It seems that you are concerned about forms of precipitation like snow, drizzle, hails etc. No doubt these are related to the atmospheric temperature, cloud mechanisms and the terminal velocity of the water droplets when fall through the atmosphere. A simple way to determine the precipitation is liquid and solid form of the water on the surface.

In order to answer this question it is necessary to consider the wind factor and the geometry of a rain gauge:

“A rain gauge acts as an obstacle that disturbs the incoming airflow originating turbulence and eddies around, resulting in deflected trajectories of some hydrometeors away from the funnel region and missing the gauge orifice [7,8]. The reduction of particle flow over the gauge (under-catch) due to wind effect depends on wind speed, precipitation type, rainfall intensity, and the aerodynamic design of the rain-gauge [9–12]. It is especially true for solid precipitation collected by unshielded gauges,…”

https://mdpi-res.com/d_attachment/water/water-13-02576/article_deploy/water-13-02576-v2.pdf

I did not work in heavy snow country. If a weighing recording rain gauge collects rain, snow, hail, etc., the weight is recorded as it accumulates. A tipping bucket raingauge would probably clog up with excessive snow or hail unless heated. But with rain, the bucket tips back and forth as rain is collected, and the number of tips are recorded As a measure of volume. A standard rain gauge does not record, it just collects rain, snow, etc. Unless serviced fequently, such as daily, antifreeze and/or a small amount of oil may be added so the rain or snow does not freeze and break the raingauge, or evaporate too much in hot weather. Unless one had the time recording of precipitation with concurrent temperatures, it may be difficult to detect precipitation type. And ground temperature can be substantially different from cloud temperatures where hail, sleet or snow may be supercooled. And it is possible to have rain that freezes when it hits very cold surfaces. As mentioned wind turbulence can influence rain catch, so most research stations use a circle of wind foils to help reduce turbulence around the rain gauge. Rain, snow, etc. are not necessarily evenly distributed across the landscape, varying with elevation, topography, and the positioning, speed and intensity of storm cells or cyclonic storm activity. In deep snow country, snow depths may be cored and depth accumulation measured, and melted Or weighed to convert to amount of water. The separation of rainfall into its various types may not be so easy. And might be helpful to have dew point with the temperatures, as some areas have documented condensation inputs to forests.

Thank you everyone, I have a paper I am using which details the use of psychrometric energy balance equation to determine phase. I have calculated all of the components that need to go into the equation BUT Psat(Ti). I do not understand how to determine Psat(Ti) when I am solving for Ti.

Equation: Attached.

PAPER: Bias corrections of precipitation measurements across experimental

sites in different ecoclimatic regions of western Canada

Xicai Pan1, Daqing Yang2, Yanping Li1, Alan Barr2, Warren Helgason1, Masaki Hayashi3, Philip Marsh4, John Pomeroy5, and Richard J. Janowicz6

https://tc.copernicus.org/articles/10/2347/2016/

>>I do not understand how to determine Psat(Ti) when I am solving for Ti.

I have attached a brief description of the solution. If you have a closed form of the expressions for all specific terms in your equation I can write a simple Matlab program to solve this problem.

you can reference following work：

Bourgouin P, 2000. A method to determine precipitation types[J]. Weather and Forecasting, 15(5): 583-592. DOI:10.1175/1520-0434(2000)0152.0.CO;2

Lowndes C A S, Beyon A, Hawson C L, 1974. An assessment of some snow predictors[J]. Meteorol Mag, 103: 341-358.

Bernstein B C, 2000. Regional and local influences on freezing drizzle, freezing rain, and ice pellet events[J]. Weather and Forecasting, 15(5): 485-508. DOI:10.1175/1520-0434

I would like to recommend the reference, i.e., Ding B, Yang K, Qin J, et al. The dependence of precipitation types on surface elevation and meteorological conditions and its parameterization. Journal of Hydrology, 2014, 513: 154-163.