The Units of Spectral Radiance

Imagine you had a bathroom scale that gave you consistent results, but the numbers it returned had no units, seemed arbitrary in magnitude and varied based on how long you stood on the scale.

This scale might still be useful for tracking your own weight over time (up or down) but the results would not be easily comparable to other people with different scales. If everyone’s bathroom scales were calibrated to return results in kilograms or pounds that would be a lot more useful.

The raw pixel values that come out of a camera sensor are like this wonky bathroom scale, they are internally consistent between photos but they have no meaningful units and depend on variable dimensions like how long you exposed the image for. These raw values are unceremoniously referred to as as Digital Numbers (DNs).

In the world of remote sensing (satellites/planes/drones/space probes etc), images are often released in a radiometrically calibrated format. Radiometric calibration normalizes the DN of each pixel so that each value represents a calibrated measurement with SI units. This makes imagery from different cameras comparable without knowing much about cameras. The unit of the calibrated values is often spectral radiance, which is a derived SI unit defined as:

Watt per Meter² per Steradian per Micrometer

Nearly all of the imagery available through the Planet API is available in a radiometrically calibrated format with unit spectral radiance.

This blog post is about breaking down each component unit to understand what spectral radiance tells us about the light being measured by a radiometrically calibrated camera.

Let’s talk about units — counting hail stones

It’s hailing outside and you decide you want to quantify just how much it’s hailing. You set out a bucket to collect hail stones and your friend across town does the same thing.

You set out a bucket and leave it for 60 seconds. Afterwords you check and see that you’ve collected 50 hailstones. Your friend uses a larger bucket, collects for 30 seconds, and reports back 80 hailstones.

The results are not directly comparable because they don’t just describe the rate of hail, the results depend on several variable dimensions:

  • The collection time is different, with unit seconds
  • The surface area of the buckets is different, with unit meter²

It would be more useful to have the rate of hail with respect to these dimensions.

First we can divide each result by the number of collection minutes, giving us a new unit: Hail Stones per Minute.

If we divide by that by the surface area of a bucket, giving us an even narrow unit: Hail Stones per Minute per Meter².

Our new unit allows us to directly compare the two measurements.


hail stones per minute per meter² is directly analogous to spectral radiance. spectral radiance quantifies the energy in the light coming from an object with respect to several dimensions that are usually variable across image captures:

  • Time (of the exposure)
  • Surface Area (of each pixel)
  • Distance (from the light source, sort of)
  • Spectral Sensitivity

Counting Photons

For the purposes of this post it’s sufficient to think of light as being made of infinitely tiny spheres, called photons. Unlike hail stones, which fall in the same direction, photons shoot out from a light source in all directions.

Similar to our hail counting buckets, an image sensor of a camera is an array of photosites (buckets) that effectively count photons[1]. Since photons from the same light source stream in at different angles, cameras use a lens to redirect photons from the same light source to the same photosite.

Like the hail buckets, each image (photon counting experiment) lasts for a fixed amount of time, this is called the “exposure time” of the image.

Energy

Each photon carries with it a fixed amount of energy. Explaining what energy is is a little beyond my understanding so you will have to accept that photons have energy and the unit of that energy is the joule. This is our starting point.

Since this post is not about how to calibrate an image, let’s assume that we have a calibrated camera that outputs how many joules of energy were in all the photons captured by a single photosite during an exposure.

Every radiometrically calibrated image could be released with units joule but it would be as useful as reporting on how many hail stones you collected without telling anyone how big your bucket was.

Time

Step one is to divide our energy measurement by the exposure time of the image so that images with different exposures can be compared. Joule per second is an SI unit that already has a name, the watt.

Watt = Joule per Second

The watt accounts for the time dimension of taking a photo and is the first component unit of spectral radiance:

Surface Area

Just like our friend’s hail bucket was larger, cameras have varying sizes for individual photosites. The photosite surface area on my Nikon dSLR sensor is 8.42 micrometers².

Dividing our Watt by Meter² accounts for the different surface areas of pixels across cameras:

Distance

This gets a little complicated. As photons stream out in all directions from a light source they can be thought of as a sphere that increases in size the farther away they get. Represented here are two identical camera sensors, one is farther away from a light source than the other.

Even though these sensors are the same size, the one closer to the source intersects (“subtends”) a much larger % of the circle formed by the light emitting from a light source. If light sources were emitting the same amount of photons the sensor further away would receive fewer.

In addition to distance, sensor size is a factor. If the sensor further from the light source were made larger it could subtend more of the circle.

A way to account for differing capture distances between images would be to divide our result by the angle of this circle subtended by each photosite of the sensor. This is exactly what is done except instead of using a two dimension unit of angle, like a radian we have to use a three dimensional unit of solid angle, called a a steradian.

2D Angle vs 3D Solid Angle:

As a camera gets farther from a light source, its sensor subtends a smaller % of the sphere formed by that light source’s photons, measured in steradians.

When you divide by steradians you get a result that is comparable with images take at any distance.

If we stopped here, the unit we would have is called Radiance. We have to take it one step farther to get to the commonly used Spectral Radiance.

Spectral Sensitivity

Photons exist on a spectrum of different wavelengths, measured in meters. Silicon based sensors like CCD and CMOS are sensitive to photons with wavelengths in the range 300nm to 1100nm, human vision ranges from about 400nm to 700nm.

This diagram represents two filters that are sensitive to two different contiguous sections of spectrum, one is 40nm wide the other 70nm wide. The orange line is light that contains equal energy at every wavelength.

If you imaged this light with these filters, the photosites under the wider filter will report a higher radiance than those under the short filter, because there are more photons making it through. If we divide the radiance reported by the width in micrometers we end up with radiance per spectrum, spectral radiance!

In reality the filters that cover photosites are often not equally sensitive to a full section of spectrum; they have a variable sensitivity to different bandwidths. The same goes for the light being imaged; there are usually not an equal amount of photons in all parts of the spectrum.

When this is the case the meaning of spectral radiance becomes less clear. It can tell you the average per-spectrum radiance under a certain filter but not the spectral radiance of any specific wavelength.

All of the other units are an average too, joule per second just gives you the average joule per second, not the instantaneous joule per second at any given second. The more variable that dimension is the less meaningful the average is.

Other Radiometric Units

Some of the intermediate units we passed through on our way to spectral radiance also have radiometric terms associated with them. For example:

These radiometric terms serve to distinguish these concepts from other concepts that have identical SI units but don’t relate to light. Horsepower and radiant flux both have the SI unit watt but they are used in different contexts.


Hopefully this gives you a better understanding of what you’re dealing with then you work with radiance calibrated imagery.