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Sometimes my monitor picks up solar flares as decreases in signal strength rather than increases (i.e. my graph goes down instead of up). Why?

The signal received is a sum of several different "modes" bouncing back and forth between the Earth and ionosphere. The modes effectively travel differently -- some are more horizontal, others are more zigzagged.

At any location, the signal is a simple sum of all these modes, but because those modes have different phases, they may well destructively interfere. This particular solar flare likely caused the ratios between the different modes to change in such a way that they interfered with each other at that specific location to a greater degree, and hence the signal strength appeared to drop. Had they taken the same measurement several miles away, they may well have gotten the opposite result.

For lightning-induced electron precipitation events, we also sometimes see positive and sometimes negative perturbations. The complicated mode structure of the signal is the reason. The result can change if we change the location of the receiver (or the transmitter, or the perturbation, for that matter).

To help you visualize destructive and constructive interference, see this Powerpoint presentation on Waves and Vibrations

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