It is very simple to calculate the exposure time when using ND Filters when the light is constant. Long exposures captured during changing lighting conditions make it much more difficult to calculate proper exposure.

Let's say that you are preparing to shoot 10 mins after sunset. The lights is still good. Your exposure without filter is about 1 second and if you want to use 10 stops filter, you would need to take about a 15 minute exposure. But during those 15 minutes the light is dramatically changing and it is a lot darker by the end than at the beginning. That means that your calculated exposure will give an underexposed image. If at the start of shooting, the normal exposure is 1 second, by the end of shooting it will be, let's say 4 seconds. Further, the function of that change is not linear. So it is difficult to just average the exposure time between start and end of shooting and make calculations.

Are there any predefined formulas, based on personal experience, for calculating exposure time in those hours of the day, when light is rapidly changing?

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    "Welcome to the world of The Calculus!" - Sir Isaac – Michael C Jul 6 '16 at 23:28

Since the speed of the sun going below horizon depends on date, I suppose you could create a table measuring exposures starting before sunset and then for example every other minute up to 30 minutes. In the table include column with angle of the sun (2, 1, 0, -1, -2...). When you would be taking the actual shot, you could lookup actual angles of the sun at the time of shooting and estimate from the table what the decrease in light will be.

  • This is a good idea of creating that table. But this is a further work. I am wondering is there already build some kind of softwear for that – Ivo Jul 6 '16 at 20:29
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    @Ivo probably because the exposure is so dependent upon actual conditions at the moment: what is the cloud cover like? Is it a mix of high and low clouds (to provide additional gold lighting after sunset)? What's the surrounding scenery (i.e., dark evergreens all around can suck the light out of the scene like black studio flags/gobos)? The best you can really get is just rules of thumb. – scottbb Jul 6 '16 at 23:28
  • The speed of the sun going down is fairly constant (At the equator it is always within a few seconds of the same - at latitudes near the poles it is much more variable. But there are a lot more folks living within 25-30º of the equator than there are living within 25-30º of the earth's poles.). The time of day it starts is more highly variable depending on date. – Michael C Jul 6 '16 at 23:33
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    @MirekE Timelapse photographers deal with this all the time. They create "bramping" (bulb ramping) profiles to account for the setting/rising sun. Using a similar idea, one could go out and take a bramped series of shots, using different profiles/slopes, and empirically determine the best profile for their location. Still a lot of work like Ivo notes, but what is photography without putting in the long experimental hours? =) – scottbb Jul 6 '16 at 23:33

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