The theoretical basis of SLP retrieval using the DAR system is that environmental conditions, especially atmospheric O₂, have significant impacts on radar signals. The basic relation between the radar signals and environments, as demonstrated originally by Lin and Hu (2005) and later in many other studies (Lawrence et al., 2011; Lawrence et al., 2012; Millan et al., 2014; Lin and Min, 2017; Lin et al., 2021), can be approximately expressed as P r f = σ 0 f 4 π R 2 P t f A f T 2 f , (1) where P _t (f) and P _r (f) are radar transmitted and receive signal powers at frequency f, respectively. A(f) is a two-way system gain constant, mainly for the antenna, while σ ⁰ (f) and R are the wavelength-dependent sea surface reflectance and the range from radar to the surface, respectively. T(f) is one-way atmospheric transmittance at the radar frequency. It critically depends on atmospheric absorption from O₂, water vapor, cloud water, and potentially rainwater. Here, we only consider non-rain or light rain (rain rate <1 mm/h) weather conditions. A recent OSSE study (Previ et al., 2024) found that sea surface pressure retrievals of spaceborne DAR instruments provide significant weather forecast impacts even with a light rain threshold of 1 mm/h. For moderate to heavy rain cases, absorption and scattering from precipitation-sized large hydrometeors and their related large column total water amounts could increase atmospheric attenuation considerably (Lin and Rossow, 1997), which would reduce the signal-to-noise ratio (SNR) of the radar measurements by an unacceptable amount. Thus, this study avoids significant atmospheric scattering (or substantial rain) cases to maintain high SNR and retrieval precision. Note that large hydrometeor scattering situations could be used for atmospheric pressure profiling by measuring radar returns at backscattering layers such as bright bands of rain. Since accurate range capability, besides regular DAR needs for SLP retrieval, is also required for the profiling, this kind of DAR system is beyond the scope of this study and left for future studies.

The atmospheric transmittance term in Eq. 1 is the key to relating radar returns with atmospheric agents. For an absorption-dominated atmosphere, the transmittance T equals T f = ⁡ exp − τ O f + τ V f + τ c f + τ L f + τ r f / μ , (2)

Here, μ is the cosine of radar viewing angle, and τ _m represents the absorption optical depth of atmospheric agent m (m = O₂, vapor, dry air continuum, cloud liquid water, and rainwater or O, V, c, L, and r for short, respectively) at frequency f. Cloud ice water is not considered in this transmittance because, for non-rain or light-rain conditions, ice particulars are very small compared to considered wavelengths and have no significant absorption at microwave wavelengths. The extinction effects of both scattering and absorption from these particulars are thus negligible (Lin and Rossow, 1994; Lin and Rossow, 1997; Lin et al., 1998a; Lin et al., 1998b). Similarly, because of the small particle sizes of light rain droplets, scattering effects can be ignored, which leaves only rain absorption effects in the transmittance term.

After calibration and normalization for the system gain A and transmitted power P _t , the ratio, γ, of two return powers with closely spaced frequencies f ₁ and f ₂ becomes γ f 1 , f 2   = C 2 C 1 k 2 k 1 P r f 1 P r f 2 = C 2 C 1 P t f 1 P t f 2 σ 0 f 1 σ 0 f 2 k 2 A f 1 k 1 A f 2 T 2 f 1 T 2 f 2   . (3)

In this equation, k values are normalization constants that mean the DAR system has the same gain for all frequencies and can be obtained from ground system tests before flight observations or be estimated with vicarious calibration. Similarly, C ₁ and C ₂ are radar power calibration and normalization coefficients at frequencies f ₁ and f _2, respectively. They are used to achieve a uniform radar transmitted power across all frequencies. Unlike constant k values, these power calibration coefficients are measured instantaneously from an internal calibration loop of the DAR system in monitoring potential radar power drafts at all wavelengths. For simplicity, hereafter we assume all DAR powers transmitted are calibrated; thus, all calibration coefficients are no longer shown in equations.

Further simplifying Eqs 2, 3 shows that Ln γ f 1 , f 2 = Ln P r f 1 P r f 2 = Ln σ 0 f 1 σ 0 f 2 + 2 Ln T f 1 T f 2 , (4) or , Δ τ O f 1 , f 2 + Δ τ V f 1 , f 2 + Δ τ c f 1 , f 2 + Δ τ L f 1 , f 2 + Δ τ r f 1 , f 2 = − 0.5 μ Ln γ f 1 , f 2 − Ln σ 0 f 1 σ 0 f 2 , (5) where Δτ _m (f ₁ , f ₂ ) means the optical depth difference between frequencies 1 and 2 for the m agent. Since the dry air continuum absorption (absorption not related to the O₂ absorption) is very weak, its differential absorption would be even smaller and could be potentially ignored. The first term in this equation is assumed to be positive (i.e., channel 1 has much stronger O₂ absorption than channel 2) and can be calculated as: Δ τ O 1 , 2 = ∫ 0 Z α O 1 z − α O 2 z n o z d z = δ O 1 , 2 N O . (6)

Here and hereafter, the numbers 1 and 2 represent f ₁ and f ₂ , respectively. Z is the flight altitude for an airborne system, and for spaceborne systems, Z is the top-of-atmosphere (TOA) height. n _o (z) is the O₂ molecular number density at z altitude, while α is the gas absorption cross-section at the given frequency. δ _O1,2 and N _O , respectively, are vertically weighted O₂ differential absorption cross-sections of the two frequencies and total O₂ molecular number in the column, or δ O 1 , 2 = ∫ 0 Z α O 1 z − α O 2 z n o z d z / ∫ 0 Z n o z d z , (7) and N O = ∫ 0 Z n o z d z = Δ τ O 1 , 2 / δ O 1 , 2 . (8)

Equations 6–8 for O₂ can also be applied to atmospheric water vapor, cloud water, and rainwater as in the calculations of this study. However, vapor generally concentrates in the lower troposphere, and cloud water and rainwater exist in certain atmospheric layers. Thus, very similar to Eq. 6, their absorption optical depth is usually expressed as a simple product of their absorption coefficient, a, and their corresponding water amount such as column water vapor (CWV). For cloud water and rainwater, they can be combined as a liquid water path (LWP) because of the same nature of liquid water absorption (e.g., Lin and Rossow, 1994; Lin et al., 1998a; Lin et al., 1998b; Wentz, 1997; Wentz, 2015). Thus, these integrated values are expressed as Δ τ V 1 , 2 = a V 1 – a V 2 C W V ; (9a) Δ τ L 1 , 2 = a L 1 – a L 2 L W P ; (9b)

Since these agents show only continuum absorption at the current spectral band, linear or quasi-linear variations of the coefficients with frequency are expected. That is, this expression clearly illustrates that the differential absorption optical depth (DAOD) values of these agents should linearly vary with frequency differences.

Since surface dry air pressure is a measure of the total dry air mass of the entire atmospheric column, it could be derived from a combination of the dry air pressure at flight altitude P _Z with the pressure caused by the dry air mass of the DAR measurement column. For space observations, P _Z = 0. Furthermore, the partial dry air pressure of the measurement column would be the product of the measured column dry air molecular number N _d and molecular mass M _d along with Earth’s acceleration of gravity g, which can be estimated based on latitude with altitude adjustment (Wenzel, 1989; van Dam et al., 2010; Campbell et al., 2020). Since O₂ is well mixed in the dry air with a constant mixing ratio χ_O, the dry air partial pressure, P _dp , can be derived from the O₂ amount or its directly related Δτ _O measurement: P d p = N d M d g = N O M d g χ O = Δ τ O 1 , 2 M d g χ O δ O 1 , 2   . (10)

Thus, SLP can be obtained from P _dp and P _Z , along with the atmospheric column water amount M _w , that is S L P = P d p + P Z + M w g . (11)

These Eqs 9–11, combined with Eqs 4, 5, provide the theoretical foundation for SLP retrieval using DAR techniques. Note that the water amount is mainly decided by atmospheric column water vapor with small (at least an order of magnitude smaller) contributions from cloud water and rainwater in non-rain or light rain conditions. Additionally, the atmospheric column total ice water amount or ice water path (IWP) for these weather conditions is usually <0.15 kg/m² (Lin and Rossow, 1996; Minnis et al., 2007). With such a small amount of ice, it would produce pressure much less than 0.1 mb. Furthermore, these moisture components, including the IWP, have been observed over oceans for decades (e.g., Lin and Rossow, 1996; Lin and Rossow, 1997; Minnis et al., 2007). The water mass uncertainties would be only about 0.3 kg/m² (Mears et al., 2015), which is well within 1 kg/m² or equivalently the air pressure uncertainty within 0.1 mb.

The analysis here clearly shows that besides O₂, other environmental variables such as humidity, cloud, rain, and sea surface could have potential impacts on the measured power ratio, though the spectral space between the two channels is very small. Theoretically, even the dry air continuum absorption could potentially affect retrieval because of the high precision required. However, it is very small and basically linearly varies with frequency, particularly for the differential absorption. Thus, the impact of dry air continuum absorption on SLP retrieval is negligible, i.e., only brief discussions are given here. The sea surface reflection term in Eq. 5 should be close to one for normal sea surface wind conditions (Lin et al., 2023). The impact of sea surface water temperature, salinity, and incidence angle was found to be similar. Thus, this term could be removed after certain calibrations. In other terms, an advantage of the differential absorption technique is that the absorption cross-section and optical depth of these variables would not only exert small changes but also near linearly vary with frequency in those closely spaced spectra. This leads to the consideration of adding a third channel with even weaker O₂ absorption than channel 2 in the DAR system design. If the spectral space between this channel and channel 2 would be the same as that of channels 1 and 2, the grand ratio, G, of the power ratios of γ ₁ , ₂ and γ ₂ , ₃ would be: G = γ 1,3 / γ 2,3 = P r , 1   P r , 3 P r , 2   P r , 2 . (12)

From this grand ratio, the difference of the differential absorption optical depth (DAOD) values, also called the DAOD of the 3-channel approach, Δτ _3C, is derived: Δ τ O 1 , 2 − Δ τ O 2 , 3 + Δ τ V 1 , 2 − Δ τ V 2 , 3 + Δ τ c 1 , 2 − Δ τ c 2 , 3 + Δ τ L 1 , 2 − Δ τ L 2 , 3 + Δ τ r 1 , 2 − Δ τ r 2 , 3 = − 0.5 μ Ln G − Ln σ 1 o σ 3 o / σ 2 o σ 2 o = Δ τ 3 C . (13)

As found from a previous sea surface reflection study (Lin et al., 2023), the sea surface reflection term in this equation can be considered as zero with sufficient accuracy. Because of the spectrally evenly spaced channels and quasi-linear absorption coefficients with a frequency of moisture agents and dry air continuum absorptions, all differential absorption terms, except O₂ in Eq. 13, are effectively removed. Thus, O₂ DAOD from the 3-channel approach, Δτ _O3C , can be derived from the grand ratio: Δ τ O 3 C = Δ τ O 1 , 2 − Δ τ O 2 , 3 = τ O 1 + τ O 3 − 2 τ O 2 = − 0.5 μ Ln G . (14) with the proportionality of sea surface dry air partial pressure and O₂ DAOD, sea surface dry air pressure and SLP can be retrieved based on the DAOD values, Δτ _O3C , of the 3-channel approach. Eqs 13, 14 show no adjustment or compensation for environmental variables is required for this 3-channel approach.

This study addresses the SLP retrieval algorithm development and assesses the impacts of measurement and environmental uncertainties on retrievals. The instrument design, structure, operation procedure, and system performance of studied DAR systems are beyond the scope of this study. The system character introduced here is to provide a basic DAR system overview for algorithm developments, assuming the systems can reach their hardware development and measurement objectives. Thus, the analysis and simulation discussed later focus on retrievals and not on instrument operation and performance.

The current design of the DAR instrument will have three channels: the first one is at a substantial O₂ absorption frequency, the second has a frequency very close to the first but with weak O₂ absorption, and the third is at the same spaced wavelength as the second to the first and has even weaker absorption. These channels are selected at the central frequencies of 65.5, 67.75, and 70 GHz, respectively, with a spectral bandwidth of 100 MHz for each channel. Unlike using frequencies at the lower sideband of the 50–75 GHz O₂-absorption complex in the original work (e.g., Lin and Hu, 2005; Lawrence et al., 2011), these upper sideband frequencies are considered to eliminate the possibility of active instruments interfering with passive microwave sensors such as Advanced Technology Microwave Sounding (ATMS), which has the same lower sideband for atmospheric temperature profiling. With these three channels, two ratios of DAR sea surface power returns, one for the pair at frequencies 65.5 and 67.75 GHz and another at 67.75 and 70.0 GHz, can be measured. As previously discussed, the key reason to have the second pair is to eliminate the impact of environment variables other than O₂ for pressure retrieval.

Figure 1 shows the spectral character of gas absorption optical depth (OD) around the selected frequency bands for climatological tropical (black curves) and midlatitude winter (blue curves) atmospheric vertical profiles (McClatchey et al., 1972). The OD values of total gases (solid curves), O₂ (dashed curves), and water vapor (dashdot curves) from the top-of-atmosphere (TOA) to the surface are plotted. As indicated, the total and O₂ gas absorption curves for the tropical profile are shifted up by 0.5 to avoid overlapping with winter ones. Similarly, the water vapor OD values are enlarged 20 times to show them clearly because they are too small compared to others. The stars in the figure represent the three channels used in current instrument. These channels are all selected at relatively smooth spectral places, which minimizes retrieval uncertainties caused by unknown spectral changes.

The dominant part of total gas absorption OD values is from O₂, especially at strong absorption wavelengths (lower frequency side of the figure). The third channel has weaker O₂ absorption compared to channel 2 and generates limited O₂ DAOD because the purpose of selecting this channel is not to achieve significant DAOD values but instead the removal of environmental impacts other than O₂. Influenced by the O₂ absorption line shapes, certain asymmetries in O₂ absorption ODs occur around individual center frequencies, particularly for channel 1. This O₂ absorption feature also leads to uneven total gas OD values around channel center frequencies. Thus, OD values at center frequencies may not represent spectral characteristics of entire channels even if the bandwidth is only 0.1 GHz or only about 0.15% of the center frequency.

Within the selected closely spaced spectral range, the other environment variables practically linearly vary with frequencies as indicated by the water vapor optical depth τ _V . Requiring enlargement to be visible in the scale, even extremely abundant water vapor, the tropical τ _V values (black dashdot line) are small compared to the O₂ optical depth. When the difference of the differential absorption at channels 1 and 2 and channels 2 and 3 is calculated or the 3-channel approach is applied, the vapor DAOD for the 3-channel approach, Δτ _V3C , is expected to be very close to zero, i.e., Δτ _V3C = τ _V1 + τ _V3 − 2τ _V2 ≈ 0 (see Figure 1). Similar DAOD results can also be obtained for cloud water and rainwater.

Figure 2 shows the absorption spectral characteristics of rain and cloud water calculated at 15°C water temperature. Precipitation-sized droplets are calculated based on classical Marshall-Parmer (1948); black lines) rain and Joss-Waldvogel (1969); magenta lines) drizzle size distributions with 1 km precipitation depth. The two size distributions are considered to account for the potential range of size spectral and total mass variations of large and/or small precipitation-sized particles under various weather conditions. The absorption and extinction of these light rains are estimated based on Mie scattering. The dashdot, dashed, and solid curves are for 0.5, 1, and 3 mm/h rain rates, respectively, which covers the rain screen need upon retrieval. For a light rain of about 3 mm/h and ∼1 km rain thickness, the extinction optical depth could reach values of 0.6 or larger, which would produce about 5 dB attenuation for DAR return signals. Thicker rain layers with weaker rains (e.g., 3 km thick rain with 1 mm/h rate) would have similar attenuation. Cloud OD (blue line) is estimated with a cloud liquid water path of 0.2 kg/m². The stars show the frequencies of the current instrument. These OD values practically vary linearly with frequency in this V-band. The near linear variations of cloud water and rain absorption with frequency cancel their DAOD values in a 3-channel approach. Additionally, this 3-channel approach removes sea surface reflection impact as shown in Lin et al. (2023). All these illustrate the crucial advantage of the 3-channel approach.

The designed DAR instruments will have transmitted powers ranging from 1 W to 10 W. The high-end power is for space applications and others for systems onboard aircraft, particularly ER-2. With these strong power levels, the instruments potentially gain the ability to scan Earth’s surface along a cross-track direction with ±15° viewing angles. The uncertainty of the viewing angle or attitude knowledge for airborne systems is within 0.01°, while that for space systems will be even smaller. Since SLP is a measure of column air mass, an adjustment of slant path DAR scanning measurement to nadir direction is required as indicated in previous equations, e.g., Eqs. 5, 13, 14. Thus, attitude uncertainty should be controlled to a reasonable level for precise SLP retrieval.

Both airborne and spaceborne systems would integrate received signals to reduce the uncertainty in partial column dry air mass pressure retrievals or the relative error ε of Δτ _O3C : ε(Δτ _O3C )/Δτ _O3C within about 0.1%–0.2%. Since the signal S for pressure retrieval is exp(−2Δτ _O3C ) or S _dB = −20Δτ _O3C log ₁₀ (e) when using decibels, the error of the signal is ε S d B = 20 ⁡ log 10 e ε Δ τ O 3 C = 20 ⁡ log 10 e Δ τ O 3 C / 1000   . (15)

Calculations show that Δτ _O3C is about 2.3 (see Figure 1), thus, ε(S _dB ) in Eq. 15 is about 0.020 dB for 0.1% retrieval uncertainty. This requires the uncertainty in the retrieval signal or in the DAR case the grand ratio to be about 0.02–0.04 dB, which sets a target in the DAR development for an SLP retrieval error of about 1–2 mb. In this simulation, noise levels with these uncertainties are added to the signals. Note that for the pair of channels 1 and 2, the Δτ _O1,2 is about 2.7 (see Figure 1) and its corresponding 0.1% error about 0.023 dB.

For received signal powers and considering the basic nadir viewing situation, based on the first-order error propagation theory (Ehret et al., 2008; Lin and Liu, 2021), the expected standard error in DAOD values of the 3-channel approach, ε(Δτ _O3C ), due to standard errors (STD) in individual signal power quantities in Eq. 14 can be estimated as ε   Δ τ O 3 C   = 1 2 S T D 1 P 1 2 + 4 S T D 2 P 2 2 + S T D 3 P 3 2 1 / 2 . (16)

The 3-channel approach could have very different retrieval errors for various signal integration schemes used during DAR operation. One scenario is the weakest channel, channel 1, has considerably lower received signal powers or bigger standard errors than the other two channels even when it reaches the targeted 0.02 dB or 10^0.002 − 1 = 0.46% precision requirement through DAR signal integration. In this case, the retrieval precision for the 3-channel approach would be ε(Δτ _O3C )/Δτ _O3C = ε(P ₁ )/(2P₁ )/Δτ _O3C ≈ 0.46%/2/2.3 ≈ 0.1% or about 1 mb SLP retrieval error, as expected. However, this may require a long integration time because of the likelihood of the high requirement for all channels. An effective way would provide more time or independent samples for weak channels. In this case, channels 1 and 3 could be assumed to be weaker than channel 2. This may be realistic since channel 2 is located at the spectral center of the DAR amplifier, resulting in higher gain. Thus, the retrieval uncertainty would be ε(Δτ _O3C )/Δτ _O3C = ε(P ₁ )/( 2 P ₁ )/Δτ _O3C ≈ 0.46%/ 2 /2.3 ≈ 0.14%, which satisfies the retrieval requirement of 1–2 mb. Another typical scenario is that all channels have the same relative standard error, which would result in ε(Δτ _O3C )/Δτ _O3C = 1.5 ε(P ₁ )/P ₁/Δτ _O3C ≈ 0.24%, which is a factor 3 higher in retrieval uncertainties compared to the previous scenario. It may need increasing samples to reduce the error to within 2 mb. This study will further investigate the two latter scenarios in retrieval simulations alongside the error analysis using Eq. 16 here.

This discussion shows that the key to achieving precise SLP retrieval is signal integration during DAR operation. For the designed DAR instruments, the integration periods will produce spatial resolutions or effective field-of-views (EFOVs) of 50 km and 4 km for spaceborne and airborne platforms, respectively. The antenna for space systems is assumed to be similar to those onboard CloudSat and GPM with a size of 1–2 m in diameter, while that for airborne DAR systems is 0.3 m in diameter.

In addition to transmitting regular tones for O₂ sounding, DAR sensors can also send another 70 GHz chirped tone with longer and increased pulse energy for rain monitoring. The sensitivity of this tone for rain detection should be between regular operational meteorological radar systems at the Ka-band and W-band. If a 3–5 dB reduction in DAR received signals is tolerable, the light rain would be within about 3 mm/h for ∼1 km rain thickness (see Figure 2). Thicker rain layers, such as those 2 km–3 km thick, with about a 1 mm/h rain rate would produce a similar impact. Thus, a threshold of 1 mm/h was used as the cutoff threshold for this analysis. This conservative threshold is also consistent with the OSSE study of Prive et al. (2024) and could result in good retrievals for NWP models as mentioned previously.

One advanced feature in SLP retrieval with designed DAR instruments is its associated microwave radiometer, which operates at the same upper sideband of the O₂ absorption complex for temperature profiling. This temperature sounder has a total of eight channels with frequencies from 63.90 to 69.80 GHz. With this temperature profiling capability, the weak temperature dependency of O₂ differential absorption will be accounted for and the potential SLP retrieval uncertainty due to temperature uncertainties will be mitigated. Furthermore, this capability will provide atmospheric pressure vertical profile retrieval based on the barometric formula with an anchor of SLP retrieved. This pressure profiling capability will be analyzed in future studies.

Table 1 shows the basic information of the assumed DAR Systems. Since the DAR systems are still in the development stage and supported under the NASA Instrument Incubator Program, detailed specifics of the instruments and flight environments are not available at present, especially for spaceborne systems. Full DAR system descriptions, component/subsystem performances, and hardware simulations will be given in future reports. The simulations of this study focus on SLP retrievals based on the system described in the table.

The microwave radiative transfer (MWRT) model used for atmospheric absorption and attenuation calculations in current simulations is the same as that of our previous studies for DAR concepts (Lin and Hu, 2005; Lawrence et al., 2012; Lin et al., 2021). This model has been used in various environmental applications like precipitation, clouds, water vapor, and land surface for satellite microwave measurements (Lin and Rossow, 1994; Lin et al., 1997; Lin et al., 1998a; Lin et al., 1998b; Ho et al., 2003; Huang et al., 2005; Min et al., 2010). To maintain sufficient SNR for SLP retrieval, rain rates higher than 1 mm/h were not considered, which avoids excessive microwave scattering and attenuation caused by large precipitating hydrometeors (Lin et al., 1997). Thus, the transmission, absorption, and weak light rain scattering of radar signals within individual atmospheric layers were the major radiative transfer processes for forward model calculations. For the absorption process, this MWRT model carefully accounted for the temperature and pressure dependences of cloud water and atmospheric gas absorptions (Lin et al., 2001). The gas absorption was calculated based on Liebe et al. (1993), except water vapor continuum absorption has been treated separately (Rosenkranz, 1998). For liquid water absorption, the liquid water refractive index model of Turner et al. (2016) was used as it accounted for water absorption across a broad temperature range, especially for conditions below 0°C. The sea surface reflection was obtained with the V-band sea surface reflection model developed by Lin et al. (2023).

During DAR retrieval simulations, environmental data from the Goddard Earth Observing System Model version 5 (GEOS-5) of the NASA Goddard Modeling and Assimilation Office were used. The GEOS-5 model is a weather-and-climate-capable model used for atmospheric analyses, weather forecasts, uncoupled and coupled climate simulations and predictions, and coupled chemistry-climate simulations (Rienecker et al., 2008). The model simulates all necessary meteorological conditions, such as sea surface wind, sea surface temperature (SST), precipitation, and atmospheric temperature, pressure, and humidity (T/p/q) profiles. Key integrated or averaged variables such as CWV, LWP, and rain rate were also obtained. The weather case used in the global spaceborne DAR retrieval simulations was obtained from the GEOS-5 nature run (G5NR; Gelaro et al., 2015). The time of the globally simulated case was 10 July in the second year of G5NR, at 0000Z. The horizontal resolution of this simulated data was 50 km. Vertically, this nature run reported its simulated results at 48 specific pressure levels, reduced from the actual 72 modeling levels, from the surface up to a 0.01 mb level. For retrieval simulations of aircraft ER-2 flights, the horizontal resolution was interpolated to 4 km around the ER-2 flight tracks.

Besides the GEOS-5 simulated global data, climatological profiles from McClatchey et al. (1972) were also used in this study. Previous studies on MWRT modeling and retrieving of atmospheric, oceanic, and land properties had used these profiles extensively (Lin and Rossow, 1997; Lin et al., 1998a; Lin et al., 1998b; Lin and Minnis, 2000; Li et al., 2009; Min et al., 2010). In this study, four specific climatological profiles—the tropics, midlatitude summer, midlatitude winter, and US standard atmosphere of McClatchey et al. (1972)—were used. The tropical and midlatitude winter profiles with surface temperatures about 300 K and 272 K, respectively, may represent the two extreme weather conditions of warm and cold over open oceans. The midlatitude summer profile is commonly used during summer field campaigns, and the US standard atmosphere is more or less a global mean. Generally, these profiles are used in the assessment of retrieval impacts caused by differential absorption residuals of environmental variables other than O₂.

The retrieval of partial column dry air pressure or P _dp in Eq. 11 using DAR systems was based on the direct comparison of the O₂ DAOD measurement Δτ _{O3C_measured} with the modeled O₂ DAOD value Δτ _{O3C_modelled} obtained from the meteorological profile of simulation models like GEOS-5: P d p = P d p _ m o d e l Δ τ O 3 C _ m e a s u r e d / Δ τ O 3 C _ m o d e l l e d (17)

SLP, then, is retrieved from the combined dry and moist air pressures (see Eq. 11).

As mentioned in the previous section, there are small uncertainties in an instrument’s attitude. Since the attitude effect (or the cosine of radar viewing angle, μ, in Eq. 5) is directly proportional to DAOD, the relative errors of μ could cause the same amounts of relative errors in DAOD. The goal of controlling the attitude uncertainty would be to make it negligible for SLP retrieval. To analyze the impact of the attitude uncertainties on SLP or DAOD retrievals, Euler rotations (or, equivalently, Tait-Bryan rotations for engineering) are used in the adjustment of the slant viewing (or scanning) angle to nadir. This adjustment is to calculate the vertical component of the DAR viewing vector in the Earth coordinate, which can be obtained by an Euler angle transformation that considers the orientation of the instrument with respect to the inertial measurement unit (IMU) and the rotation order of the IMU. A typical case involves no installation offsets in attitude angles. For this condition, μ is equal to cosβcosγ, where γ and β are roll and pitch angles, respectively. This result shows that when the IMU uncertainty is the same for individual roll and pitch conditions, larger potential errors could occur for larger roll and pitch angles and the nadir view would have higher (about 2°) tolerance for attitude errors. Usually, the pitch angle is small for space platforms and cannot cause significant mechanical errors for SLP retrieval. However, a few degrees of pitch angle off the nadir could introduce a Doppler frequency shift of a few hundred KHz for spaceborne systems. Though the small shift in this case may not produce considerable retrieval errors (see spectral discussions next), changes and/or variations in pitch angle for space applications should be monitored well, and potential spectral errors introduced by Doppler shift should be accounted for when needed. The Doppler shift may not be an issue for airborne systems due to a much slower aircraft speed.

For aircraft such as the ER-2, the pitch angle is generally small (within a few degrees). The roll angle depends usually on the scanning, which can be 15° for the current DAR design. At a roll angle of 15°, potential errors in μ could be about 0.05% or 0.5 mb for 0.1° incidence angle uncertainty, which is significant. However, SLP retrieval errors introduced by currently designed instruments with an incidence uncertainty of 0.01° will be negligible. Current DAR systems may have sufficient attitude knowledge for scanning applications. For space applications, the attitude uncertainty may not produce significant retrieval errors either. Anyway, attitude angle uncertainties need to be considered in system mechanical design for precise DAR observations.

For precise SLP retrieval, spectral characteristics within individual channels need to be accurately accounted for in MWRT model calculations. To achieve the required number of independent samples of surface NRCS, multiple tones are transmitted at each channel using frequency diversity (McLinden et al., 2021). While the overall bandwidth of each channel is only 0.1 GHz and very narrow compared to its center frequency, there is potentially considerable non-linear variation in O₂ absorptions within the bandwidth, particularly for Ch.1 (65.5 GHz channel) as seen in Figure 1. Calculation of column absorption optical depths of O₂ and total gases using only center frequencies for the climatological profiles of tropics, midlatitude summer, midlatitude winter, and US standard atmosphere of McClatchey et al. (1972) results in errors that must be accounted for. This is particularly the case for Ch.1 where the error of assuming only the center frequency is close to 0.1%, compared to those calculated from entire spectra within the bandwidths. The DAOD errors for the 3-channel approach could be even greater than 0.1%, which clearly demonstrates that center frequency only is not sufficient in O₂ and total gas MWRT calculations. Tests of radiative transfer calculation for the four atmospheric profiles suggest that five frequency points with each one at the quarter position of spectral bandwidth would provide good results and reduce the calculation error to a negligible level for individual channels, channel pairs, and the 3-channel approach. For other atmospheric agents such as water vapor, cloud water, and light rain absorption and attenuation, calculations using only the center frequency give very good results with errors of less than 0.01% due to the quasi-linearity of absorption with frequency in the selected narrow band for these agents. Hereafter, this study uses the five spectral point approach in MWRT calculations for O₂ and total gases except for only center frequency for other atmospheric agents.

Another spectral characteristic that needs to be accounted for is spectral stability. During typical radar system operation, the potential drifting of channel frequencies is one of the key considerations in instrument design. The drifting issue is particularly important for DAR systems where differential absorptions of atmospheric agents such as O₂ depend on the channel frequencies actually used. Accurate knowledge of the channel frequencies is required.

As discussed previously, the sensitivity of O₂ absorption OD values on frequency variations varies largely at the considered DAR V-band, partially around O₂ absorption line centers. Figure 3 plots the relative OD change per MHz frequency drift for radar signals that have a 100 MHz bandwidth as a considered system (see Table 1). The solid and dashed curves are for climatological profiles of midlatitude winter and tropics, respectively. The stars show the channels of the current DAR instrument. For current DAR channels, OD changes are non-negligible for 1 MHz spectral drift. The potential error of channel 1 is the biggest and could reach about 0.09%. To limit the retrieval error caused by spectral uncertainties, the current DAR design targets a frequency stability of less than 0.1 MHz for short-term variations and 0.01 MHz for long-term drifting with temperature control. This means that the DAR spectral stability will be at least one magnitude smaller than those for 1 MHz frequency changes shown in the figure, which will satisfy SLP retrieval objectives in spectral characterization.

Atmospheric water, including water vapor, cloud liquid water, and precipitation-sized hydrometeors of light rain, is among the most important environmental factors that potentially affect O₂ differential absorption measurements for SLP retrieval.

The selection of DAR channels purposely in a narrow spectral band with evenly spaced frequencies may have great potential to remove the impacts of atmospheric water on O₂ DAOD due to the quasi-linear variability of its absorption and extinction with frequency as qualitatively explained in previous sections (Figures 1, 2 and their related discussions). A quantitative study is provided here. Since the extinction and/or attenuation of light rain at the selected DAR wavelengths is dominantly decided by absorption, the effect of light rain on O₂ DAOD can be estimated from the absorption of rainwater as that of cloud water (Eq. 9b and its related discussions). So, cloud water and rainwater are combined, and their absorption impacts are assessed with cloud liquid water. It is emphasized that the actual effect of clouds and light rain is not limited to their DAOD residuals on O₂ DAOD measurements but also DAR signal attenuations and atmospheric moisture. For example, 1 mm/h light rain with 2 km thickness would have an OD of about 0.4, which would produce about 3.5 dB extra attenuation in DAR power returns on top of regular O₂ and other atmospheric attenuation. This could reduce the SNR of the returns and significantly affect the accuracy of SLP retrieval. Thus, for DAR SLP retrieval products, 1 mm/h or higher rain rates should be removed and the retrievals for other light rain rates should be flagged. Similarly, cases with a sea surface wind speed higher than 15 m/s are not considered. The calculations in this study are mainly for the assessment of the relative effects of DAOD values of water vapor and cloud liquid water on O₂ DAOD measurements for SLP retrievals. Note that for rainwater mass impact on pressure retrieval, this rain rate threshold also limits the pressure error considerably due to small rainwater mass (ref., discussions on Eq. 11; Figure 2).

Figure 4 shows the relative errors of O₂ DAOD caused by DAOD residuals of (a) atmospheric water vapor and (b) cloud liquid water in the DAR DAOD estimates (see Eqs 5, 14). The water vapor impact is calculated for the four climatological profiles mentioned previously (Figure 4A). As expected, tropical weather conditions would cause larger O₂ DOAD errors due to high vapor content compared to colder winter weather situations. For channels 1 and 2, water vapor DAOD could introduce about −0.5% residual compared to O₂ DAOD in a differential absorption estimation (corresponding to approximately −5 mb). The negative sign means that the vapor DAOD decreases with decreasing frequencies while that for O₂ is the opposite (see Figure 1). For other climatological conditions, the relative errors could be still considerable (about a few 10ths of a percent). Thus, when this pair of two channels is directly used to retrieve SLP, the water vapor DAOD has to be compensated by modeling its value from either measured or simulated meteorological profiles with humidity observations. Channels 2 and 3 have very large relative errors because the O₂ DAOD of the pair is small. The reason for plotting the results of this pair is to demonstrate that this pair (or channel 3) is not used for retrieving O₂ amounts directly but is instead used for the removal of environmental contributions including water vapor, cloud water, rain, sea surface, etc. on O₂ DAOD. The results of the 3-channel approach exactly exhibit this effect. Errors of all four climatological profiles from the 3-channel approach are reduced to within a few ten thousandths—about 0.023% and 0.011% for the tropical and winter profiles, respectively, which is below the preferable level (0.05%). Besides the DAOD impact, errors in column water vapor are also an error source in moisture air pressure estimation and then, SLP retrieval (ref. Eq. 11). However, the uncertainties in water vapor amount over oceans are small and cannot produce significant SLP retrieval error as mentioned previously.

The impact of cloud liquid water DAOD on O₂ DAOD (Figure 4B) is estimated for a cloud with 0.2 kg/m² LWP, which is very thick for non-precipitating clouds. Different cloud temperatures in the tropical (black curves) and midlatitude winter (magenta curves) profiles are considered because of certain variations of cloud water absorption with temperature at studied wavelengths. The channel 1 and 2 pair (the two very close dashed curves) could produce about −0.2% uncertainties in O₂ DAOD estimates if the cloud water impact on DAR return attenuation was not removed, which is significant for SLP retrieval. For similar reasons to water vapor, cloud liquid water DAOD for the channel 2 and 3 pair (dashdot curves) could cause large relative uncertainties in O₂ DAOD estimation. However, this cloud DAOD residual is used directly to cancel the cloud water DAOD of the channel 1 and 2 pair in the 3-channel approach (solid lines). Thus, the impact of cloud DAOD for the approach is negligible, resulting in an overlap of the relative errors of the two climatological profiles plotted in the figure. Although cloud DAOD may not have a substantial influence on SLP retrieval, the attenuation of cloud water and rainwater could reduce radar SNR significantly (ref., Figure 2). For example, with 0.5 kg/m² cloud LWP, DAR returns would experience an extra more than 2 dB attenuation compared to clear or regular cloudy conditions. Furthermore, at this LWP, light rain is likely (Lin and Rossow, 1994), which would introduce even higher attenuation as discussed previously. Thus, the light rain screen and flag of high cloud LWP are required during retrieval data processing.

Atmospheric temperature is crucial priori information needed for forward MWRT calculations similar to estimating modeled O₂ optical depth for retrievals. Errors in temperature profiles could potentially produce non-negligible errors in retrievals due to the dependence of the O₂ absorption cross-section on temperature. This is one of the key reasons that temperature-sounding capability has been implemented in the current DAR instrument design and would be beneficial for a spaceborne instrument (ref. Table 1). This study focuses on the impact of temperature bias on O₂ DAOD calculations. The influence of random errors in temperature profiles could be reduced by vertical and horizontal averages of both the temperature and the calculated O₂ absorption cross-section profiles.

Figure 5 illustrates the temperature impact on O₂ DAOD estimates, including (a) normalized weighting function, (b) relative sensitivity of the weighting function for 1K temperature increases in vertical profiles, and (c) O₂ DAOD changes on temperature errors. The first two panels provide basic characteristics of O₂ absorption with meteorological conditions, and the last one gives a quantitative estimate of potential O₂ DAOD errors introduced by temperature bias. Two extreme climatological conditions of tropical and midlatitude winter profiles are analyzed. As expected, the lower atmosphere weighs significantly more in O₂ differential absorption compared to the upper atmosphere, providing a strong ability in surface air pressure retrieval (Figure 5A). The weighting functions of the high-frequency channel pair (dashdot curves) more or less linearly increase with pressure, while those of the low-frequency channel pair (dashed curves) show a certain non-linear effect. The lowest frequency channel is within the wings of strong O₂ absorption lines (ref. Figure 1), and its absorption cross-section could be affected by line broadening depending on atmospheric states. This non-linear feature of the low-frequency channel pair propagates to the 3-channel approach (solid curves). Although the differential absorption cross-sections of this approach are strongly affected by those of low-frequency channel pair, they are the combined results of the two pairs and, thus, have the largest deviation from linearity among the three. This also yields the highest weighting function sensitivity of the 3-channel approach on temperature.

Figure 5B shows the relative weighting function change or the sensitivity of the 3-channel approach (solid curves with black for tropics and blue for midlatitude winter), along with those from the two pairs, on 1 K temperature bias. The weighting function of this approach generally has a consistent temperature influence for various meteorological conditions as seen from the closeness of tropical (black curves) and winter (blue curves) results, particularly within the middle atmosphere. This is basically inherited from that of the channel 1 and 2 pair (dashed curves). The separation of tropical and winter profiles in the upper atmosphere is caused by the large relative change in the channel 2 and 3 pair (dashdot curves). For upper layers, the weighting function of this approach could change by as much as close to 0.5%, and at lower layers, a change of −0.2% would be possible. The opposite sign in the sensitivity of these weighting functions on potential temperature bias errors creates certain error cancellations on the vertically integrated weights and DAOD values. This character can also be seen in the other two pairs, except the direction of the change for the high-frequency channel pair is switched. This error cancellation benefits differential absorption measurements of O₂ DAR. The switched direction on the temperature sensitivity of the two pairs makes the 3-channel approach have slightly higher temperature sensitivity than the channel 1 and 2 pair, though the differences are small, as mentioned previously, due to the very small differential absorption of the pair of channels 2 and 3.

Figure 5C shows the relative errors of O₂ DAOD caused by temperature bias errors for the two pairs and the 3-channel approach. The maximum temperature bias considered is 4 K, which is very large for a marine environment. SST errors are generally very small. For infrared and microwave observations, SST uncertainty is within 0.5 K (Emery et al., 2001; Chelton and Wentz, 2005; Gentemann et al., 2010). With certain averages, the error could be even smaller. These SST observations are key inputs for numerical weather prediction and climate models that well control SST error within 0.3 K (Minnett et al., 2019). Thus, atmospheric temperature profiles simulated from models such as GEOS-5 over open oceans would not have large errors as they are anchored to accurate SSTs. The relative errors for the 3-channel approach (solid lines) for different climatological profiles are very close and overlap each other. For 4 K temperature error, the maximum DAOD errors would be within about 0.035%, which is within the preferable level. Furthermore, the synergy of assimilated model meteorological profiles with DAR temperature-sounding results will further mitigate the impact of temperature errors on SLP retrieval. This analysis indicates that the impact of temperature uncertainty can be controlled within a negligible level. To illustrate the advantage of the 3-channel approach, the relative DAOD errors introduced by temperature bias for the DAOD values of the two pairs are also plotted in the figure. The relative errors of the channel 2 and 3 pair (dashdot curves) could be large (about absolute 2.0%) because of their small DAOD values, while those of the channel 1 and 2 pair (dashed curves) are significantly larger than for the 3-channel approach, though considerably smaller than those of the channel 2 and 3 pair, because of a similar weighting function but less cancellation. Winter cases have larger temperature sensitivity than tropical cases due to the relatively larger temperature errors for the same temperature biases under low winter temperatures. These errors can be reduced by the temperature synergy as well, strongly supporting the temperature-sounding capability of DAR systems.