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    Re: Dip uncertainty
    From: Trevor Kenchington
    Date: 2004 Dec 7, 19:46 -0400

    With respect to the text which follows, you wrote:
     > Would one or more opponents of my view, expressed above, kindly summarise
     > what's wrong with it, in simple terms that I can understand?
    I will try to oblige.
    > "I think Bruce has got it wrong, about anomalous dip affecting observations
    > from small vessels rather than from large ones.
    > On my small boat, the cockpit sole is no more than inches above sea-level:
    > it's only just self-draining. So my height of eye is no more than 6 feet,
    > which puts my visible horizon 2.8 miles away. Let's say there's a marker
    > North of me, just 2.8 miles away, just there to mark my horizon, appearing
    > to float right on it.
    > When I see the horizon to the North, the light-ray marking that horizon has
    > skimmed just over the sea-surface, at a tangent to it, close to that marker
    > vessel. The dip that I see is the angle between that light ray as it passes
    > my head, and the direction of my own local horizon.
    > You can think of that dip as being made up of three parts. The major part,
    > by far, being simply the curvature of the sea surface, the "geometrical"
    > component of dip, which is easy to calculate precisely.
    > As you ascend above sea-level, the air gets less dense, according to a
    > well-known law for a "standard" atmosphere. This density-gradient refracts
    > the light from the horizon into a predictably curved path. The curvature is
    > in a downward direction, as is the curvature of the surface, and the dip,
    > depending on the difference between the two, is reduced by about one part
    > in 12 as a result. So far all is predictable, and given in the standard
    > "dip tables".
    > But the region of the atmosphere, within a few feet of the sea-surface,
    > through which the light has been passing, is rather special, because of
    > exchange of heat with the surface. Depending on some complex factors, such
    > as wind (or lack of it), temperature difference, surface roughness, complex
    > layers may exist at different levels. The light, in passing through those
    > layers, is refracted in an unpredictable way. That unpredictable component
    > is "anomalous dip"; it can add to or subtract from the predicted dip.
    > Without a dipmeter, nobody knows of its existence.
    > What happens to the light which has passed my head, from the horizon 2.8
    > miles away? It still carries on rising, and may be seen from the bridge of
    > a larger vessel, from 24ft high, which is 5.6 miles from that marker on his
    > horizon (because the horizon distance increases with the square root of
    > eye-height) and 2.8 miles to my South.
    > And what dip affects observations from that vessel? The standard dip, from
    > the almanac, is doubled. The anomalous part of the dip will be, initially,
    > the same as affected me at 2.8 miles, but now the light goes on, still
    > close (from 6 to 24 ft) to the sea surface, so there's extra curvature
    > added as a result of that additional 2.8 miles of its path.
    All OK to there.
    > What Bruce is arguing, it seems to me, is that in the latter part of its
    > path, that light is  sufficiently far (6 to 24 ft) from the sea-surface,
    > that no anomalies of temperature-gradient will occur to affect the dip.
    > This seems to me an unlikely proposition, but even if it were true, the
    > anomalous dip would be no less than it was for me at 6 ft height-of-eye. So
    > I don't accept his argument that anomalous dip will be more of a problem to
    > smaller vessels; it's the other way about, as I see it."
    Your problems, I think, all lie in that paragraph and I think I have
    laid them out in previous messages. However, to go through them again:
    1: I don't agree that either Bruce or I have suggested that no anomalies
    of refraction occur between 6 and 24 feet above the sea surface. What we
    have argued is that, in typical situations, the anomalies along that
    part of the light path to the observer at 24 feet are less than those in
    the part of the light path from zero to 6 feet. [I have, in effect,
    argued that despite its extra length, the anomalies along the light path
    between 6 and 60 feet are less than those between zero and 6, though the
    ones between 6 and 600,000 feet are greater.]
    However, to better understand the qualitative difference between heights
    of eye of 6 and 24 feet, one could choose a model with no anomalies
    above 6 feet. That would exaggerate the quantitative advantage of being
    higher above the water, of course.
    2: Where I think you go seriously wrong is in saying that "the anomalous
    dip would be no less than it was for me at 6 ft height-of-eye".
    You are, I think, supposing that, in the absence of anomalous
    refraction, the respective rays of light from the horizon to each of
    your two observers would be parallel. Were that so, all that would
    matter of the anomaly would be the angle at which the light reaches the
    observer. Once the perturbed ray was distorted in its passage through
    the first 6 feet of height, the disturbance to its angle would remain
    even if there was no further anomalous refraction at greater heights and
    hence that ray would cut the line of the expected refracted ray from
    horizon to observer by the same angle, regardless of the observers
    height of eye. The amount of anomalous dip would be the same for both.
    If, however, there was some anomalous refraction above 6 feet, the lower
    observer would have the advantage (except in rare cases where the
    anomaly above 6 feet counter-acted that below).
    However, the horizon is not at an infinite distance from observers with
    heights of eye of 6 or 24 feet. Thus, rays of light from the horizon to
    their eyes will not be parallel to one another -- except in the case
    where they are both placed on the same ray. You, however, have defined
    the model as one in which they are both on the same
    anomalously-refracted ray. That being so, they cannot both be on the
    same ray that would be expected with standard refraction.
    If the actual light ray suffers anomalous refraction primarily when very
    close to the sea's surface, then the further the observer backs away
    from the horizon and hence the higher his eye, the smaller than angular
    anomaly will become. I don't know how to prove that mathematically but
    it is easy to see if you draw out the light rays and easy to understand
    if you follow Bruce's argument and reduce the anomalous refraction to
    the effects of a single prism placed in the light path.
    You also wrote:
    > I would hope that any explanation would deal with the influence of
    > air-layers on the dip as seen by our two mariners, at heights of eye 6 ft
    > and 24 ft. Comparing three situations, as follows-
    > Normal atmosphere, right down to sea level.
    > A layer with abnormal temperature gradient, confined to within 6 feet of
    > sea level.
    > A layer with similar abnormal temperature gradient, confined to within 24
    > feet of sea level.
    My remarks above encompass your second scenario, though they are not
    confined toe exactly those conditions.
    With a normal atmosphere at all levels, there is of course no anomalous dip.
    With a similar (both vertically and horizontally) abnormality of
    temperature gradient, encompassing both observers and their mutual
    horizon, I think the direction and magnitude of the anomalous dip would
    be equal for both observers in almost all cases. If the anomaly was
    really massive, there might (I think) be enough non-linearity in the
    system for observable differences in the anomaly but I would be really
    surprised if that could occur with observers at 6 and 24 feet. (Maybe it
    could with them at 6 and 600 feet, but by then the assumption of a
    smooth anomaly in the temperature gradient would be unrealistic.
    How often are these anomalies similar across a few miles (horizontally)?
    With still, stable air (likely warm air passing over a cold sea), they
    might be. Where the anomaly is caused by intense solar heating under
    cold air, they will be highly variable and so all discussion of these
    matters should really be in stochastic terms. The deterministic ones
    used here are just to simplify the problem and to ease a qualitative
    Trevor Kenchington
    Trevor J. Kenchington PhD                         Gadus{at}iStar.ca
    Gadus Associates,                                 Office(902) 889-9250
    R.R.#1, Musquodoboit Harbour,                     Fax   (902) 889-9251
    Nova Scotia  B0J 2L0, CANADA                      Home  (902) 889-3555
                         Science Serving the Fisheries

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