Effects of Target Distance

Introduction

The effects of smoothing the edges of the duct-producing thermal inversion vary with both the height of the eye and the distance to the target. The simulations are made with the 2° thermal inversion between 50 and 60 m height that I have used elsewhere. As usual, this inversion is represented by a shaded band in the ray diagrams; remember that the duct actually extends below the bottom of this band.

The target-distance effects are shown on this page, for a couple of different heights: one just inside the bottom of the duct, at 46 m height; and the often-used height of 54 m, near the middle of the duct.

As on the eye-height page, most of the examples here will contain a link to the corresponding unsmoothed (or zigzag-profile) model simulation. Also, the discussion here concentrates on the additional effects produced by smoothing the edges of the inversion, rather than the gross effects such as towering, stooping, and image inversions like the superior mirage and Wegener's Nachspiegelung. For an explanation of these effects, see the zigzag-profile page.

Eye height: 46 m

As for the zigzag models, we begin just inside the bottom of the duct, but below the bottom of the thermal inversion that produces it.

Target at 10 km, eye at 46 m   (smooth-duct model)   [align]

We again start with the target at a distance of 10 km. At this short range, the effects of smoothing the temperature profile are negligible; you can compare by using the link to the zigzag case to verify this.

Instructions for using the [align] links are found here.

Target at 20 km, eye at 46 m   (smooth-duct model)   [align]

At 20 km, we have the same image that is discussed on the page where eye-height effects are shown for this target range. Again, nothing differs significantly from the corresponding zigzag model.

(To see the effect of varying the eye height at this distance, look here.)

Target at 30 km, eye at 46 m   (smooth-duct model)   [align]

At 30 km range, we have a fine 3-image mirage. And at this distance, we see a little difference from the zigzag case, just at the top of the inverted (miraged) zone.

The difference is best understood by looking at the transfer curve. Just above the astronomical horizon, at about +1′ altitude, the zigzag model has a short vertical segment there; but here, the smoothing of the temperature profile has made the transfer curve smooth, instead. So, smoothing the thermal profile has eliminated that segment of the transfer curve, and the corresponding horizontal shelf at the top of Wegener's miraging strip.

Target at 40 km, eye at 46 m   (smooth-duct model)   [align]

At 40 km range, we see further differences from the zigzag model: a narrower blank (or miraging) strip, and smoother corners.

This case is discussed further on the page dealing with the effects of eye height.

Target at 50 km, eye at 46 m   (smooth-duct model)   [align]

At 50 km, the main difference from the zigzag model is a narrower strip.

Target at 60 km, eye at 46 m   (smooth-duct model)   [align]

At 60 km, the strip is not only narrower than in the zigzag model, but has developed a prominent sharp upper lip.

The distinctive effect of the smooth upper edge on the thermal inversion is to capture rays near the upper edge of the miraging strip, so that they approach Kummer's circulating condition, where the ray curvature matches that of the Earth.

A nearly-circulating ray at the top of the strip meets the target just at the top of the inversion. But just below this critical level, a ray remains near the top of the inversion for a considerable distance, and then bends back down, meeting the target considerably lower. A little lower still, a ducted ray eventually bends down so that it meets the target at the bottom of the duct, near 46 m. This last ray is the one at the tip of the lip in the present image; it meets the target lower than any other ducted ray. As the target is widest (within the duct) at the bottom of the duct, such rays reach the widest slice through the target.

Rays that are slightly lower at the observer don't get close enough to the circulating level to meet the target at the bottom of the duct; so they show the observer a higher level on the target, where it's narrower.

Target at 70 km, eye at 46 m   (smooth-duct model)

At 70 km, a similar lip appears on the bottom edge of the miraging zone. This lower lip is produced by ducted rays that are below the observer's horizon, but approach the circulating condition at the top of the smoothed inversion before reaching the target, at this large target range.

Notice that Wegener's miraging strip is completely separated from the direct view of the target, below it. At this range, we see only the tip of the target above the apparent horizon. An inverted image (Wegener's Nachspiegelung) of its top appears just underneath the miraging strip, which is filled with an enormously stretched image of a thin slice of the target, and decorated with enormously compressed images of the 50-m level of the target (i.e., the sharp lips). Finally, the strip is surmounted by a very compressed erect image of the top of the target.

(Because of a quirk of my software, I can't produce a corresponding simulation for the zigzag model at this distance.)

It's also useful to compare these mirage simulations with sunset simulations for the same model and eye height (46 m). Of course, Wegener's miraging strip appears blank in the sunset images.

Eye height: 54 m

As with the zigzag model, I'll start this sequence at a target distance of 10 km.

Target at 10 km, eye at 54 m   (smooth-duct model)   [align]

It's not surprising that there's negligible difference between the smoothed model here and the zigzag model, at this short range.

Target at 15 km, eye at 54 m   (smooth-duct model)   [align]

Even at 15 km range, the three image (or transfer-curve) zones that were so well-defined in the zigzag model are almost equally distinct here, with the edges of the inversion smoothed. That's because we don't yet have rays that become nearly horizontal at the critical heights near the top and bottom of the inversion.

Target at 20 km, eye at 54 m   (smooth-duct model)   [align]

At 20 km, the three image zones are still quite prominent: the short, undistorted section (loomed), sandwiched between the stretched — here, miraged — lower zone and the squashed (stooped) one above it.

The image still resembles that for the zigzag model; but at this range, we finally see some smoothing of the image due to smoothing the inversion.

(To see the effect of varying the eye height at this distance, look here.)

Target at 25 km, eye at 54 m   (smooth-duct model)   [align]

At 25 km distance, the image is still smoother than at 20 km. (Compare with the zigzag model.) The raised (loomed) zone is now rather ill-defined here, while it remained distinct in the zigzag simulation. So the shoulder above the mirage is rounded here, instead of sharp-cornered as in the zigzag simulation.

Target at 30 km, eye at 54 m   (smooth-duct model)   [align]

At 30 km, a different effect appears: smoothing the inversion smoothes out the little ripples in the zigzag simulation near the astronomical horizon.

Another interesting phenomenon here is the difference between the shape of the ray at +3′ here and in the zigzag model, where it's bent down considerably more. This ray is nearly horizontal when it reaches the top of the inversion, so it spends a long time just below that height (60 m). Because of smoothing the temperature profile, the ray curvature between 59 and 60 m is appreciably weaker here than in the zigzag model. Other rays are much less affected.

This difference in the +3′ ray remains a prominent feature of the next several simulations, at increasing ranges.

Target at 40 km, eye at 54 m   (smooth-duct model)   [align]

At 40 km, we start to notice the sharpening of the lip (at the upper edge of Wegener's miraging strip) that was also seen from below the inversion — though at a considerably larger range. (Again, compare with the zigzag model to see the effect.) The difference is more obvious in the transfer curve than in the image.

The transfer curve also shows why the +3′ ray is so strongly affected by smoothing the inversion: it strikes the target at a height near 120 m, which is in the upper part of the nearly-vertical section of the transfer curve.

(To see the effect of varying the eye height at this distance, look here.)

Target at 50 km, eye at 54 m   (smooth-duct model)   [align]

As the target distance increases, the sharpening of the lip becomes quite marked. Compare this simulation of the smoothed inversion with the zigzag one for the same geometry.

While the lip has become a spike here, other features at lower altitudes that were sharp and angular in the zigzag model have become smoother and less obvious here.

As usual, smoothing the inversion has made the miraging strip slightly narrower. This allows more of the target to be seen just above the strip.

Target at 60 km, eye at 54 m   (smooth-duct model)   [align]

At 60 km, the spike has become needle-sharp. Compare with the zigzag simulation. The narrowing of the strip is quite obvious, both in the image and in the transfer curve.

Target at 70 km, eye at 54 m   (smooth-duct model)

As at 46 m, it's possible to carry the simulations for the smoothed-inversion model out to greater ranges than for the zigzag model. Here's the simulation for 70 km.

Notice the spike developing at the lower edge of the miraged strip here, and the increasing complexity of detail within the strip — especially, the extreme vertical stretching (towering) in its lower half.