My hunch is that the location of the satellites can be deduced from the weather data. These satellites would be a target in a time of war.
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But a bigger problem comes before the above issue:most of the current human meteorological satellites do not have stealth capabilities.You can see them directly.Perhaps your idea will become a practical problem when satellite stealth technology matures.
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I do think you might able to deduce the orbit even if the data release is delayed by a random time period. If you’re a foreign adversary that has its own satellites, you can measure the same information from a known orbit. Then one could compare the published data with one’s known dataset to deduce things like the angle from which the data was measured.