You can see that when the earth is cold, it don't snow much and Snow Accumulation is very low. You can see that when the earth is warm, it snows more and Snow Accumulation is much higher. NOAA's data does show this.  NOAA's climate theory and models do it different. They increase ice volume during the very cold periods of the ice age when it is not snowing much.
They decrease all the ice volume while the earth is warming rapidly. 
Earth does not warm just because ice volume is decreasing. Earth only warms when ice is retreating  and ice albedo is decreasing. The ice sheets keep the earth cool during the long cold of a major ice age while they are thinning. Ice albedo is a positive feedback to temperature when ice is retreating. When the earth warms to the point it starts melting Arctic Sea Ice, the snow accumulation increases enough to cause the ice albedo to become a powerful negative feedback  to temperature.  This is the phase that earth has entered during the most recent decade. Ice volume is increasing and the oceans are going down. This stable cycle of warm, then cool, then warm, then cool,..  has dominated for the past ten thousand years.  Ice Melts, Earth warms, Arctic Sea Ice Melts, it Snows, Earth cools, Arctic Sea Ice freezes, Snow stops,  Ice Melts ............

Infrared Radiation from the Greenhouse Gas, Water Vapor, does most of the cooling of Earth, with a Trace of help from from CO2, but Albedo is used to do the fine tuning and the very tight bounding of temperature.
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