Exposure Topics

Exposure Values: Where’s the math?

The original reason for posting this was that I found that the formula for calculating exposure values (EV) given in my photography references and posted all over the web was wrong.   Subsequently, I found similar problems with other lighting calculations such as lighting ratios.   So, I decided to research the math and put it in a spreadsheet for future reference.  

I would like to start with the observation that many photo lighting references seem to imply that the exposure value (EV) changes with film speed (ISO).   This is fundamentally wrong.   The EV is a measure of illumination.   If a scene is illuminated at EV15 (sunny day) changing the ISO, aperture, or shutter speed does not change the illumination.   It is still EV15.   All we can do is change the ISO, aperture, and/or shutter to match the light that is available.  

The correct formula for calculating exposure values is shown here.

EV = Log2(aperture2 x (1/shutter_speed) x (100/ISO))

Every reference I saw showed ISO/100 instead of 100/ISO for the EV calculation. Of course at ISO 100 it does not matter.   And all of the references show a table of calculations at ISO 100.   But at other ISOs I could never calculate the proper EV.   My goal was simply to create a spreadsheet that I could use to validate the math.   I have now correlated the spreadsheet results with my light meter and my camera meter and find them consistent.   The bottom line is that I can trust this spreadsheet.  

From the formula for EV:

Aperture = sqrt(2EV x shutter_speed x (ISO/100))
Shutter_speed = (1 / 2EV) x aperture2 x (100/ISO)

Part of the madness was the basic issue that photographers have their own rules for math.   The physics of light follow mathematics at log base two.   Since photography had a 100-year head start on the popularity of personal computers and hand held calculators, photographers rounded base two numbers in all kinds of bizarre ways.   This leads to a progression of f-stops, shutter speeds, and ISO values that defy any logical numeric progression.   For example, the sunny f/16 rule is based on the fact that ISO 100 at 1/100 second and f/16 would be the correct exposure for EV15.   I have yet to see a camera that will allow you to set the shutter speed to 1/100.   In fact, the standard is based on a shutter speed of 1/125 (that actually should be 1/128).   This does yield EV 15.   If this wasn’t enough, my cameras and light meters allow 1/2, 1/3, or 1/10 stop increments.   In any case, the numbers used in photography have become industry standards and aren’t going to change now.  

The spreadsheet shows the calculated EV for shutter speeds from 1/16000 to 30 seconds and apertures from f/2.8 to f/32 in ½ stop increments.   You can easily add cells to cover any range that you want.  

It has helped me understand how to use the fundamental “sunny f/16” rule better.   I can simply use my light meter or camera to get the correct aperture setting at a given ISO by setting the shutter speed at the reciprocal of the ISO I am using.   That is normally 1/125 at ISO 125 for me.   Subtract one from the aperture value and you have a pretty good handle on the corresponding EV.   Using the printed spreadsheets I can quickly see what ISO I am going to need to meet my shutter/aperture objectives and leave a little breathing room.   That way I don’t find myself shooting at the extreme limits of the lens when the lighting changes slightly.  

Here is a sample full-stop fragment of the results at ISO 125.   The red cells are the normal daylight range (EV11-15).   The formulas are available in ExposureMatrix.xls (sheet 1).  

shutteraperture
f/2.8f/4.0f/5.6 f/8.0f/11.0f/16.0
1/1600016.617.718.6 19.720.621.7
1/800015.616.717.6 18.719.620.7
1/400014.615.716.6 17.718.619.7
1/200013.614.715.6 16.717.618.7
1/100012.613.714.6 15.716.617.7
1/50011.612.713.6 14.715.616.7
1/25010.611.712.6 13.714.615.7
1/1259.610.711.6 12.713.614.7
1/608.69.710.6 11.712.613.7
1/307.68.79.6 10.711.612.7

I need to make it clear that I do not use this to set my shutter and aperture for a particular shot.   My camera and light meter are faster and more accurate for that.   I do use this for my own education and understanding of the challenges I keep running into with lighting.   For example, from the numbers above I can see that although my camera will go down to a shutter speed of 1/16000 second there is no way that I am going to get that hummingbird shot at ISO 125, especially if I want some DOF.   Even at f/2.8 I need almost two stops more light than “sunny day EV15”.   I need to crank the ISO up to 800 just to get to f/4 with full noon sunlight.   If I prepare myself mentally for that fact, I’m more likely to get that elusive hummingbird shot.  


Lighting Ratios: Where’s the math?

The next challenge was to understand the math behind calculating light ratios.   This was driven by a photography certification test that had several questions requiring calculations for lighting ratios when using multiple flash heads.   I had to guess because I long ago gave up and rely exclusively on my light meter.   I saw that I needed to revisit this topic.  

The problem was that the more I researched, the more confused I became.   I found many tables but they contradicted each other.   None of the references provided the formulas used.   I have researched at least five hardbound books on lighting as well as the Kodak, Minolta, Canon, and Nikon web sites.   They all provide incomplete and contradictory information.  

There are at least four ways to evaluate lighting ratios.   They can all be considered a measure of contrast.   The most common definition refers to the combined intensity of the main and fill lights as compared to the intensity of the fill light alone.   Another definition compares the intensity of the main and fill lights independently. Sometimes it is a measure of the light striking the highlight side of a subject verses the light striking the shadow side.   This can even be a single light source.   The fourth technique measures the reflected light in the highlight, mid tone, and shadow areas.   These measurements can be mapped into the photographic zone system.   It is important to understand which definition is being used and for what objective.  

For this exercise I will be using the first definition, main plus fill verses fill alone.   Thus, two equal lights at equal power settings and distance yield a lighting ratio of 2:1.   This answer seems counter intuitive.   This setup describes flat lighting with essentially no contrast.   At this point, the published tables start to differ.   Some start with two lights at equal f-stops being a ratio of 1:1.   If you start with two lights of equal power and remove one, you have a one-stop difference and a ratio of 2:1, not 1:1.   A 1:1 ratio would be a single light source with no difference to compare.  

From this point on, the ratios quoted vary even more radically.   Three f-stops difference is frequently shown as a ratio of 4:1.   Sometimes it is shown as 3:1.   Which is right?   It turns out that neither is mathematically correct.   Three f-stops difference is a ratio of 8:1.   The formula is simply:  

ratio = 2(EV2-EV1)

The exposure value (EV) at each f-stop can be calculated assuming a shutter speed of one second and ISO 100.   The answer is a linear progression from f/1=0 to f/64=12.   Once again, several references incorrectly state that EV1 is equivalent to f/1 for one second at ISO 100.   The difference in EV values between two light sources is the same as the difference in f-stops except that EV values follow a linear progression.   If you assume one second at ISO 100, to convert from an f-stop to an EV you can use: EV=sqrt(2f).   To convert from an EV back to an f-stop you can use: f=sqrt(2EV).   To use these as linear expressions of light intensity, the EV can be converted to Lux.   Lux=2.5x2EV.   To convert Lux back to EV use: Log2(EV/2.5).  

If we assign the variables m=main, f=fill, and r=ratio, the lighting ratio is calculated via: r=(m+f)/f.   Thus, f=m/(r-1) and m=(f*r)–f.   Where m and f are exposure values.   The same math and tables for lighting ratios also apply to filter factors.   These formulas are available in ExposureMatrix.xls (sheet 3).   They yield the following table of lighting ratios and f-stop differences:  
Ratiof-stop difference
10
21
31.5
42
52.3
62.6
72.8
83
164

Based on our accepted definition, this is the difference between main plus fill and fill alone.   It is not the difference between the main and fill lights individually.   This can be simplified to:  

exposure or f-stop difference=Log2(ratio)

There are two primary ways to change the lighting ratio between two light sources.   One is to change the distance and the other is to change the power.   Either results in a change in the f-stop of the lights.   The math should yield the same answers with any of the metrics.   Otherwise, the definition would be flawed.  

Power has a linear relationship to light intensity but light from a point source follows the inverse square law of energy over distance (energy = 1/distance2).   Technically, the distance is the radius of a sphere and the energy would be measured at a spherical plane.   For photography the image plane is flat.   This is one reason that we can experience vignetting in some images.

Using 100% power as the base exposure value (f-stop).  

The relationship of power to intensity (EV) is linear but the resulting f-stops still follow a logarithmic progression.  

Flash to subject distance has an exponential relationship to exposure.  

Thus, both the distance and the f-stops follow logarithmic progressions.

Some references state that the distance in these equations should be one half of the distance from the flash to the subject to the camera, not just the flash to subject distance.   This is not correct.   The inverse square law is based on a point source of light illuminating a plane, not the diffused refraction.   If you shoot a subject eight feet away and illuminated by sunlight, your exposure is based on the sunny 16 rule.   If you shoot a full moon at night, your exposure works out to be the same (unless you are exposing for the dark sky instead of the moon).   The light source is still the sun at 93 million miles.   But the subject distance has changed from 8 feet to 240 thousand miles.   On the other hand, if you shoot a subject illuminated by the moon (a mirrored reflection of the light source), the inverse square law does apply.   Point your camera at that same subject eight feet away and you need 12-15 stops more exposure.   The moon is now your light source.  

After all this excruciating math, all we really care about is the artistic (subjective) result.   Unless you are taking an exam, some simple rules of thumb and a little experience (practice) will suffice.   Lighting ratios of 3:1 and 4:1 are 1.5 and 2 f-stops difference respectively, between the main and fill lights.   Anything less than 3:1 produces generally flat lighting.   Anything over 3 f-stops (8:1) difference will be almost impossible to see in the image.  

For daylight fill, if the flash power is greater than the ambient light the flash becomes the main light.   In most cases, you will want the flash to be one or two stops below the ambient light.   In my opinion, the actual setting will be more of an artistic decision than a technical one.  

For studio portraits, the sum of the fill lights should collectively be 1.5 to 2 stops below the sum of the main lights.   I do not believe that the use of more than one flash for the main light source is objectionable, especially for group portraits.   Simply use their collective output as your base calculation for the main light.   I would include background and hair lights in this.   The fill lights should also be considered collectively if there is more than one.   Remember, the objective of the fill lights should be to soften shadows, not to create new ones.  

Of course, there will always be exceptions for specific artistic objectives.  

The calculations are available for your use in extreme or critical situations, as well as for general educational value.  


Flash Guide Numbers

Since we’ve gone this far it seems reasonable to wrap up this discussion with a few comments on flash guide numbers.  

A flash guide number is simply an index value (distance) that can be used to calculate the light from a flash at some new distance.   The guide number is actually a distance based on an f/1 aperture and usually ISO 100.   The distance may be stated in feet or meters, so you need to be consistent with your measurements.   One meter is 3.048 feet.   If you are not using ISO 100, the new guide number would be calculated via:

GN2 = GN1 x sqrt(ISO2/ISO1)
After these adjustments (if needed), you can calculate:
a distance as GN/f-stop or an f-stop as: GN/distance

The published guide numbers are generally useful for quick rough estimates.   As a rule, a diffusion screen will remove one stop of light.   A softbox will usually remove two stops.   Using a silvered umbrella for bounce has a negligible effect but a white surface will drop up to two stops.   With close up macro photography guide numbers may not work if the light is too close to the subject.   This is because the light may be so close that it no longer acts as a point source but instead becomes a large, extended light source.   As you can see, there are many variables.   I would strongly recommend that you do some independent testing before relying on the published numbers.   These formulas are available in ExposureMatrix.xls (sheet 3).  

TTL flash metering is another option, but it does not work with most external flash units.   And, like matrix metering the microchip doing the calculations can easily be fooled and cannot understand your real objectives.   For critical work I would always recommend the basic light meter.  


Summary

It is easy to get tripped up calculating exposure values.   Exposure values are a series of integers that represent doubling or halving the amount of light for each unit of change.   Each unit is known as a stop and also tracks the standardized f-stop progression.   The amount of light is related to several factors.   Some of these factors follow a linear progression while others follow exponential progressions.   Shutter speeds, ISO sensitivity, and flash power follow linear progressions.   Therefore, doubling or halving the values produce a one-stop (EV) difference.   Apertures, Lighting distance, lighting ratios, and filter factors follow progressions based on the square root of two.   Therefore, doubling or halving the values produce a two-stop (EV) difference.   Reflectance measurements do not change with distance unless the reflected light is considered as a point source of light.  


The Spreadsheet

The above referenced MS Excel spreadsheet (in a zip file) can be accessed from this link:   Exposure: Excel Spreadsheet.   Sheet 1 contains the basic EV calculations.   Sheet 2 has calculations for EV, Lux, and foot-candle conversions.   Sheet 3 has calculations for lighting ratios and flash guide numbers.   Input cells are blue and calculated cells are bold.  

For EV calculations, the ISO is set in sheet 1, row 2, column R.   The shutter speeds are shown as text in column B and again as a number for the calculations in column A.   Rows 45 and 46 are included only to allow quick testing and comparisons.   You can alter the shutter speed in column B and the ISO in column R for this.   If you have any comments, or suggestions, I would welcome your input.   Please send me an  Email.


The Measurements of Light

If you would like to read more about the basic measurements of light:


Rags Gardner
Rags Int., Inc.
204 Trailwood Drive
Euless, TX 76039
(817) 267-2554
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www.rags-int-inc.com
May 13, 2004

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