To model ice nucleation through AgI particles, we follow the parameterization described in Xue et al. (2013a). This parameterization was originally developed based on cloud chamber experiments from DeMott (1995) and Meyers et al. (1995). Four ice nucleation modes are considered. For deposition nucleation, which is valid when the saturation ratio with respect to ice is greater than 1.04 and the temperature is colder than 268.2 K, the fraction of seeded AgI that can nucleate ice is expressed as follows: 1Fdep=aSi-1+b273.16-TT0+c(Si-1)2+d273.16-TT02+e(Si-1)3, where T0=10 K, a=-3.25×10-3, b=5.39×10-5, c=4.35×10-2, d=1.55×10-4, and e=-0.07. Si is the saturation ratio with respect to ice.

The fraction of AgI that can nucleate ice through condensation-freezing nucleation, which applies when temperatures are colder than 268.66 K, can be calculated using 2Fcdf=a268.66-TT03(Sw-1)2, where a=900. Sw is the saturation ratio with respect to water.

The third mode is contact freezing, which is minor compared to the other nucleation mechanisms. This mode is given by 3Fctf=Fscava+bSi-1+cSi-12+dSi-13+eSi-14+fSi-15+g(Si-1)6, where a=0.0878, b=23.7947, c=52.3167, d=2255.4484, e=568.3257, f=2460.4234, and g=263.1248. Fscav is the fraction of the total number of AgI particles scavenged by cloud droplets (Caro et al., 2004). Contact freezing is valid when Si>1.058 and T<269.2 K.

And the last mode is immersion freezing, which is valid for temperatures colder than 268.2 K. This mode is expressed as 4Fimf=aFimm268.2-TT0b, where a=0.0274 and b=3.3. Fimm is the fraction of AgI particles that are immersed in cloud droplets but not nucleated. Based on the equations above, we can calculate the concentration of ice generated through cloud seeding at a given AgI particle concentration. For simplicity, this 1D model does not incorporate the 3D turbulent dispersion of AgI particles. In each numerical experiment, seeding is performed at a given temperature (height), and seeding takes place only at the beginning of each run. The total AgI particle concentration decreases with each time step (1 s) due to continuous ice nucleation. The nucleated ice crystals that form at the seeding level are either columnar or plate-like, depending on the background temperature, and they remain that way as they descend to the surface.

In mixed-phase stratiform clouds with relatively weak turbulence, the WBF process, during which ice grows at the expense of liquid water, is the most important mechanism for ice growth. Therefore, if there is sufficient liquid water in the cloud, diffusional growth contributes the most to ice mass and size. However, it should be noted that the WBF process is not always valid; if the updraft is strong enough, the cloud will become supersaturated with respect to water, leading to the simultaneous growth of droplets and ice crystals. Korolev (2007) showed that the WBF process only applies in weak downdrafts and weak updrafts; therefore, we emphasize that the model used here only applies to stratiform clouds with weak turbulence. In fact, even weak turbulence may occasionally result in supersaturation with respect to water, and radiative cooling at the cloud top reduces stability, resulting in weak vertical motions at the cloud top that may also enhance supersaturation. These factors are not considered and could, in part, contribute to the model uncertainties in the results shown in Sect. 3. In addition to the WBF process, we also consider the riming process, but the model does not incorporate aggregation or secondary-ice-production (SIP) mechanisms, such as the rime-splintering process and the shattering of freezing drops, which may alter ice particle size distributions and radar reflectivity (Yang et al., 2024b). By not including these processes, the model may underestimate reflectivity in conditions where aggregation is a dominant growth mechanism or in clouds with either abundant supercooled water or high ice crystal concentrations as a result of SIP.

According to Mason (1953), the diffusional-growth rate of an ice crystal is given by 5dmdt=4πCSi-1fvLd2/KRvTa2+1/Dvρs, where Rv is the specific gas constant for vapor, Ld is the latent heat of deposition, K is the thermal conductivity of air, Ta is the ambient temperature, Dv is the diffusion coefficient of vapor, ρs is the density of saturated vapor, and Si is the saturation ratio with respect to ice (which depends on the liquid phase and temperature). Moreover, fv is the ventilation factor, expressed as 6fv=1+0.14X2,X<10.86+0.28X,X≥1, where X=Sc1/3Re1/2, with Sc representing the Schmidt number and Re representing the Reynolds number (Hall and Pruppacher, 1976).

In Eq. (5), the capacitance (C) reveals the shape of the ice crystals, which is dependent on temperature and supersaturation. Many laboratory experiments have investigated the shapes of ice crystals during diffusional growth (e.g., Fukuta and Takahashi, 1999; Bailey and Hallett, 2009). Typically, plate-like ice crystals (hexagonal plates, sectored plates, dendrites, etc.) form at temperatures warmer than -4 °C and at temperatures between -8 and -22 °C, whereas column-like ice crystals (columns, needles, etc.) form at temperatures between -4 and -8 °C (Fukuta and Takahashi, 1999). At temperatures colder than -22 °C, ice crystals could be either columns or plates (Bailey and Hallett, 2009). Based on laboratory experiments, Chen and Lamb (1994) developed a parameterization of C. For plate-like ice crystals,

7C=aεsin⁡-1ε,8ε=1-c2/a2=1-ϕ2, and for column-like ice crystals, 9C=cεln⁡1+εϕ,10ε=1-a2/c2=1-ϕ-2, where a and c are the particle radii along the a and c axes, respectively. For plate-like ice crystals, the a axis is longer than the c axis, while for column-like particles, the c axis is longer. Moreover, ϕ=c/a is the aspect ratio of the ice crystal. The change in the surface volume of the ice crystal is expressed as 11dV=1ρdepdm, where ρdep is the mass density at the time of deposition. Chen and Lamb (1994) showed that ρdep (g cm⁻³) can be expressed using 12ρdep=91exp⁡-3max⁡Δρ-0.05,0ΓT, where Δρ is the excess vapor density. Γ(T) is the inherent growth ratio of ice, which is parameterized based on data from multiple laboratory experiments (Harrington et al., 2019): 13ΓT=dcϕda. The initial ice particle is assumed to be spherical. The volume can be expressed as 14V=43πa2c=43πa3ϕ,15dln⁡V=3dln⁡a+dln⁡ϕ. The differential representation of ϕ is given by 16dϕ=1adc-ca2da=dcda-ϕdln⁡a=ϕΓ-1dln⁡a,17dϕϕ=dVVΓ-1Γ+2. Based on the equations above, we can estimate the changes in ϕ, a, and c over time as follows:

18ϕt+Δt=ϕtVt+dVVtΓ-1Γ+2,19at+Δt=34πVt+dVϕt+Δt1/3,20ct+Δt=ϕt+Δt×at+Δt. Although diffusional growth is the most important mechanism for ice growth in mixed-phase stratiform clouds, accretional growth (riming) is also considered in the model. According to Heymsfield (1982), the mass growth rate due to riming can be parameterized as 21dmRdt=AVtE(Dd)LWC, where A is the particle cross-sectional area normal to the fall; Vt is the terminal fall velocity of ice crystals; LWC refers to the liquid water content, estimated assuming adiabatic-cloud conditions; and E(Dd) is the collection efficiency between an ice crystal with a diameter of D and a droplet with a diameter of d. For simplicity, we assume a droplet diameter of 10 µm, which is a typical size for droplets in continental clouds (Wallace and Hobbs, 2006; Wang et al., 2021).

To account for the impact of turbulence on the growth of ice crystals, we use a Gaussian normal distribution with a zero-mean vertical air velocity and a standard deviation of 0.75 m s⁻¹. A similar method has been used in previous studies (e.g., Korolev and Field, 2008; Zhang et al., 2014). The vertical velocity of an ice crystal is the sum of the particle's terminal fall velocity and vertical air velocity. As the ice crystals fall from the seeding level, the temperature below the seeding level is calculated using a lapse rate of 5.5 K km⁻¹, and the pressure is calculated using the hydrostatic equation. Sensitivity tests using lapse rates of 5 and 6 K km⁻¹ reveal minor differences in the modeled results of the ice mass.

The terminal velocity of an ice crystal depends on its mass, size, and habits, as well as on the ambient air density. In the present study, we follow a method developed by Heymsfield and Westbrook (2010) to estimate the terminal velocity (Vt) of ice crystals, which has been evaluated against observations (Heymsfield and Westbrook, 2010; Zhang et al., 2014), as follows: 22Vt=ηReρairD, where η is the dynamical viscosity and D is the maximum size of the projection of the particle perpendicular to the direction of fall. According to Böhm (1992), D=2a for a plate, D=2ac for column, and D=2r for a sphere. Re is the Reynolds number, given by

23Re=δ0241+4X∗δ02C0, where C0=0.35 and δ0=8.0. 24X∗=ρairη28mgArk, where the value of k is 0.5, according to Heymsfield and Westbrook (2010), and Ar is the ratio of the projected area (A) of the particle to the area of the external circle, expressed as 25Ar=A/π/4D2. Following Harrington et al. (2013), for plate-like particles, 26A=πa2ρi/ρbi2, and for columnar particles, 27A=πac, where ρi is the density of the ice crystal and ρbi is the bulk density of solid ice (0.91 g cm⁻³). The ice shape (plate-like or column-like) is assumed to remain unchanged with height as the ice falls.

In this model, we assume that the initial ice size distribution follows the modified gamma distribution (Mace et al., 1998): 28Ni(D)=NXexp⁡(α)DDXexp⁡-DαDX, where DX=10 µm and α=2. Sensitivity tests using different values for these coefficients will be shown later (Fig. 3). Ni is the ice concentration per unit of length, while NX is the number concentration per unit of length at the functional maximum and is determined by the ice nucleation induced by AgI. This initial size distribution is partitioned into 80 bins with isometric radius intervals, and we loop through all the bins to simulate the growth of ice crystals with different initial sizes. Based on the modeled ice mass and size, we can calculate the IWC and Ze as follows: 29IWC=∫0∞m(D)Ni(D)dD,30Ze=Ni∫0∞f(Dm)Dm6dD, where Dm is the melted-ice diameter, f(Dm) is the normalized ice size distribution, and Ni is the ice concentration. In the present study, Ze is calculated assuming Rayleigh scattering; thus, the radar signal is more sensitive to the size of the particles. For radars with shorter wavelengths, such as the W-band cloud radar, the signal may be sensitive to ice concentration. In SNOWIE, both W-band and X-band radars were used. The W-band radar observed clear Ze enhancement near the seeding level, while the X-band radar detected weak Ze in the ice plume near the seeding level, which increased downwind toward the surface (Friedrich et al., 2021). We will show that the modeled results (Sect. 3) are consistent with the observed Ze profile using the X-band radar.

In turbulent clouds, ice crystals may disperse horizontally over time, resulting in a reduced ice concentration. In the 1D model, we can estimate the change in Ni(D) by solving the 1D dispersion equation as follows: 31∂Ni∂t=ν∂2Ni∂x2, where ν is the turbulent kinematic-diffusion coefficient and depends on the turbulent intensity. In mixed-phase stratiform clouds with relatively simple dynamics, ν is typically smaller than 10 m⁻² s⁻¹ (Pinsky et al., 2018). In clouds with strong wind shear or embedded convections, ν could be larger, and the ice growth trajectory could be much more complicated. Therefore, we emphasize again that the model described here only applies to stratiform clouds with relatively weak turbulence. In the present study, the value of ν is randomly selected between 1 and 10 m⁻² s⁻¹. In clouds with strong turbulence or embedded convection, the microphysics and dynamics are much more complicated (Yang et al., 2016a, b).

First, we evaluate the growth of a single ice crystal simulated using the 1D model. Figure 1 compares the modeled mass and size of a single ice crystal with the laboratory experiments from Takahashi et al. (1991). The laboratory experiments were conducted at various temperatures under standard atmospheric pressure. The initial ice crystal is assumed to be spherical and has a radius of approximately 4 µm. In the model simulation, we use the same thermodynamic conditions and initial ice size as in the laboratory experiments (Eq. 28 is used instead of Fig. 1). It can be seen in the figure that the modeled results are fairly consistent with the observations. The mass growth rate has two peaks, at -15 or -6 °C, and is at its highest at -15 °C. The ice crystal formed at -15 °C is plate-like, and the one formed at -6 °C is column-like (Eqs. 7–10 in Sect. 2.2). After 25 min, the mass of the ice crystal grown at -15 °C is about an order of magnitude greater than that grown at -20 °C. The sizes along different axes depend on the shape of the ice crystals. At -6 °C, the ice crystal is column-like, meaning the c axis is the largest, while at -15 °C, the a axis is the largest as the ice crystal is plate-like. The ice size can rapidly increase from 4 µm to more than 1 mm within 20 min at -15 and -6 °C. At temperatures lower than -20 °C, the ice growth habit is less sensitive to temperature. In general, the uncertainty in the modeled ice mass and size is less than 20 % compared to the observation. This gives us the confidence needed to use the model to estimate ice growth.

According to Fig. 1, it is expected that a detectable seeding signature is more probable at -15 °C or colder temperatures. To investigate the seeding signature at a given height, it is necessary to explore how ice concentration, IWC, and Ze change vertically as the ice crystals fall from the seeding level. Figure 2 shows the vertical profiles of ice concentration, IWC, and Ze for seeding temperatures of -15, -18, and -20 °C (at seeding heights of 5, 5.6, and 6 km altitude, respectively). The results are obtained using 10 numerical experiments for each seeding temperature. The model runs for 60 min in each experiment. The shaded area indicates the 20th–80th percentile range, and the solid lines represent the mean profiles. The uncertainties are due to the random numbers used to model the turbulence effect. Seeding at a relatively low temperature results in a higher ice concentration, and ice concentration decreases with height. For an AgI particle concentration of 35 cm⁻³, the ice concentrations are 10 times lower. Regardless of the AgI particle concentration, it can be seen that the vertical variations in IWC and Ze are consistent. Below 4.8 km altitude, IWC and Ze values are at their greatest when the seeding temperature is -15 °C and at their lowest when the seeding temperature is -20 °C. Although more ice can be nucleated through AgI at lower temperatures, its mass growth is much slower than that of ice initiated at -15 °C, leading to lower IWC and Ze values, which are more sensitive to size than concentration. The vertical variations in IWC and Ze are also related to the terminal fall velocity of ice crystals with different shapes. Ice crystals initiated at -20 °C are of a similar scale along the a axis and c axis; thus, the ice particles are more spherical compared to the plate-like ice crystals initiated at -15 °C. The plate-like ice crystals have a low terminal velocity, meaning they can remain at temperatures near -15 °C for a relatively long time, resulting in a substantial increase in ice mass and size. However, the ice particles initiated at -20 °C have a higher terminal velocity when they fall through the -15 °C level. Based on the results shown in Figs. 1 and 2, it is suggested that seeding at a temperature of -15 °C can provide a higher probability of detecting the seeding signature using radar. This conclusion is interestingly consistent with the fact that the seeding temperature was about -15 °C (or slightly warmer in non-precipitating clouds) in many of the cases with unambiguous seeding signatures detected (e.g., French et al., 2018; Yue et al., 2021; Wang et al., 2024), while at colder or warmer seeding temperatures in precipitating clouds, no clear seeding signature was observed.

In the model, we assume that there is sufficient supercooled liquid water and a continuous water supply. However, in real clouds, this is not always true, and there is often an upper limit to the LWC in clouds. This would certainly affect ice growth and the Ze profiles. We conducted several sensitivity tests with a seeding temperature of -15 °C, including different upper limits for the LWC (assuming no continuous liquid water supply; Fig. 3a), different time durations for ice growth (Fig. 3b), and different AgI particle concentrations for a limited LWC (Fig. 3c). It can be seen in the figure that for a model time of 90 min, Ze decreases with decreasing LWC (Fig. 3a). For a given LWC of 0.2 g m⁻³, ice nucleation and growth over a longer time would consume more liquid water, leading to lower Ze values (Fig. 3b), which would mean that ice formed later on would lack sufficient liquid water and vapor for growth. For a given LWC and temporal duration, a higher AgI concentration does not always produce larger Ze values (Fig. 3c); ice crystals may compete for the limited liquid water and suppress the ice crystal size. In addition, we conducted sensitivity tests with different turbulent-dispersion coefficients and different initial ice particle size distributions (Fig. 3d–f). These are also sources of uncertainty in the model, though the Ze profile is less sensitive to these factors compared to LWC. These results provide us with a better understanding of how modeled ice mass and Ze may vary due to different environmental conditions.

To further evaluate the 1D ice growth model, it is compared with a 3D large-eddy simulation (LES) of stratiform clouds observed on 1 January 2022 in northern China. The cloud was originally shallow, with a depth of approximately 600 m. The cloud top temperature was approximately -16 °C (Wang et al., 2024). This cloud was seeded at around -15 °C, and evident Ze enhancement was detected by radar after seeding (Wang et al., 2024). The cloud was almost entirely liquid before seeding, and the seeded area quickly glaciated within an hour. More details of the measurements can be found in Wang et al. (2024). In this study, we use the LES mode of the Weather Research and Forecasting (WRF) model, employing periodic boundaries to simulate the cloud. The solid lines in Fig. 4 show the initial profiles of temperature, potential temperature, the vapor mixing ratio, and wind velocity above the ground level. Adiabatic liquid water content (LWC) is assumed in the cloud, and the wind shear in the cloud layer (dark shaded area in Fig. 4) is relatively weak. To evaluate the performance of the 1D model in simulating deeper clouds, we modified the temperature and vapor-mixing-ratio data (dashed lines) to allow for the formation of a deep saturated layer. The modeled cloud has a base height of 1.3 km altitude and a top height of 2.9 km altitude (shaded area in Fig. 4). The initial LWC increases from 0 to 0.2 g m⁻³ from the cloud base to the top, and it rapidly decreases to 0 g m⁻³ above 2.9 km altitude.

In the experiment for the shallow cloud, the model has a horizontal resolution of 50 m, and the domain size is 10 km × 10 km × 4 km. In the deep-cloud experiment, the same horizontal resolution and size are used, but in the vertical dimension, we extend the depth to 5 km. In both experiments, 180 vertical levels are used to resolve the vertical structure of the cloud. A ±0.1 K temperature perturbation is added at the cloud base to initiate turbulence. A spin-up time of 30 min is used (Yang et al., 2024a), and the ice process is turned off during the spin-up time. Seeding is conducted for 30 min along a straight line (with a length of 10 km in the longitudinal direction) at 1.8 km altitude (-15 °C) for the shallow case and at 2.6 km altitude (-21 °C) for the deep case. We tested two different AgI particle concentrations (35 and 350 cm⁻³) in the simulations. The physics schemes used in the simulation include the revised MM5 surface layer scheme (Jiménez et al., 2012); the Noah land surface model (Tewari et al., 2004); the Rapid Radiative Transfer Model (Mlawer et al., 1997); and the Fast Spectral Bin Microphysics scheme (Khain et al., 2004), which can explicitly simulate more complicated microphysics than the 1D model. Cumulus and boundary layer parameterizations are turned off in the LES. The cloud condensation nuclei (CCN) concentration is expressed by NCCN=N0Swk, where N0 refers to the CCN concentration at a supersaturation level of 1 %, Sw represents the supersaturation with respect to water (expressed as a percentage), and k is the slope of the CCN size distribution. For continental China, Qu et al. (2017) suggested that N0 = 4000 and k = 0.9. The warm-rain process is not possible with such a high aerosol concentration and weak wind shear in the cloud since collision–coalescence cannot be initiated. Since the observation suggests that the cloud is entirely liquid before seeding, the natural ice nucleation process is turned off in the model. Thus, ice crystals are only generated due to cloud seeding. Similar to the 1D model, the AgI nucleation parameterization is from Xue et al. (2013a).

Figure 5a–d shows Ze values, cloud water, and ice mixing ratios corresponding to 30 min after seeding in the 3D LES model for the shallow case. Ice crystals, which are initially nucleated near the cloud top, can rapidly grow and fall out of the cloud within 30 min. With a lower AgI concentration (35 cm⁻³), the ice mixing ratio is lower than 0.06 g kg⁻¹ after 30 min, while at a higher AgI particle concentration (350 cm⁻³), the modeled ice mixing ratio exceeds 0.15 g kg⁻¹. Since the cloud is entirely liquid in areas unaffected by seeding, the seeding signature can be clearly identified once the ice plume forms, using either in situ measurements or remote sensing measurements (Wang et al., 2024; French et al., 2018). A higher ice mixing ratio implies a higher radar reflectivity. As shown in Fig. 5a, at a lower AgI particle concentration (35 cm⁻³), the maximum Ze value near the cloud base 30 min after seeding is approximately 10 dBZ, while at a higher AgI particle concentration (350 cm⁻³), Ze exceeds 20 dBZ. Ze is relatively small near the cloud top and increases from the cloud top to the base. Below the cloud base, Ze decreases due to ice sublimation. The magnitude of Ze in the core of seeding plumes (5–30 dBZ) is fairly consistent with observational studies in which cloud seeding was operated at -15 °C (Wang et al., 2021; French et al., 2018), indicating that the LES model can capture the characteristics of ice growth habits. The ice concentration in such a shallow cloud, with top temperatures as warm as -15 °C, is typically low (Zhang et al., 2014), resulting in weak Ze in the natural cloud. Ze attributed to cloud seeding (Fig. 5) is large enough to be detected by operational weather radars.

For the deeper case, as seen in Fig. 5e–f, the modeled Ze and IWC values are much lower than those for the shallow case because seeding was performed at a temperature of -21 °C. We present the results from 40 min after seeding because the ice crystals needed more time to reach the cloud base in this case than in the shallow case. In general, Ze and IWC values increase from the seeding level toward the cloud base, and the maximum Ze value is only -10 (1 dBZ) for an AgI particle concentration of 35 (350 cm⁻³). These low values are often smaller than those of the signals of natural precipitation. This conclusion is consistent with Fig. 2; therefore, both the 1D model and the 3D model suggest a high ice growth rate at -15 °C and a lower ice growth rate at colder temperatures.

Figure 6 compares the radar reflectivity from the 1D model simulation with that from the 3D model simulation. The left and right boundaries of each blue-shaded area indicate the 95th percentile and the maximum Ze value from the 3D model, respectively. As shown in the figure, the 1D model is generally consistent with the 3D model, and the oversimplified dynamics do not change the magnitude and vertical variation in Ze. Near the cloud base, an AgI concentration of 35 cm⁻³ leads to a maximum reflectivity of 11 dBZ (Fig. 6a), and Ze increases to 21 dBZ as the AgI concentration increases to 350 cm⁻³ (Fig. 6b). This magnitude is similar for both the 1D and 3D models. Therefore, the 1D model is valid for investigating the vertical variation in Ze below the seeding level. Although the 1D model is consistent with the 3D model in the cases presented here, it should be noted that uncertainties are inevitably present when modeling Ze in both models. Improving the ice nucleation and growth parameterizations in the models is vital for the purpose of this study. Previous observational studies have shown that the ice nucleation efficiency of AgI particles varies across different experiments (Marcolli et al., 2016), and recently, Ramelli et al. (2024), using in situ and remote sensing measurements, showed that ice growth rates exhibit large variations in seeded clouds, indicating that in real clouds, the dynamics and microphysics are complicated. Such observational datasets are useful for evaluating and improving ice growth models (Omanovic et al., 2024).

Since the 1D model can simulate the growth habit of ice crystals reasonably well, we conducted 2500 numerical experiments with a variety of seeding temperatures (-30 to -7 °C), seeding pressures (350–850 hPa), AgI particle concentrations at the seeding level (1–10⁶ cm⁻³), and depths from the seeding level (i.e., the vertical distances between the seeding levels and target levels, where we aim to detect unambiguous seeding signatures; 500–4000 m). We include very high AgI particle concentrations at the seeding level in the experiments because it is helpful to have a large parameter space to constrain the parameterization. In addition, in many observational studies, normal AgI particle concentrations (10–1000 cm⁻³) are insufficient for detecting the seeding signature, though AgI particle concentrations with a magnitude of 10⁵–10⁶ cm⁻³ are probably not possible in reality. Again, it should be noted that we assume there is sufficient supercooled liquid water in the cloud. Figure 7 shows modeled Ze values (colored) at the target levels as a function of different seeding temperatures, AgI particle concentrations, and depths from the seeding level. Statistically, we can see that both temperature and AgI particle concentrations have important impacts on the seeding impacts. The modeled Ze values are relatively high at -15 °C, with higher AgI particle concentrations. Clouds with cold top temperatures can be either shallow or deep, and at a given seeding temperature, the modeled Ze values increase with depth (similar to the Ze profiles in Fig. 2). Cloud top pressure also influences the growth rate of ice crystals, but its impact is minor compared to that of other factors.

Using the results from the 2500 numerical experiments, we parameterize the AgI particle concentration that is needed to detect the seeding signature. We train the data using polynomial regression based on 2000 experiments randomly selected from all simulations, and we then test the parameterization using the remaining 500 experiments. The inputs are seeding-level temperature (Tseed), seeding-level pressure (Pseed), cloud depth from the seeding level (Dseed), and Ze attributed to seeding at the target level (Ze_target). The output is the AgI particle concentration at the seeding level (NAgI). 32NAgI=fTseed,Pseed,Dseed,Ze_target Thus, for a given cloud, we can decide how much AgI is needed to seed at different heights to produce a radar signal exceeding the natural variability at the target level. Figure 8a compares the modeled and parameterized AgI particle concentrations using the training data. Generally, the data points lie along the 1 : 1 line, and this works well for the test data (Fig. 8b), indicating that the parameterization can reasonably reveal the relationships between AgI particle concentrations and Ze under different cloud conditions. In the polynomial regression used for Fig. 8a, a degree of 6 (i.e., the maximum power for the input variables) is used. We also tested different degree values in the regression, as shown in Fig. 8c. A degree of 6 gives the smallest root mean square error (RMSE) and the highest correlation coefficient between the modeled and parameterized AgI particle concentrations. The correlation coefficient increases as the degree increases from 1 to 6. For degrees larger than 6, the polynomial regression becomes overfitted.

In this section, the parameterization is applied to a mixed-phase stratiform cloud with moderate natural precipitation (0.2–0.3 mm h⁻¹). The purpose is to estimate the concentration of AgI that is needed to detect the seeding signature in this case. Figure 9 shows the temperature profile and Ze values measured by a Ka-band cloud radar. The cloud was observed in the region of Hulun Buir, northeastern China, on 3 August 2023. It formed during the passage of a warm front (near z = 1.7 km in Fig. 9a). The cloud was stratiform and deep, with a top temperature of about -25 °C. The freezing level was at about 4.2 km altitude. It can be seen that there are large natural variabilities in observed Ze values. Above the freezing level, the largest Ze value is about 15 dBZ, and the average Ze value is 4 dBZ.

To determine the AgI particle concentration, firstly, we need to choose a threshold for Ze induced by cloud seeding; i.e., seeding-induced Ze values should be higher than this threshold so that composite Ze values (combining natural and seeding Ze) can exceed the natural variability after seeding. Figure 10 shows a contoured frequency-by-altitude diagram (CFAD) of observed Ze values. The solid red line indicates the average Ze value plus 1 standard deviation, and the dashed line indicates the mean plus 2 standard deviations. These two thresholds are probably still not large enough if seeding is performed in regions with relatively high values of natural Ze (e.g., at 16:40 LT), but they may be fine if seeding is conducted at 16:10 LT, when values of natural Ze are low. To better detect the seeding signature, a larger value of Ze is necessary, such as the mean plus 3 standard deviations (dotted line), which well exceeds the maximum value of natural Ze.

Figure 11 shows the AgI particle concentrations needed at different heights (-22 to -7 °C) to detect Ze values attributed to seeding above the freezing level (4.2 km altitude), assuming there is sufficient supercooled liquid water to support ice growth. We test three different thresholds for seeding-induced Ze values (5.9, 11.2, and 16.7 dBZ). These values correspond to composite Ze values (mean values of natural Ze and seeding-induced Ze) of 8, 12, and 17 dBZ above the freezing level (see the three red lines in Fig. 10). It is expected that more AgI is needed to obtain larger Ze values, but the vertical variations in the three profiles of AgI particle concentration are very similar (Fig. 11). The required AgI concentration is lowest at about 7.3 km altitude (-15 °C), where 20 cm⁻³ of AgI is sufficient for cloud seeding to induce a composite Ze value exceeding 17 dBZ. Therefore, seeding at this level provides the highest probability of detecting the seeding signature. However, airborne seeding at -15 °C is not always possible because of flight limitations (e.g., airframe icing). Seeding at colder or warmer temperatures may also exceed the Ze threshold if the AgI particle concentration is high enough; however, since a normal AgI concentration in cloud-seeding operations has a magnitude of 10–1000 cm⁻³, it is unlikely that seeding below 6.5 km altitude will yield a detectable seeding signal near the freezing level in this case.

In precipitating clouds, the presence of natural ice would diminish the detectability of the seeding signature because both natural and seeded ice crystals would compete for the supercooled liquid water. We do not consider the interaction between seeded and natural ice in the 1D model because we assume that there is sufficient supercooled liquid water for their growth, and aggregation between natural and seeded ice is expected to further enhance the ice crystal size, which is favorable for seeding-signature detection. Therefore, we are able to apply the parameterization to a precipitating cloud; at the very least, it provides a threshold of AgI particle concentration that is needed to detect unambiguous seeding signatures. However, the assumption of sufficient liquid water is not always valid, especially when the natural ice concentration is high. Figure 12 shows the liquid water path (LWP) between the seeding level and the target level (4.2 km altitude) needed to support the growth of ice at different seeding temperatures. Seeding at -15 °C consumes the least liquid water, i.e., less than 30 g m⁻². The LWP in this cloud, observed by a microwave radiometer, was mostly larger than 300 g m⁻². However, both rainwater and cloud water contribute to the LWP, meaning we do not have direct measurements of the LWC in clouds. Accordingly, it is not known whether there is sufficient water for ice growth when seeding occurs at temperatures between -21 and -11 °C. Seeding at temperatures warmer than -10 °C requires much more liquid water due to the high AgI particle concentration; therefore, the size of ice crystals and Ze may be even smaller, as suggested by the model. Previous observational studies suggest that the LWP in shallow mixed-phase stratiform clouds often exhibits a value lower than 100 g m⁻² (Zhang et al., 2014). Such a low LWP value prevents the growth of ice crystals if the ice concentration is high. In addition, if there is no continuous supply of liquid water (e.g., in orographic updrafts), seeding can cause complete cloud glaciation in the seeded region (Dong et al., 2021).