There are three aspects to the weather: temperature, precipitation, and wind. A weather forecast will give information for the next 1 to 7 days.
The temperature determines how quickly overland travel makes you fatigued, and very high or low temperatures will inflict damage. Heavy rain causes you to become fatigued more quickly, and if you're wet, you treat the weather as one step colder. Winter clothes offer protection against all but the most severe cold damage.
Notes:
Cold weather clothing reduces the damage from extreme cold to severe cold, negates the damage from severe cold, and stops mild cold from fatiguing you more quickly. If the clothing is wet, it does not offer these benefits.
Rain or snow affects how easily you can see (or hear, if the rain is very heavy). Very heavy rain can also make creatures count as concealed to one and other.
As mentioned in temperature, rain increases the speed at which you become fatigued, and prolonged exposure will make you soaked - which makes the temperature count as one step colder!
Notes:
The speed of the wind applies circumstance penalties to auditory Perception checks and physical ranged weapon attacks (such as arrows). Very high winds force creatures to make Athletics checks to move; if they critically fail, they are knocked back by the wind and become prone. Winds also make aerial terrain count as difficult or greater difficult terrain, and can force flying creatures to make Maneuver in Flight checks. If they critically fail these checks, or do not make at least one every round, they are blown away. The most powerful winds prevent all physical ranged attacks and flight, and make ground movement very difficult.
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