When operated from a spaceborne platform in a nadir-looking configuration, radar systems transmit pulses towards the Earth's atmosphere and receive backscattered signals from atmospheric targets. As the radar pulse traverses the atmospheric column, it experiences two-way attenuation due to two primary mechanisms: (1) absorption by atmospheric gases, and (2) scattering and absorption by hydrometeors such as cloud and precipitation particles. This cumulative signal loss is referred to as Path-Integrated Attenuation (PIA). The total PIA can be decomposed into two components: gaseous attenuation (PIA_gas) and hydrometeor attenuation (PIA_hydro), as expressed in Eq. () 1PIA=PIAgas+PIAhydro.

The two terms are particularly large for millimeter wavelength radars that have traditionally been used to study clouds and precipitation systems . The two-way gaseous attenuation component (PIA_gas) is estimated using atmospheric absorption models based on thermodynamic profiles (see Sect. ), whereas the PIA_hydro accounts for two-way integrated extinction caused by hydrometeors, assuming negligible effects from multiple scattering , and is expressed as: 2PIA[dB]=20log⁡(10)∫0Hkext(z)dz, where kext is the height-dependent extinction coefficient (in m⁻¹) due to cloud and precipitation particles .

At 94 GHz, the Earth's surface acts as a strong radar reflector, returning signals with an intensity that is often several orders of magnitude greater than that from atmospheric targets. When attenuating hydrometeors (e.g., rain, snow, or cloud particles) are present in the radar beam, they reduce the strength of the surface return. This diminished surface echo, or surface signal depression, can be analyzed to quantify the attenuation introduced by hydrometeors .

At 94 GHz, attenuation by cloud liquid water is well described by Rayleigh scattering theory, resulting in an approximately linear relationship with the liquid water path (LWP) and exhibiting moderate sensitivity to temperature. In contrast, attenuation caused by rain must be modeled using Mie scattering theory, as the interaction of radar waves with larger hydrometeors depends on both temperature and the drop size distribution (DSD) of the precipitation. Despite these complexities, the total Path-Integrated Attenuation (PIA) is largely dominated by the column-integrated liquid water content, making it a robust proxy for estimating LWP .

Numerous studies have highlighted the effectiveness of using PIA for improving rainfall estimation. Traditional ground-based radar rainfall retrieval algorithms often rely on the Rayleigh approximation, which assumes that raindrops are small relative to the radar wavelength. These methods typically employ an assumed DSD to derive a simplified relationship between radar reflectivity and rainfall rate. However, for space-borne radars operating in the microwave spectrum, relying solely on reflectivity is insufficient due to the significant influence of attenuation. High-frequency radars, such as the CloudSat and EarthCARE 94 GHz radars experience significantly greater attenuation than lower-frequency radars for the same rain intensity . Hence, for high frequency radars, PIA is a useful measurement in rainfall and LWP retrieval . To address this, proposes a retrieval method specifically designed for attenuating radars, emphasizing the use of PIA or estimates of LWP as constraints. A comparative analysis between 14 GHz (Ku-band) and 94 GHz (W-band) radars demonstrates that reflectivity-based rain rate estimates at 94 GHz become highly uncertain beyond 1 mm h⁻¹, whereas the 14 GHz radar provides reliable estimates up to 40 mm h⁻¹. When LWP is incorporated as a constraint, the retrieval accuracy improves significantly, underscoring the critical role of PIA in rainfall retrieval for high-frequency radar systems. The CloudSat warm rain retrieval algorithm utilizes this method and employs a hybrid approach, using reflectivity-based retrieval at lower rain rates and switching to an attenuation-based method at higher rain rates, where attenuation becomes more pronounced. Quantitative analysis of the algorithm reveals that this transition between reflectivity-dominant and attenuation-dominant retrieval occurs within the rain rate range of approximately 0.1 to 0.5 mm h⁻¹ .

Given the importance of PIA in microwave remote sensing, its accurate estimation largely depends on the reliable characterization of the effective surface backscattering cross section (σ0e). Over the ocean, σ0e can be parameterized as a function of incidence angle, sea-surface temperature, and wind speed . In contrast, over land, σ0e is highly variable and depends on factors such as vegetation type, soil moisture, and terrain roughness, making characterization of σ0e far more difficult . Consequently, PIA estimation and PIA-based rainfall retrievals are generally restricted to oceanic regions, where geophysical models can provide robust estimates of σ0e. Section , explores in detail how σ0e over the ocean varies with wind speed and assess the ability of different geophysical models to reproduce the EarthCARE σ0e climatology.

CloudSat's 2C-PRECIP-COLUMN product employs two complementary methods for estimating PIA. In regions with scattered clouds, where adjacent clear-sky observations are available, the Surface Reference Technique (SRT) is used. This method estimates PIA by interpolating the clear-sky surface backscattering cross section from nearby cloud-free profiles over the cloudy areas. However, in the presence of extensive, continuous cloud cover, where no nearby clear-sky profiles are available, SRT cannot be applied. In such cases, PIA estimation relies entirely on geophysical models (see Sect. ).

In this paper, we propose a PIA retrieval scheme that mirrors CloudSat’s dual-path strategy but is tailored to the early operational phase of EarthCARE, when instrument calibration is still evolving. Section  outlines the methodology in detail. The modeling of gaseous attenuation used to derive PIA_gas is detailed in Sect. , while Sect.  describes the procedure for estimating the normalized radar cross section (NRCS). Section  explains criteria used in selecting calibration points for the SRT. Section  explores how the σ0e varies with wind speed and assess the performance of different geophysical models. Section  illustrates our retrieval's performance through several case studies. Comparative analysis with CloudSat's PIA estimates is provided in Sect.  and section “Statistical comparison with the CloudSat PIA estimates”, highlighting the consistency and potential advantages of the proposed method.

Hereafter the PIA_hydro (PIA_gas) indicates the two-way PIA associated to the hydrometeors (atmospheric gases). The unknown is PIAhydro(x) at a given location x (where clouds and/or precipitation are presents). The σ0e, which is effective back scattering cross section represents the expected surface backscatter signal under no atmospheric attenuation (gases and hydrometeors). The measured NRCS at point x denoted by σ0m(x) will be related to the σ0e at the same point by: 3σ0m(x)=σ0e(x)-PIAgas(x)-PIAhydro(x). If in the vicinity of x there is a location x1 characterized by calibration condition, then: 4σ0m(x1cal)=σ0e(x1)-PIAgas(x1). Section  provides a detailed explanation on how calibration points are chosen. The effective surface normalized radar cross section can be either derived from a geophysical model with σ0e being a function of wind speed and SST or from measured NRCS by correcting for gas attenuation. The PIA_gas term can be derived from gas attenuation models (given the atmospheric thermodynamic profile; see Sect. ). With these components, the PIA_hydro can be inferred by inverting Eq. () as: 5PIAhydro(x)=σ0e(x)-PIAgas(x)︸σ0gas(x)-σ0m(x). On the other hand, by subtracting Eq. () from Eq. () an alternative relationship for computing PIAhydro(x) is obtained as: 6PIAhydro(x,x1)=PIAgas(x1)-PIAgas(x)+σ0e(x)-σ0e(x1)+σ0m(x1cal)︸σ0gas(x,x1)-σ0m(x), where in both Eqs. () and () σ0gas(x) and σ0gas(x,x1) (respectively) represent two ways to estimate the NRCS that would be measured at x in the absence of hydrometeors (but with the presence of gases).

The advantage of computing PIA by Eq. () is that only differences appear on the right hand side of Eq. (), which makes the estimate very robust for radar miscalibration [affecting the values of σ0m but not of the difference σ0mcal(x1)-σ0m(x)] and for biases of the gas attenuation or in the σ0e estimation. For any given cloudy/rainy profile at position x, there may be multiple neighboring calibration points. If such N points are over the contiguous ocean free of ice they can be used as calibration points. Equation () can be generalized to: 7PIAhydrox,x1,x2…xN=∑i=1NwiPIAgas(xi)-PIAgas(x)+σ0e(x)-σ0e(xi)+σ0mxical︷σ0gas(x,xi)∑i=1Nwi︸σ0gasx,x1,…,xN≡σ̃0gas(x)-σ0m(x). In the PIA estimation algorithm used in the EarthCARE product, an optimal number of N = 5, calibration points is used to ensure that calibration points remain sufficiently close, as more distant points have negligible influence (see Sect.  for a detailed explanation of calibration point selection).

Each σ0gas(x,xi) term in Eq. () is assigned a weight wi that reflects the uncertainty introduced when using calibration point xi. This uncertainty primarily depends on two factors: the distance to the calibration point (d(x,xi)) and the wind speed, u(x), at that cloudy profile (x). To quantify this uncertainty, a large set of clear-sky profiles with measured NRCS (σ0gas) and wind speed is compiled based on one year of EarthCARE observations. Clear-sky profiles are chosen based on “profile_class” product from the Level 2 C-PRO FMR dataset . All the profiles with wind speeds uj in a given interval j ((j-1)δu < u(x) < jδu, δu = 1 m s⁻¹) have been paired with profiles whose distance dk falls in a class k ((k-1)δd < d(x,xi) < kδd, δd = 25 km); for all pairs the residuals 8Δσ0d(x,xi),u(x)=Δσ0(x,xi)=σ0gas(x,xi)-σ0gas(x) are computed based on the estimate of σ0gas(x) and σ0gas(x,xi) defined by Eqs. () and (), respectively.

For each bin of wind speeds falling in the j-bin and separation distances falling in the k-bin the standard deviation (SD) of the residuals 9Suncer(dk,uj)=SDΔσ0(d,u)|(k-1)δd<d<kδd,(j-1)δu<u<jδu is assumed to represent the uncertainty associated to an estimate of σ0gas(x,xi) at a location x with wind speed uj based on a calibration point xi separated by a distance dk. When multiple points are used for the estimation of σ0gas(x) like done in Eq. () the uncertainties of the different calibration points can be used to weight differently each σ0gas(x,xi). Hereafter the weights wi assigned to each calibration point xi are defined as the inverse of the squared uncertainty associated with each calibration point (Suncer(d(x,xi),u(x)), such that points with lower uncertainty contribute more strongly to the estimate: (wi = 1/(Suncer(d(x,xi),u(x)))2).

As an alternative to the clear-sky interpolation method, PIA can be estimated directly using Eq. (), where σ0e is derived from a geophysical model or from a climatology-based derived lookup table that estimates σ0e as a function of wind speed and SST. This approach is referred to as the model-driven method or Wind/SST method.

Figure  shows a comparison of PIA uncertainty derived from the model-driven approach and from interpolation using calibration points located at varying distances, where the uncertainty is estimated following the method described above. Figure  presents the PIA uncertainty lookup tables constructed by combining the curves shown in Fig. .

In Fig. , the black curve delineates the region beyond which model-driven method yields lower uncertainty than the interpolation-based approach. For model-driven method, σ0e is derived from a climatologically constructed look-up table based on clear-sky NRCS measurements, corrected for gaseous attenuation, as a function of sea surface temperature (SST) and wind speed over the period June 2024 to June 2025. For each SST-wind speed bin, the mean and standard deviation of σ0e are computed. The uncertainty associated with σ0e as a function of wind speed is estimated by calculating the weighted mean of standard deviations across SST bins, with weights given by the number of occurrences in each SST bin.

For wind speeds between 4 and 15 m s⁻¹, and calibration point distances ranging from ≈ 200 to ≈ 100 km respectively, clear-sky interpolation generally yields lower uncertainty than the model-driven method, often below 1 dB. However, as distance increases, the interpolation error increases and the model-driven approach becomes preferable. Although Fig.  suggests that interpolation may outperform the model-driven approach at very low wind speeds (below 4 m s⁻¹) and large calibration-point distances, uncertainties for both methods remain substantially high (Fig. ); therefore, at large distances, reliance on the model-driven method is preferable.

Hence, during periods when the radar is well-calibrated, a hybrid approach can be employed, combining the interpolation method using N calibration points (Eq. ) and model driven method (Eq. ).

The total uncertainty in the two-way PIA estimate at a given location x arises from two main sources, uncertainty in estimating the σ̃0gas(x), and from the inherent measurement error in radar reflectivity. The first component is estimated from weight associated with each calibration points as each weight wi corresponds to the inverse of the variance associated with the calibration point at xi, leading to a total uncertainty expressed as: 10Sgasuncer(x)=∑i=1Nwi-1/2. In addition to this methodological error, inherent measurement error in radar reflectivity which depends on the signal-to-noise ratio (SNR) and the number of independent samples (nsamples) also contributes to the overall uncertainty in the PIA estimate. This error is analytically estimated as : 11Sz[dB]=10log⁡101+1+1SNRnsamples. The number of samples is estimated from the pulse repetition frequency (PRF), the integration length (Lint), and the ground satellite velocity (Vsat) as: 12nsamples=PRFLintvsat For the EarthCARE Level 2 C-PRO FMR dataset, the integration length is 1 km, the PRF varies between 6100 and 7500 Hz, and the satellite velocity is approximately 7 km s⁻¹. Substituting these values (and assuming high SNR values) yields a measurement error ranging from approximately 0.15 to 0.13 dB depending on the PRF. Hence total PIA uncertainty is estimated as: 13SPIAuncer[dB]=Sgasuncer2+(Sz)2.

Figure  is a schematic depiction of the described PIA estimation methodology.

This approach is similar to the one already adopted for CloudSat but with two major differences.

In CloudSat, the interpolation method is typically applied only when clear-sky pixels are immediately adjacent to the cloudy pixel of interest, usually within a window of thirty surrounding profiles, which corresponds to ≈ 30 km . In contrast, the method used here allows interpolation even when the calibration points are ≈ 200 to ≈ 100 km from the cloudy pixel in wind speed conditions between 4 and 15 m s⁻¹ (Fig. ). This is possible because the variability in the gas absorption profile and in the σ0e due to the modulation of the atmospheric (temperature and relative humidity profile) and of the surface (SST and wind) properties, respectively, is accounted for in Eq. ().

To demonstrate the performance of the proposed PIA estimation methodology under varying atmospheric and surface conditions, several case studies using EarthCARE observations are presented in Figs. –. In each case, the first panel displays the vertical profiles of the radar reflectivity factor as a function of the along-track distance, with the calibration points selected based on the criteria outlined in Sect.  and shaded in grey. The second panel shows the measured NRCS (σ0m), which may be attenuated by hydrometeors, alongside the estimated gas-only NRCS (σ0gas). The σ0gas and corresponding PIA are computed using five clear-sky calibration points as defined by Eq. (). The third panel presents the resulting PIA estimates, with shaded regions indicating profiles where negative PIA values are obtained. The fourth panel shows the total PIA uncertainty estimate, calculated using Eq. (), which includes both the uncertainty from the PIA uncertainty look-up table (Fig. ) and a fixed contribution of 0.15 dB from inherent measurement noise, corresponding to a PRF of 6100 Hz. The shading in the panel represents calibration points where PIA is not estimated.

Figure  depicts a scene of scattered cumulus clouds over the Southern Ocean. Note that, thanks to the sharp EarthCARE point target response , the profiles of radar reflectivity are not contaminated by clutter down to 500 m. In this case, numerous calibration points are situated close to the cloudy profiles, allowing for high-confidence PIA estimates with relatively low uncertainty. The farthest calibration point is approximately 50 km away, and the maximum PIA uncertainty is 0.4 dB, which is presented in the fourth panel. Wind speeds in this scene range between 3.5 and 8.8 m s⁻¹.

Figures  and  depicts extensive, continuous cloud systems with few or no nearby calibration points. Figure  shows an extensive stratocumulus deck over the southeastern Atlantic Ocean, off the southwestern coast of Africa. These clouds are typically shallow with flat tops and are capped by a temperature inversion. Although the resulting PIA values remain relatively low (generally below 2–3 dB), accurate estimation is essential for reliable rainfall retrievals. In this case, for profiles in the middle of the precipitating system, the farthest calibration point can be located approximately 480 km away, corresponding to a maximum PIA uncertainty of 0.8 dB as shown in the fourth panel. Wind speeds in the scene range from 7 to 11 m s⁻¹, and the cloud deck extends roughly 1170 km in length. Figure  illustrates a widespread stratiform cloud system over the southeastern Atlantic Ocean near the western coast of Africa. The cloud cover stretches nearly 930 km, with limited or no nearby calibration points. The farthest calibration point is about 340 km away, resulting in a maximum estimated PIA uncertainty of 0.45 dB, which is represented in the fourth panel. Wind speeds in this region range from 7 to 10.5 m s⁻¹.

In all the cases above, the negative PIA values are small, typically fractions of a dB. These negative estimates arise from the noisiness in the measured σ0m associated with the fluctuations of the backscattering returns and from the uncertainties associated with σ0gas (e.g., associated with the ECMWF reanalysis wind speed and SST used as inputs).

These diverse case studies highlight the flexibility and robustness of the proposed PIA retrieval approach across different cloud morphologies, calibration point availability, and wind conditions.

As briefly discussed in Sect. , CloudSat employs a hybrid approach to estimate PIA, combining two complementary methods similar to the one proposed in this study. The first approach, referred to as the Wind/SST method, estimates the NRCS at the cloudy profile in absence of hydrometeor and presence of gaseous attenuation (σ0gas) as a function of surface wind speed and SST using empirically derived look-up table and the second one is interpolation-based approach, where clear-sky profiles surrounding the cloudy profile are used to estimate the σ0gas. In the interpolation-based method, a search is performed within 30 profiles (approximately 30 km) surrounding the cloudy profile for clear-sky conditions. If at least five clear profiles are found, a weighted average of their observed NRCS is computed, with weights based on the distance of each clear profile to the cloudy profile . If the minimum requirement of five clear-sky profiles is not met, the Wind/SST method is used instead.

Figure  presents a case study from 2 January 2008, using the CloudSat observations. The PIA estimation methodology proposed in this study is also applied to this case for direct comparison with CloudSat method. CloudSat provides estimates of the unfiltered PIA (i.e., without discarding negative values), along with the measured NRCS (σ0m). Therefore, the estimate of NRCS at the cloudy region in the presence of gas only (σ0gas) for the CloudSat products can be obtained by just summing the PIA and the measured σ0m.

The second panel of Fig.  shows the measured NRCS (σ0m), the σ0gas based on CloudSat method (σ0gas CloudSat), and the estimated σ0gas based on the methodology proposed for EarthCARE (σ0gas EarthCARE). Abrupt jumps of up to nearly 1 dB are observed in the CloudSat- and EarthCARE-derived σ0gas, particularly at transition points between the two estimation methods. Discontinuities in the CloudSat-derived σ0gas are highlighted with red circles, while jumps observed in both CloudSat and EarthCARE estimates are indicated with black circles in the second panel of Fig. . The variable “Diagnostic_PIA_method” in the CloudSat 2C-PRECIP-COLUMN product indicates the estimation method used for each profile, represented by shading in the second panel of Fig. . Blue shading denotes the interpolation-based method, while gray shading corresponds to the Wind/SST method.

The 30 km limit of CloudSat’s clear-sky interpolation often leads to frequent switches to the Wind/SST-based method, causing nonphysical jumps in σ0gas and PIA. In contrast, the EarthCARE approach allows interpolation over much longer distances, typically between ≈ 200 km to ≈100 km, depending on the surface wind speed, significantly reducing such transitions and yielding smoother, more consistent estimates. Although method transitions may still introduce occasional discontinuities in the EarthCARE estimates, their frequency is markedly lower compared to the CloudSat approach, as shown in Fig. .

This study presents a robust methodology for estimating path-integrated attenuation (PIA) over oceanic regions, which is currently under implementation and will be incorporated into the PIA estimation component of the EarthCARE CPR Level 2A C-PRO product. The approach is specifically designed to be resilient to potential radar calibration biases, such as those that may arise during the early phases of the mission, thereby enhancing the reliability of attenuation-based retrievals under non-ideal instrument conditions. It combines two complementary approaches: a clear-sky interpolation technique and a model-driven (wind/SST) method. The clear-sky interpolation method estimates PIA at a cloudy profile by leveraging surrounding calibration points selected based on criteria described in Sect. , as defined in Eq. (). Importantly, the clear-sky interpolation method, as described in Eq. (), estimates PIA by computing the difference between the measured normalized radar cross section (NRCS) (or effective surface backscattering cross section corrected for gaseous attenuation) at the cloudy profile and that at surrounding clear-sky calibration points, rather than relying on their absolute values. The method uses multiple calibration points, optimally weighted based on their distance from the cloudy profile and the surface wind speed at the cloudy profile, so that the nearest calibration points are weighted higher, and the PIA estimate at a cloudy profile at low wind conditions reports larger uncertainty.

In situations where suitable clear-sky calibration points are not available within a distance that permits accurate interpolation, the retrieval defaults to a model-based approach, as described in Eq. (). The model-based method estimates PIA using the effective normalized radar cross section (σ0e), derived from the climatology-based look-up table that relates σ0e to surface wind speed and sea surface temperature (SST), based on collocated ECMWF data. The hybrid method can be applied when the radar is well calibrated.

The performance of the EarthCARE method was evaluated by applying it to CloudSat observations over a four months and by comparing the resulting PIA estimates to those reported in CloudSat's 2C-PRECIP-COLUMN product . CloudSat uses a similar hybrid strategy, choosing between clear-sky interpolation and a wind/SST-based approach depending on the availability of nearby clear-sky profiles. However, CloudSat applies clear-sky interpolation only within a 30 km, while the EarthCARE approach allows interpolation from calibration points located ≈ 100 to ≈ 200 km away from the cloudy profile, depending on the surface wind speed. This extended interpolation capability reduces the number of transitions between estimation methods and improves the spatial consistency of the retrieved PIA.

A detailed case study and global statistical analysis confirm the effectiveness of the proposed EarthCARE methodology. For profiles where CloudSat and EarthCARE apply clear-sky interpolation, both methods yield highly consistent PIA values, with most differences falling within ±0.5 dB. In contrast, for profiles where CloudSat employs the Wind/SST method and EarthCARE uses interpolation, the PIA difference histogram peaks at approximately 0.3–0.5 dB, slightly above zero. This is partly because CloudSat applies the wind/SST method more frequently, over 66 % of profiles globally compared to 23 % in the EarthCARE method, which maintains a higher reliance on clear-sky interpolation. In general EarthCARE method provided PIA estimates with marginally lesser negative PIA estimates and a higher occurrence of positive PIA estimates.

Overall, the proposed retrieval scheme demonstrates strong agreement with CloudSat's established method. In future work, other EarthCARE instruments beyond the radar, such as the Multi-Spectral Imager (MSI) and the Atmospheric Lidar (ATLID), can be leveraged in order to better identify clear-sky profiles, to validate the PIA estimates, and to improve estimates of the LWP product . Furthermore, a brightness temperature product for the CPR, expected to be developed in the coming months and analogous to that available for CloudSat , could provide additional constraints on these retrievals.