BTW, I checked the Icom IC-7100 manual for the Weather Alert function and I think I should change my answer. It appears that when Weather Alert is enabled on the IC-7100, the radio periodically checks the selected weather channel for the 1050 Hz alert tone. So, you could have the radio sitting on your favorite memory channel or the VFO and still be alerted in a "dual watch" sort of operation.
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I'll point out that these basic weather alert functions have one major problem in that the function will activate whenever the NOAA station alerts whether the alert is for your county or not. The NOAA station for my area covers 14-15 counties. If all I do is monitor for the 1050 Hz alert tone, I'll hear about weather in counties that are somewhat distant from me. That's why SAME decoders are so useful, but I'm not aware of any radio made for the amateur radio market that has a SAME decoder.