrieval evaluated across the full swath and falls below the curves of the GPROF-NN retrievals for scales exceeding 12 km. This is an unexpected result since the GPROF-NN HR retrieval is designed to reproduce the results from the GPM CMB retrieval. A potential explanation could be that the GPM CMB retrieval systematically misses precipitation from shallow clouds while the PMW retrieval is able to recover some of that precipitation because the signal in the PMW observations is less dependent on the depth of the observed clouds.

The resulting effective resolutions of the retrievals are listed in Table . The effective resolution for the conventional GPROF retrieval over the CONUS are around 45 and 60 km for the GPROF V7 and the GPROF V5 retrieval, respectively. The effective resolution of the GPROF V5 retrieval is significantly lower when calculated only over the swath center, but this is caused by the curve falling slightly below the threshold for the calculation of the effective resolution and does not correspond to a significant difference in the coherence curves. With effective resolution of about 29 and 21 km, respectively, the GPROF-NN 1D and 3D retrievals significantly improve the effective resolution over the CONUS. The effective resolution also clearly reflects the overfitting behavior of the GPROF-NN HR retrieval, which achieves an effective resolution of 17 km over the central part of the swath but only 31 km across the full swath. The effective resolution of the GPM CMB retrieval is 18 km over the CONUS.

Over the Kwajalein site, the effective resolution of the GPROF V5 retrieval of 16 km is slightly better than that of the GPROF V7 retrieval with 17 km. Both GPROF-NN retrievals achieve an effective resolution of around 15 km. The effective resolution of the GPROF-NN HR retrieval is 11 km evaluated at the swath center but decreases to 15 km over the full swath. The effective resolution of the GPM CMB retrieval is only 15 km.

This section evaluates the accuracy of the GPROF and GPROF-NN retrievals for the other sensor types of the GPM constellation for which MRMS validation data is available. One year of co-locations with MRMS and K-POL precipitation estimates were extracted for one instance of each sensor type. The corresponding platforms and co-location periods are listed in Table . The resulting retrieval accuracy metrics over the CONUS and Kwajalein are displayed in Figs.  and , respectively.

The biases of the GPROF V7 retrieval are well within 10 % over both the CONUS and the Kwajalein site. Exceptions are MHS and ATMS, for which the bias for Kwajalein exceeds 30 %. For most sensors, MSE, MAE, and SMAPE are larger than for GMI, which is expected considering that the other sensors of the GPM constellation have lower spatial resolution, fewer channels, or both. An exception is TMI, which despite having fewer channels and larger footprints than GMI, achieves higher correlation over the CONUS. This is likely due to the reduced latitude coverage of TMI, which only covers the lower part of CONUS and is therefore more likely to observe precipitation with clear signatures in PMW observations.

Compared to GPROF V5, the retrieval biases of V7 tend to be smaller than those of V5. GPROF V7 tends to exhibit larger MAE but reduced MSE and SMAPE. A possible explanation for this may be the treatment of pixels with low probability of precipitation, which were explicitly set to 0.0 mm h-1 in GPROF V5. The clearest differences between GPROF V7 and GPROF V5 are observed for SSMIS, for which all accuracy metrics are improved.

The GPROF-NN retrievals exhibit significant spread in biases. Over the CONUS, the retrievals have a tendency to be biased low, with biases reaching up to 20 %. Over Kwajalein, the retrievals for the conical scanners (TMI, SSMIS, AMSR2) are biased low, whereas the retrievals for the cross-track scanners exhibit wet biases of up 70 % for MHS. GPROF-NN 1D yields similar or slightly more accurate results in terms of MAE, MSE and correlation for the conical scanners but yields less accurate results for the assessed cross-track-scanning sensors, MHS and ATMS. For GPROF-NN 3D the improvements are more robust. In terms of the SMAPE, the results are not conclusive with the GPROF-NN retrievals yielding higher errors for some of the sensors.

The results obtained for the GPROF-NN retrievals contrast with the results for GMI, for which very clear improvements over the conventional GPROF retrievals were observed. A likely cause for this is that the training of the GPROF-NN retrievals is based on radiative transfer simulations, which introduce an additional error source into the retrieval. The large biases over the Kwajalein site in the retrievals from the cross-track scanners, whose simulations use a different code than the simulations for the conical scanners, point to problems with these simulations, in particular.

The analyses of the GPROF retrievals presented in Sect.  and were limited to precipitation that is not identified as frozen by the MRMS hydrometeor classification and restricted to retrievals over snow-free and non-mountainous surfaces. As mentioned above, this was primarily motivated by both satellite retrievals and ground-based validation measurements being highly uncertain in these scenarios.

Figure  displays the relative contribution of the excluded cases to the total precipitation for all validation samples. The samples that the presented analyses focused on accounts for 80 % of all precipitation in the MRMS validation data, while mountain precipitation accounts for around 15 %. Snow and precipitation over snow-covered surfaces account for less than 5 % of the total precipitation.

GPROF V7, GPROF-NN 1D, and GPROF-NN 3D produce notably higher precipitation over mountain surfaces than MRMS, GPM CMB, and GPROF-NN HR. This is because of the orographic enhancement factors that are applied in the a priori and training database of GPROF V7. Since GPROF-NN HR is trained directly on GPM CMB precipitation it does not reproduce this correction for orographic enhancement.

Although GPROF V7, GPROF-NN 1D, and GPROF-NN 3D retrieve more precipitation over mountains, they strongly underestimate snow and weakly underestimate precipitation over snow-covered surfaces. These effects, together with the overall smaller contribution of mountain precipitation, cause the relative biases calculated over all validation samples to be similar to those calculated excluding frozen precipitation and retrievals over snow-covered surfaces and mountains.

Our validation excluded retrievals over snow-covered land surfaces and mountainous terrain because of the enhanced uncertainties in both validation data and retrievals as well as their special treatment in the a priori/training database. This special treatment would skew comparison to the GPM CMB product and thus complicate the attribution of retrieval errors. For similar reasons, we have also excluded precipitation identified as frozen by the MRMS hydrometeor classification.

We have investigated the effect of excluding snow-covered land surfaces, mountainous terrain and frozen precipitation on the retrieval biases over all of the CONUS and found them to robust to these exclusions. The orographic enhancement factors applied in the a priori database cause GPROF V7 (and GPROF-NN 1D and 3D) to overestimate MRMS precipitation in mountain regions. In regions where much precipitation occurs in mountain regions, as is the case in the NW region, GPROF V7 may thus be able to balance the underestimation found in this analysis. However, when frozen precipitation is included in the accumulations the underestimation of precipitation is more likely to be exacerbated.

Finally, it should be kept in mind that the validation data over the CONUS was not sampled uniformly, but is limited to regions where high-quality radar observations are available. Since radar coverage is generally biased towards non-mountainous areas and lower elevations, the contribution of both frozen precipitation and precipitation over mountain surfaces may be underestimated in our analyses.

Two principal differences between GPROF V5 and V7 are the inclusion of light precipitation from MIRS over ocean and the application of orographic enhancement factors, which both act to increase the amount of precipitation in the a priori database. The effect of these changes are reflected in the larger amount of precipitation retrieved by GPROF V7 over the Kwajalein site (Fig. ) and the higher amount of precipitation retrieved over mountains (Fig. ). If precipitation over mountain surfaces is included, GPROF V7 retrievals are about 5 % wetter over the CONUS than those of GPROF V5.

For the retrievals of liquid precipitation over snow-free and non-mountainous surfaces that this study focused on, the retrieval accuracy of GPROF V7 for GMI was similar to the previous version of the algorithm, GPROF V5. While the biases of GPROF V7 compared to the ground radar validation data are larger than those of V5, they are closer to GPM CMB, which the retrieval aims to reproduce. Small improvements are found in terms of MAE, MSE, correlation coefficient, and SMAPE. The effective resolutions of the two retrievals are practically identical.

On regional scales, the most significant changes are a reduction in the overestimation of warm-season precipitation over the central CONUS, reduced underestimation of cold-season precipitation in the northwest, a slightly more pronounced underestimation of warm-season precipitation in the southeast, and a more pronounced overestimation of precipitation in the tropical Pacific. Without gauges to calibrate the rainfall product, the Kwajalein site may, however, be more uncertain than the CONUS estimates. For the other sensors of the GPM constellation assessed here, GPROF V7 exhibited smaller biases than GPROF V5 compared to the ground-based validation data.

In order for this validation to inform the future development of GPROF, it is essential to determine whether the observed deviations from the validation data are due to the reference data used by GPROF or the inherent uncertainties in the retrieval and the applied method. Figure  revealed a high degree of similarity between biases in GPROF and GPM CMB in both space and time. This indicates that the GPROF retrieval inherits systematic errors from the GPM CMB retrieval. Further evidence for a notable contribution of GPM CMB to systematic errors in GPROF was found in the analysis of regional biases of GPROF, which tend to follow those of GPM CMB (Figs. , , ).

The attribution of the errors can be formalized by investigating the coefficient of determination between the errors in GPROF and GPM CMB with respect to the validation data. The coefficient of determination represents the fraction of variance of the GPROF errors that can be explained by the GPM CMB errors and thus quantifies the contribution of the GPM CMB errors to the MSE of GPROF. Table  lists the coefficients of determination of errors between GPM CMB and GPROF and validation data for instantaneous estimates, as well as the annual means at 5∘ resolution shown in Fig. . For the GPROF V7 retrievals, the coefficient of determination for instantaneous errors varies between 0.17 and 0.3 over the 4 assessed years. This means that between 17 % and 30 % of the mean squared error can be explained by the error in the GPM CMB retrieval. For the GPROF-NN retrievals this rate is consistently higher and reaches up to between 30 % and 46 % for the GPROF-NN HR retrieval. This indicates that GPM CMB error has a notable influence on the accuracy of GPROF even for instantaneous precipitation estimates.

All rates of explained variability increase drastically when annual accumulations over 5∘ boxes are considered. For GPROF V7 they range from 36 % to 67 %, while they reach between 79 % and 93 % for GPROF-NN 3D. While the fraction of explained variability in the GPROF V7 biases is around 50 % over the 4-year period, it increases to about 75 % for GPROF-NN 1D, 80 % for GPROF-NN HR, and 85 % for GPROF-NN 3D. This confirms that systematic regional errors in the GPM CMB data contribute significantly to regional biases in GPROF.

The role of GPM CMB for the systematic error in GPROF becomes even clearer when the regional biases of GPM CMB are subtracted from the biases of GPROF. The resulting maps of reference-data-corrected GPROF biases are shown in Fig. . Subtracting the GPM CMB biases reduces the overall magnitude of the biases and their temporal and spatial variability. The reduction in bias is more pronounced for the GPROF-NN retrievals, indicating that their higher accuracy puts increased weight on the reference data.

The assessment of the GPROF-NN retrievals against ground radar measurements confirmed the significantly higher retrieval accuracy of the neural-network-based implementation for GMI. Although the biases of GPROF-NN 1D are similar to those of GPROF V7, consistent improvements of the order of 10 %–20 % are observed in all other assessed accuracy metrics. Significant improvements were also found in detection of precipitation and the effective resolution of the retrieval. The effective resolution of the GPROF-NN retrievals over land is more than twice as high as that of GPROF V7. This is a very encouraging result, considering that these improvements are comparable to those between sensor generations. The fact that a simple algorithm update can unlock these improvements highlights the importance of improving the retrieval methodology for current and historical space-borne precipitation estimates.

Even higher accuracy is achieved by GPROF-NN 3D and the GPROF-NN HR retrievals. However, the analysis of the effective resolution presented in Sect.  indicates that the retrieval is overfitting to the viewing geometry at the center of the swath. It seems likely that the decrease in retrieval accuracy towards the edges of the swath is due to the parallax effect, which changes direction towards the edges of the swath. Since reference precipitation measurements are only available at the center of the swath, the neural network cannot learn that the direction of the parallax effect changes across the swath. Fully realizing the potential of the GMI sensor may therefore require incorporating this effect into the training process. Furthermore, the overfitting of the GPROF-NN 3D and GPROF-NN HR retrievals points to a general limitation of using data based on the GPM DPR to train and evaluate GMI retrievals that are based on convolutional neural networks. When only data from the center of the swath are used to train and evaluate a retrieval, its accuracy will likely be overestimated.

Nonetheless, the results of the GPROF-NN 3D and especially the GPROF-NN HR retrievals are worth highlighting. The GPROF-NN HR retrieval demonstrates that precipitation retrievals from GMI can achieve retrieval accuracy that is on par with that of the GPM CMB retrievals and significantly higher than that of the current GPROF retrievals for GMI. The analysis of the contribution of the GPM CMB retrieval errors indicates that with a suitable neural-network-based retrieval systematic errors at regional scales are limited by the accuracy of the reference data, which accounts for up to 37 % of the MSE of instantaneous precipitation estimates and up to 95 % of the variability in annual accumulations at regional scales.

Unfortunately the improvements observed for GMI did not carry over to the other sensors of the constellation. The principal difference for those sensors is that the a priori database uses simulated brightness temperatures instead of real observations. Errors in the simulated brightness temperatures thus constitute an additional error source that incurs a generalization error when the retrieval is applied to real observations. found clear improvements in retrieval accuracy of the GPROF-NN retrieval also for the MHS sensor when the retrieval is evaluated against the a priori database. This indicates that the simulation errors are to blame for the reduced accuracy of the GPROF-NN retrievals for the other GPM sensors. Nonetheless, the GPROF-NN 3D retrieval improves most error metrics compared to the GPROF V7 retrieval. This indicates that the incorporation of structural information helps to make the retrieval more robust.

The largest retrieval biases were found for the GPROF-NN 1D retrieval for the cross-track scanners over the Kwajalein site where wet biases of up to 70 % were found. This may indicate a particular problem with the simulation of the observations from cross-track scanners over ocean. While not as extreme, the GPROF-NN retrievals exhibit higher biases than the GPROF V7 retrieval over the CONUS.

It should be noted, however, that the same simulations are used by GPROF V7, which appears to be less affected by the suspected simulation errors. The improved robustness of the Bayesian scheme used by GPROF V7 is likely due to the hard-coded channel uncertainties, which take into account the forward simulation error of each sensor. In contrast to that, the neural-network retrievals learn the observation uncertainties from the training data.

Another notable difference between the GPROF and the GPROF-NN retrievals is that GPROF applies an additional calibration of the precipitation estimates that ensures that the annual mean precipitation from other sensors matches that of GMI. Such a calibration has not been included in the GPROF-NN retrievals. Furthermore, it should be noted that the measurements from the K-POL radar can be as much as 20 % lower than the GPM CMB retrievals, which highlights the difficulty of finding reliable validation measurements over the ocean.

The results presented in this study show that observations from GMI can be used to produce precipitation estimates at essentially the same accuracy as the GPM CMB retrievals with respect to gauge-corrected ground-based radar measurements. However, achieving this level of accuracy across the full swath will require including reference data from the outer edges of the GMI swath in the training data. In principle, this can be achieved through synthetic transformations of the existing training data that account for the parallax effect. However, a technically simpler approach is including reference data from other sources in the training. Given that these improvements can be retroactively applied to the full record of GMI observations, we consider this worthy of further development work.

The largest impediment to an operational adoption of the GPROF-NN retrievals is the retrieval bias observed for other sensors. The reason for this is likely the errors in the assumed hydrometeor profiles and/or radiative transfer simulations used to generate the a priori databases for these sensors. We thus identify the radiative transfer simulations used for the GPROF a priori database as a critical limitation for the accuracy of GPM precipitation estimates and suggest that future research should systematically assess and improve these simulations. Alternatively, co-locations between GPM CMB and the other sensors may be incorporated in the training process to improve the retrievals from these sensors.

However, our results also show that systematic retrieval errors of all GPROF retrievals largely follow the biases of the GPM CMB retrieval. In the longer term, improvements in global precipitation measurements will therefore require improvements in what are today considered reference measurements of precipitation, i.e., (potentially gauge-corrected) ground radar measurements and space-borne combined radar and PMW radiometer measurements. Convergence of these measurements is a prerequisite to better constrain global precipitation measurements from meteorological satellites.