From my perspective, the ideal is to set the white balance manually, and then you've only got to "polish" the video in post. But I didn't used to know that ... and/or couldn't set white balance manually.

You CAN - provided you've got a suitable editing program - do the same sort of levels correction you might do with stills. Certainly Adobe Premiere (6.5 and Pro) can do this, and the results can be very impressive. I did a few samples - from old Hi8 or DV green water footage through to blue water DV (filter and/or manual white balance) for someone on YD the other day. All done quickly in Premiere 6.5. Originals on a Sony SC100 or PDX10.

One old sample that shows the principle very well (original at 15m in UK on auto white balance):





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