I get that for the weather forecast they use barometers for air pressure, a Doppler radar for weather fronts, but these are all short term. Farmers almanac (amongst others) can claim to know weather years in advance. How?
They site solar science, climatology, meteorology and 30 year rolling average. How do account for climate change? What about climate anomalies?
In: Mathematics
In meteorological prediction there are [scales of motion](https://www.eoas.ubc.ca/courses/atsc113/applied/learning_goals-applied.html) which become more difficult to predict the smaller you get.
Where the leaves will go after a gust of wind is almost unpredictable, a tornado might have about 30 minutes of warning, and larger thunderstorm complexes on the order of hours.
The synoptic scale (the scale many think of when they think “weather”, highs and lows n stuff) can be reasonably predicted within a few days to a week, but depending on the complexity of the system this remains difficult at times.
To your question at the seasonal scale: the best we can do (and not do very well) is give broad trends like “wetter/drier” or “hotter/colder” than some baseline; this is usually done through the understanding of long-term planetary trends such as the [MJO](https://en.wikipedia.org/wiki/Madden%E2%80%93Julian_oscillation) or [PDO](https://en.wikipedia.org/wiki/Pacific_decadal_oscillation). An oscillation many might be more familiar with is the [ENSO](https://en.wikipedia.org/wiki/El_Ni%C3%B1o%E2%80%93Southern_Oscillation).
The almanac itself thrives on ambiguity, much like any other predictive pseudoscience. Say “it’ll be cold this week” in a winter month and you’re likely to verify. [Attempts to better quantify](https://ggweather.com/farmers/2005/index.htm) how “useful” it is exists, all with predictable (hah) results.
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