Fog forms when moist air near the surface becomes saturated and water vapor condenses into droplets, often reducing horizontal visibility (Vis) to below 1 km. It is prevalent in valleys and in mountainous, urban, and coastal regions, where radiative cooling, moisture advection, evaporation, or orographic lifting can cause near-surface air to reach saturation. Fog poses substantial risks to port operations, maritime traffic safety, and regional economic activities . As the core parameter for quantifying fog intensity and guiding operational decisions, Vis is defined as the maximum horizontal distance at which an observer with normal eyesight can distinguish a target against its background under current weather conditions . Despite its operational importance, accurate measurement of Vis and characterization of fog structure remain challenging because of limitations in existing observational techniques.

Various instruments have been used to conduct fog observations. Ground-based sensors such as optical visibility meters and fog droplet spectrometers provide high-precision point measurements but fail to capture the horizontal or vertical distribution of fog . Laser-based active remote-sensing instruments (e.g., lidar, ceilometers) offer vertical profiling capabilities; however, their signals suffer severe attenuation in dense fog, often leading to signal saturation or complete loss . Satellite passive imagers enable large-scale fog detection but struggle to distinguish fog from low-level clouds and lack vertical resolution to resolve internal stratification or microphysical structures . These limitations hinder reliable short-term fog nowcasting and proactive risk mitigation in transportation and maritime operations . In recent years, millimeter-wave cloud radar (MMCR) has gained prominence in fog monitoring due to its strong penetration, fine spatiotemporal resolution, and sensitivity to small hydrometeors. Field campaigns such as the 2021 Pingtan Sea Fog Experiment in China's Fujian Province have shown that Ka-band radar maintains robust signal integrity under dense fog conditions . utilized dual-frequency radar to observe multi-layer fog, and by assimilating these radar data into the Applications de la Recherche à l'Opérationnel à Méso-Echelle (AROME) model, they reduced the fog-top height error by 64 % and largely improved visibility forecasts. In the absence of large hydrometeors, MMCR has gained prominence in fog monitoring for its strong penetration, fine resolution, and sensitivity to small hydrometeors . The development of millimeter-wave radar has thus provided a new means for fog detection.

Previous studies have advanced the quantification of fog visibility. Early work proposed an inverse relationship between Vis and LWC through bulk microphysical analysis. Subsequent studies showed that DSD strongly modulates extinction and visibility . incorporated droplet number concentration into a multiparameter scheme, improving Vis estimates in numerical weather prediction models by up to 50 %. further incorporated effective droplet diameter. These studies indicate that fog microphysical properties can improve the physical consistency of visibility estimation . Nevertheless, most radar-based Vis parameterizations remain empirical and are commonly obtained by directly fitting Vis to Z . Although introduced radar attenuation correction, the resulting retrieval still relies mainly on empirical relations. Approaches that account for microphysical effects often require auxiliary attenuation or microphysical measurements, restricting operational flexibility where supporting instruments are unavailable.

In this study, we develop a physically based, observation-constrained visibility retrieval algorithm for single-wavelength Ka-band MMCR. Radar transmission theory and fog microphysics provide moment-based links among reflectivity (Z), liquid water content (LWC), effective radius (re), and extinction coefficient (βext); however, these physical relations alone cannot resolve the ill-posedness inherent in single-wavelength radar measurements. We therefore use five years of FM-120 droplet size distributions as a priori microphysical constraints to inform the retrieval. Specifically, the observed microphysical ranges are used to (i) constrain the initialization and bounds of the mass-absorption-based LWC inversion and (ii) establish an extinction-oriented power-law relationship between the geometric median radius (rm) and linear reflectivity (Zlin). Section 2 describes the instrumentation and FM-120 processing. Section 3 presents the theoretical framework, the observed DSD regimes, and the two-step retrieval. Section 4 evaluates retrieval uncertainties associated with DSD shape, the constant-Qext approximation, and perturbations in LWC and radius. Section 5 demonstrates the algorithm through a 2025 sea-fog case study and a ten-event consistency comparison. Section 6 summarizes the findings and limitations.

Dedicated monitoring campaigns have been conducted at DOB since 2019. The YLUKA2 solid-state Ka-band dual-polarization cloud radar provides reflectivity for the visibility retrieval. A DNQ3 visibility meter provides the near-surface point record for the ten-event consistency comparison, and an FM-120 spectrometer provides DSD and microphysical measurements for algorithm development and sensitivity analysis .

The YLUKA2 MMCR is installed at the highest point of DOB, with an antenna height of approximately 25 m. It operates at 34.50–35.50 GHz with an adjustable center frequency and integrates solid-state transmission, dual-transmitter/dual-receiver linear polarization, and fully coherent pulse-Doppler technology. Manufacturer specifications list 30 m spatial resolution, a 30 km maximum detection range, a 42 dB ground-clutter suppression ratio, and a minimum detectable reflectivity factor of -30 dBZ at 10 km. The YLUKA2 MMCR performs sector Plan Position Indicator (PPI) scans at a fixed elevation angle of 2°, conducting continuous azimuthal surveillance between 60 and 300°. The YLUKA2 MMCR data used in this study are primarily affected by clear-sky ground clutter. As this method places high demands on data continuity, we recorded the ground clutter under clear-sky conditions and subsequently subtracted this clutter contribution to complete the ground clutter removal .

The FM-120 operated continuously from 2019 through 2023. It measures droplets from 2–50 µm at 1 Hz in 30 non-equidistant size bins. The DSD observations are used to derive fog microphysical parameters, including βext, LWC, and rm, for algorithm development and sensitivity analysis.

The DNQ3 visibility meter operates on the forward-scattering principle. It emits infrared pulses with a central wavelength of 0.87 µm, and the sensor receives light scattered at a forward angle. The instrument is approximately 153 m from the radar along azimuth 150°. Because this position is near the radar's minimum range, a nearby radar point whose temporal variation is most consistent with the DNQ3 trend is selected for matching. The ten spring-2025 cases were included when DNQ3 visibility remained continuously below 1000 m for at least 30 min and no other precipitation weather occurred during the interval.

The DOB location and MMCR installation are shown in Fig. .

The classical Koschmieder law expresses Vis as 2Vis=-ln⁡εβext=-ln⁡0.02βext where ε is the brightness-contrast threshold, typically set to 0.02, and βext is the extinction coefficient (unit: m-1) at 550 nm. According to Mie scattering theory, 3βext=∫0∞Qext(r,λ)n(r)πr2dr where Qext is the extinction efficiency for a droplet of radius r, and n(r) is the fog DSD (unit: m-4), which is commonly represented by a lognormal distribution : 4n(r)=NT2πrσexp⁡-ln⁡2(r/rm)2σ2 Here, NT is total droplet number concentration, rm is geometric median radius, and σ is logarithmic spectral width. For a lognormal distribution, the kth DSD moment can be written as 5Mk=NTrmkexp⁡k2σ22

MMCR operates in the millimetre-wave band, where most fog droplets are in the Rayleigh regime. The attenuation-corrected radar reflectivity factor Z is proportional to the sixth DSD moment: 6Z=64M6=64NTrm6exp⁡18σ2 Because Z and βext depend on different DSD moments, they cannot be linked uniquely without intermediate microphysical information. LWC and effective radius are defined by 7LWC=43πρwM3=43πρwNTrm3exp⁡92σ28re=M3M2=rmexp⁡52σ2 where ρw is water density. Eliminating NT and σ from the moment relations gives 9re=rm49πρwZ48LWC527 In the visible band, the extinction efficiency of sufficiently large fog droplets approaches 2 . Figure  shows its radius dependence at 550 nm.

For Qext≈2, extinction is 3LWC/(2ρwre). Substitution of Eq. () gives 10βext=3LWC2ρwre=32ρwrm-49πρwZ48-527LWC3227. All quantities are evaluated in consistent SI units, including ρw, LWC, radius, and the conventional mm6m-3 reflectivity factor. Combining Eq. () with Koschmieder's relation yields 11Vis=3.912ρw3rm49πρwZ48527LWC-3227. Equation () requires both LWC and rm, neither of which is uniquely determined by one MMCR reflectivity measurement. We therefore use two observation-constrained steps. The mass-absorption/TRR routine retrieves LWC and applies liquid-water attenuation correction, and the FM-120 development data provide the conditional radius–reflectivity relations.

The retrieval framework constructed in Sect.  reduces reliance on direct visibility–reflectivity fitting, but its accuracy is ultimately governed by two algorithmic assumptions. The first is the constant extinction-efficiency approximation (Qext=2), which replaces the size-dependent Mie efficiency shown in Fig.  with its large-particle asymptote. The second is the single-lognormal closure, which constrains the droplet size distribution to a parametric form that links reflectivity, liquid water content, and effective radius through shared moments. Both assumptions introduce structural biases whose magnitude depends on droplet size and spectral modality. Section 4.1 attributes the extinction-residual patterns revealed in Fig.  to these two sources: the weak-reflectivity bias is traced to the Qext=2 approximation, while the broader residual reflects the lognormal closure and is partially mitigated by the extinction-oriented parameterization. Section 4.2 quantifies the analytical sensitivity of the retrieved extinction to perturbations in LWC and radius.

Figures  and illustrate the retrieved temporal and spatial evolution of fog microphysical parameters and visibility during a criterion-selected case on 2 March 2025. Influenced by a southern-branch trough, warm and humid southerly airflows advected over the observation site, producing a typical advection sea fog. The fog developed rapidly after 18:00 local standard time (LST) and reached maximum intensity around 21:00 LST. At the radar elevation angle of 2°, fog returns were detectable to a maximum range of approximately 10 km, and the fog-top height exceeded 350 m. Between 18:00 and 24:00 LST, the radar reflectivity factor Z generally decreased with increasing range, indicating a stable fog structure whose density decreased with altitude. During 20:00 to 22:00 LST, numerous range gates with Z exceeding -10 dBZ were observed, with peak values approaching 0 dBZ. As shown in Fig. , the polar scan at 20:29 LST reveals that high-Z regions were concentrated to the south and southwest of the observation site, where topographical undulations and man-made structures are distributed. The uneven underlying surface generates turbulence through surface friction, which promotes droplet coalescence and the formation of larger droplets, thereby enhancing Z.

To assess retrieval performance, the present method was cross-validated against DNQ3 forward-scattering visibility-meter observations and three direct empirical visibility–reflectivity relations (Table ). Near-range radar data are susceptible to ground-clutter contamination, and the minimum detection range is approximately 150 m; the comparison point was therefore selected at 300 m along the 150° azimuth, where the temporal variation correlated most closely with DNQ3. A systematic height difference exists between the DNQ3 sensor at 1.5 m a.g.l. and the radar detection altitude of approximately 30 m at this range, arising from the 25 m antenna height and the beam elevation geometry. Notably, at 21:30 and 22:30 LST, DNQ3 visibility increased markedly while the radar retrievals did not show a corresponding rise. The intense radar echo around 21:00 LST indicates vigorous fog processes that induced droplet sedimentation; the resulting larger particles reduced the total droplet number concentration and raised near-surface visibility as measured by DNQ3 at 1.5 m, while the radar, sampling at a higher altitude, was less affected. Furthermore, the present retrieval is based on backscatter whereas DNQ3 exploits forward scattering, so inherent principle-related differences also contribute to the observed discrepancies. Comparison of FM-120-derived Mie-scattering visibility against DNQ3 shows the same tendency: the retrieval is lower than DNQ3 during the early-to-mature fog stage and converges during dissipation. It should also be noted that the FM-120 fog spectrometer has a limited observation range for large droplets, so variations in the median radius cannot be fully captured in the fitting process when large fog droplets are present, leading to larger biases in the fitted median-radius values.

The local and Liu et al. (2019) prefactors are 0.02015 and 0.040 km, respectively, for the kilometre axes in Figs. e and . The local relation was derived by weighted nonlinear least-squares fitting of 5 min aggregated visibility–reflectivity pairs from FM-120 DSD observations, with elevated weight applied to the low-reflectivity regime where visibility sensitivity is highest (Table ). Because the radar sampling volume and the DNQ3 point measurement differ in geometry and representativeness, and DNQ3 data also conditioned the event selection and range-gate matching, the comparison in Fig.  assesses local consistency and reasonable range rather than ranking methods.

Taken together, Figs. – present the algorithm outputs used only for a DNQ3-conditioned reasonable-range and local-consistency comparison in the criterion-selected coastal low-visibility cases. Owing to systematic differences between the radar sampling volume and the DNQ3 point measurement, as well as the conditioning of event selection and range-gate matching on DNQ3 data, this evaluation reflects local consistency under conditioned samples rather than independent retrieval validation. A stricter quantitative evaluation requires more completely collocated observations over additional fog regimes and explicit characterization of inter-instrument systematic differences.

This study presents a physically constrained, observation-calibrated visibility retrieval algorithm for single-wavelength Ka-band millimeter-wave cloud radar. DSD moment analysis confirms that single-wavelength reflectivity cannot uniquely determine both droplet number concentration and geometric median radius, motivating the introduction of two intermediate microphysical constraints. The first is a segment-wise mass-absorption LWC power law that operates on contiguous 4–10-gate segments with reflectivity-dependent starting vectors, a strong-branch retry, and branch-specific attenuation accumulation. The second is an extinction-oriented radius–reflectivity parameterization calibrated against same-source FM-120 DSD observations, enabling closed-form calculation of the extinction coefficient and hence visibility. Compared with purely empirical visibility–reflectivity relations, the proposed method provides clearer physical underpinnings and better internal self-consistency. The main findings are summarized as follows.

During algorithm development, the dominant error sources are identified as the constant extinction efficiency (Qext=2) approximation and the single-lognormal DSD assumption. Multi-year FM-120 observations reveal high morphological diversity in coastal fog spectra: broad-unimodal spectra dominate, whereas bimodal spectra become increasingly prevalent below 200 m visibility. Both LWC and Z exhibit strong discriminability across fog-intensity regimes, and this contrast is exploited to optimize the mass-absorption retrieval framework following .

The single-lognormal closure introduces inherent uncertainty into the radius–reflectivity relation. The extinction-oriented parameterization mitigates systematic extinction retrieval errors without additional observational inputs, with more pronounced improvement in dense fog characterized by relatively high reflectivity, achieving a mean absolute fractional extinction error of 9.37 % on the development dataset. Both intermediate quantities (rm and LWC) carry retrieval uncertainties, and LWC perturbations exert a stronger influence on the final visibility estimate than median-radius perturbations. Within the perturbation range of 0.50–2.00 times the baseline values, the relative change in extinction spans from -67.7 % to +209.4 %. Nevertheless, the covariation of rm and LWC can partly compensate for individual errors, as reflected in the analytical sensitivity response.

Evaluation on ten criterion-selected spring low-visibility fog events shows that the near-surface visibility retrieved by the proposed method achieves the lowest error statistics among the four compared relations, with a mean absolute error (MAE) of 0.092 km and a relative error (RE) of 42.86 %, followed by the local site-specific FM-120 relation (MAE = 0.123 km, RE = 50.53 %) and Liu et al. (2019) (MAE = 0.160 km, RE = 85.30 %), whereas the Gultepe et al. (2009) relation departs most strongly (MAE = 0.533 km, RE = 310.63 %). The present retrieval, the local FM-120 relation, and Liu et al. (2019) yield broadly comparable visibility ranges, consistent with their shared origin in Fujian Province coastal-fog observations, while the larger departure of Gultepe et al. (2009) reflects regional and latitudinal differences in fog microphysics. However, owing to the spatial mismatch between the radar sampling volume and the ground point measurement, as well as the conditioning of event selection and range-gate matching on DNQ3 data, this evaluation reflects local consistency under conditioned samples rather than independent retrieval validation.

Several limitations of this study must be acknowledged. First, independent in situ LWC measurements and standardized hydrometeor phase data are unavailable, so the retrieved microphysical parameters cannot be validated against reference truth. Second, near-range retrieval results may still be affected by residual ground clutter, as the full quantitative performance of the clutter-filtering algorithm cannot be completely verified. Third, the study samples are limited to spring warm fog at a single southeast coastal site, so the generalizability of the algorithm to other fog types (e.g., radiation fog, orographic fog) and other climate regions remains to be tested.

Future work will pursue three directions. First, dual-wavelength cloud radar observations will be incorporated to further constrain LWC retrieval using differential attenuation characteristics, reducing the non-uniqueness of microphysical parameter estimation. Second, a multi-source synergistic visibility retrieval scheme will be developed by combining lidar, microwave radiometer, and fog-spectrometer observations to add phase identification, vertical structure, and microphysical constraints. Third, observational samples from diverse climate zones and fog types will be accumulated to improve parameterization robustness and generalizability, and to carry out targeted error evaluation for transportation and civil aviation operational applications.