## Imaging and Spectral Dependence of Contrast

A good rule of thumb is that most terrestrial objects (e.g., snow, soil, water, vegetation) are approximately black (have spectral emissivities near 1) over the range of frequencies that encompass much of the Planck function for typical terrestrial temperatures. Because of this, the brightness temperature (for infrared frequencies) of terrestrial objects is a good approximation to their thermodynamic temperature. This provides a means for determining temperatures remotely: measure the amount of radiation (in some frequency interval) emitted by an object and convert this radiation into a temperature by way of the Planck function. We can do even

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Figure 1.7: Solar spectrum (after Kurucz and Clough).

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Figure 1.7: Solar spectrum (after Kurucz and Clough).

more. Temperature differences give rise to contrast, which allows for the possibility of imaging radiation we cannot see but that an instrument can.

The image of any object is a one-to-one transformation: every point on the object corresponds to one and only one point on the image. This transformation is the function of lenses in cameras, in slide projectors, and in your eyes. Remove the lens from a camera and light from objects still illuminates the film, but a single point on the film receives light from many object points. Remove the lens from a slide projector and it still projects light, but no image, onto a screen. A pinhole is the simplest kind of imaging device. One and only one line can be drawn from any point on an object, through the pinhole, to an image point.

Our eyes are imaging devices that respond to different amounts of radiation coming from different directions. We can get about in the world because of these (relative) differences, which is called contrast. A whiteout is the absence of contrast. You can experience whiteouts in blizzards or while descending through thick clouds in an airplane (see Sec. 5.2). If you have ever been in a whiteout you know that it can be frightening. You can't tell up from down, right from left. You are lost in a field of radiation the same in all directions.

Our eyes form images using visible radiation, the (indirect) source of which is the sun or lamps. That is, we usually image scattered light (we rarely look directly at the sun or at light bulbs). But imaging devices are not restricted to visible frequencies. The radiation emitted by terrestrial objects at different temperatures provides contrast between them, the greater the temperature difference, the greater the contrast. All imaging devices have a contrast threshold below which they cannot distinguish one object from an adjacent one.

Several years ago we got a telephone call from a very frustrated scientist. While poring over infrared images of sea ice he had noticed that the contrast was better in images at shorter wavelengths. He wanted to know why. So he called remote sensing experts and asked them. They all agreed that contrast was indeed better at the shorter infrared wavelengths. But that wasn't his question. He wanted to know why, but ran into a blank wall. No one disagreed with his observation but no one could explain it. And the more incomprehension he encountered,

Figure 1.8: Two adjacent objects at different temperatures can be distinguished one from the other if the relative difference in the radiation emitted by them is sufficiently large. This contrast depends on the relative temperature difference AT/T and the chosen frequency of the radiation.

the more frustrated he became. The following is our answer to his question, which is a simple application of the Planck function and also illustrates the importance of beginning with fundamentals when faced with a problem.

Suppose that we have two objects, side by side, at different temperatures T and T + AT (Fig. 1.8). We may define the contrast between these two objects (assumed to be nearly black over the frequency range of interest) as

The first term in the numerator can be expanded in a Taylor series and truncated to yield an approximation for the contrast:

where x = hu/kBT. Not surprisingly, the contrast between the two objects depends on their relative temperature difference, but also on frequency by way of the quantity x. The contrast enhancement

is approximately 1 for small x and increases approximately linearly with increasing x. Thus, all else being equal (always an important caveat), greater contrast is obtained for higher frequencies (shorter wavelengths). At typical terrestrial temperatures (^300 K) x > 1 at wavelengths in the range 4-40 pm and so the contrast is approximately

If inherent contrast were the only criterion for choosing one infrared frequency over another, the highest possible frequency would be the best choice. Of course, at sufficiently high frequencies there might not be enough emitted radiation to image.