Kevin T. Kilty

Copyright © 1998, All
Rights Reserved

For those readers who do not know what inverse theory is,
perhaps a quick summary is in order. The term *inversion* in
the world of geophysics means finding parameter values for a
model from observed or experimental data. Meteorlogists use the
term *retrieval* because *inversion* has a separate
meaning. The parameters that a person finds through inversion are
optimal in some sense. Because we ordinarily use a model with
specified parameters to simulate a process through a forward
calculation, people often refer to finding the parameters as
*inverting the data*. In essence finding a mean value and
standard deviation from sample data is an inverse problem. More
commonly, though, when people use the term *inverse
problem*, they mean something more complex than mere
statistical calculation; for example, everyone would agree that
finding the parameters of a differential equation or a function
or functional in an integral equation are inverse problems. In
fact, inverse theory most likely refers to the general
methodology of doing the estimation, rather than the specific
application itself.

Inverse theory is a discipline all its own, and its practitioners like to focus on the esoteric aspects of their subject. They love the complex and beautiful; they shun the simple and crude. Thus, a person who views finding a mean and standard deviation as a inverse problem is ridiculed as an inverse theory troglodyte. Exhaustive search often provokes ridicule as well. Anyone using it seems compelled to apologize, refer to their analysis as barely adequate, and conclude that a more complex method would lead to better results (See for example ref. 1).

Exhaustive search has advantages, though. When a problem
involves only a few parameters, exhaustive search provides a
complete view of parameter space; and provides an immediate idea
of the parameter confidence intervals, and how parameters are
correlated with one another. In 1978 I took a class in inverse
theory from the late, redoubtable, Gerald Hohmann. As a class
project I decided to examine the application of inverse theory to
a problem in astronomy; and, because the problem was non-linear,
I decided to use exhaustive search ^{2}. Hohmann chided
me at the time by saying that he learned more about astronomy
from my paper than he did about inverse theory; and, of course he
didn't learn any astronomy from my paper. However, this
particular application makes a more interesting story now. It is
a rare example of a problem in which the true parameter values
were unknown at the time I wrote the paper, but are now known
exactly.

Having accurate diameters of satellites of planets in our solar system is important for a variety of reasons, not the least of which is to test theories of their formation. However, diameters were difficult to measure accurately at one time simply because the satellites are far away and appear so small. Prior to space missions that could fly-by these satellites for a close look, astronomers used a variety of other means to make diameter measurements. One method put to use just prior to the Voyager missions making their way to Saturn, was to analyze light curves from lunar occultation of Saturn's satellites.

An occultation occurs when a nearby astronomical body eclipses
another one that is farther away. For example, earth's moon often
eclipses a distant star or planet. As the eclipse proceeds the
amount of light coming to earth from the distant body diminishes
steadily until the eclipse is total. On March 30, 1974 there
occurred an occultation of Saturn by the earth's moon.
Astronomers using the Mauna Kea observatory in Hawaii, observed
the occultation of Saturn and its satellites through two large
telescopes ^{3,4}. They obtained light curves by counting
the number of photons that arrived at a photomultiplier at the
focus of the telescope in 10 millisecond intervals. The
telescopes were 0.61m and 2.24m in diameter, which are so large
that diffraction of the satellite images from the limited
aperture was negligible. The moon was near first-quarter which
provided an extensive dark limb for the occultation. The
occultation occurred shortly after sunset with the moon quite
high in the sky. Thus, in all respects this occultation was ideal
for finding satellite diameters through analysis of the resulting
light curves.

A light curve in this context means the time series of photon counts in 10 millisecond intervals beginning at a time when a satellite is fully visible above the limb of the moon, until some time after it has disappeared completely behind the moon. Thus, the light curve exhibits a decline in intensity or photon count throughout the event. The time taken to decline from full intensity to minimal intensity is related to the diameter of the satellite. However, there are other factors to consider. Figure 1 shows the geometry of an occultation, and explains the meaning of a few factors.

Geometry of an occultation explaining the meaning of the angles e and a.

Satellites reflect light according to their size, the amount
of their image that is covered by the occulting body, and
according to the distribution of apparent brightness across their
surface. This last factor has a great effect on the diameter that
a person infers from the light curve. Most planets and satellites
are not uniformly bright from center to limb. One model to
describe this varying brightness is Minneart's law where the
brightness is proportional to

Cos^{2k-1}(e)

The parameter e is the
angle between the observer on earth and a vector normal to the
satellite's surface (see Fig. 1). Dead center on the satellite
e=0, and the scattered light is a maximum. At the limb of the
satellite e=PI/4, and the scattered light is zero. The angle a is
that between the earth and Sun as seen from Saturn. Obviously if
a is not zero, then the satellite is illuminated asymmetrically.
Saturn is so far from earth that a is very close to zero, and its
satellites appear almost fully lit at any time. At the time of
this particular occultation the satellites each had a small
crescent zone near one limb that was not lit; but, the resulting
distortion amounted to less than 0.3% of any satellite's full
diameter.

An occultation which is one-fourth completed, but which has less than a fourth of the satellite's face covered.

The parameter k in the Minneart law is called a limb darkening coefficient. If k<1/2 the limb appears brighter than the center of the satellite. At k=1/2 the satellite is uniformly bright like a flat disk. Lambert scattering corresponds to the case of k=1; and, a highly limb-darkened body is one for which k>1. There are other models of limb darkening, but this one describes ideal bodies well, especially for limb darkening coefficients between k=1/2 and k=1.0.

It is not easy to identify the duration of an occultation from the initial decline of the light curve to a point of complete eclipse; reason being that the event begins and ends gradually. It is more easy to identify a mid point for the event and produce a light curve from specified values of duration and k that best explains the observed light curve. Complicating this situation is the fact that the clock drives for telescopes run at a sidereal rate-the rate of the background stars. Thus, once the telescope is oriented to observe a particular satellite the dark limb of the moon slowly covers the field of view. This causes background light to increase with time.

Elliot considered the background brightness to increase linearly with time, and his photon counting plots for all satellites except Titan show exactly this. Titan, however, shows quite clearly a decreasing count before occultation and an increasing count afterward (see fig. 2 in Elliot I). Thus, two parameters may not be sufficient to completely describe background effects-a factor I will elaborate upon later.

Finally, to convert the duration of occultation to diameter of the satellite we need a few astrometric quantities, such as the distance from the earth to the satellite, the angular rate of the moon against the background sky, and so forth. Luckily these are known to great accuracy by other means.

In summary, occultation photon counts or light curves should depend on six parameters.

- The brightness of the satellite when viewed as a whole.
- The duration of the occultation.
- The limb darkening parameter.
- A constant and slope for the background light.
- The mid-time of the occultation (or any other suitable time reference.)

Elliot described his method as a least squares inverse procedure, seeking 6 parameter values by fitting data to a theoretical curve. However, the fitting wasn't done simultaneously for all parameters. Instead, Elliot found background light parameters from light curves that obviously precede and follow the occultation. The exact inversion method is unnamed but is a non-linear least squares program available at the time through the Cornell Computing Center (NLLS). The method was unstable if both k and duration were free to vary; so the strategy was to fix k arbitrarily and allow the program to find the best estimate of duration of the event by fit to the data. Elliot did this for a series of assumed values of k. Finally, Elliot considered photon counting to be his main source of random error, so he weighted photon counts by 1/count. Table 1, below, summarizes Elliot's results for satellite diameter and limb darkening coefficient.

Satellite | Diameter (km) | Limb Darkening |
---|---|---|

Titan | 5832 ± (53) | 1.25 (+0.75,-0.25) |

Iapetus | 1595 ± (139) | 0.75 ± (0.25) |

Rhea | 1576 ± (86)^{2} | 1.0 ± (0.5) |

Tethys | 1042 ± (115) | -NA- |

Dione | 825 ± (148) | -NA- |

Notes:

^{1}Elliot considered the 5832 km diameter of Titan as a
minimum, despite the (() figure, because larger diameters
actually provided a better fit.

^{2}A lower limit of 1325km for the diameter of Rhea was
available from its occultation of a background star in August
1974.

Since 1978 two Voyager missions have flown past Saturn,
Voyager I in Autumn 1980 ^{5} and Voyager II in Autumn
1981 ^{6}; and the Hubble space telescope has made
additional observations of Saturn's moons. We have very accurate
values for the diameter of the satellites. Table 2 shows actual
diameters as obtained from the Los Alamos National Laboratory web
page on Saturn.

Satellite | Diameter | k |
---|---|---|

Titan | 5150 | 0.5 |

Iapetus | 1460 | 0.75 |

Rhea | 1530 | 1.0 |

Tethys | 1060 | 1.0 |

Dione | 1120 | Large, 2.5 perhaps |

Note:The value of k in this table is the one that would produce the requisite diameter using Elliot's model and analysis.

Keep in mind that Elliot regarded the small satellites as dusty or frost covered bodies that would scatter light according to Lambert's law, and would each have a limb darkening coefficient close to 1.0. He also regarded Titan as highly limb darkened, and would have rejected a value for k of 1/2 as too small. A cursory inspection of the occultation light curve in Elliot's paper shows that k=1/2 does not describe the situation at all.

I analyzed only the data for Titan because it is the most detailed light curve of the set. Rather than use a non-linear inversion method, I chose to calculate the misfit for the entirety of parameter space by brute force. Figure 2 shows two of the parameters, k and duration, with contours of the sum of squared residuals. The residuals are related to confidence intervals; in particular the 90% and 99% confidence intervals are equivalent to the contours labelled 580 and 600, respectively.

Note that contours are elliptical in a slightly oblique direction. This shows that k and diameter are correlated, but only slightly so. Therefore, a person should be able to find values for k and duration independently of one another. By choosing to call the point at the minimum residual my best fit, I found that a diameter of Titan of 5686 (114km and the limb darkening coefficient of 1.75 (.15. This is a substantially smaller diameter for Titan than Elliot found, and lies below his estimated uncertainty interval. My estimate of the mid-time of the occultation is 0.07seconds later than Elliot's, which is to say about 3.5% later when compared to the 1.94 second duration of the event.

A cross section of parameter space for occultation duration against limg darkening coefficient. The best possible pair of parameter values are found in the center of the contours.

My estimate of k is tighter than Elliot's because I used the entire light curve for fitting to the data. The toe and shoulder of an occultation curve are much more sensitive to the value of k than is the central portion of the curve. More important, if a person uses the entire light curve to estimate k, then k and duration are independent of one another. By using only the central portion of each light curve to estimate k, Elliot caused k and duration to become highly anti-correlated with one another. This contributed to the instability that prevented Elliot from using k and duration simultaneously in his fit; and shows an advantage of exhaustive search.

Figure 3 shows ideal light curves for varying values of k. An inspection of these curves show that the light curve has more shape at its extremes, which makes the extremes more sensitive to the value of k. Note, too, that in the central half of the light curve there is almost no difference between increasing k and shortening the duration. This explains the anti-correlation of these two parameters that plagued Elliot.

A family of curves for various values of limb darkening coefficient. The duration is normalized to 1.0 in each case. Note how K and duration are anticorrelated between -¼ and +¼, and correlated outside that interval.

Why do Elliot's results fare so well for three satellites (Iapetus, Rhea, and Tethys) and fare poorly for the other two? The first issue is Titan, which is so large, and has so detailed an occultation light curve that the method should have worked well. It seems that there are three reasons for failure, here.

First, the model of a constantly increasing background light
curve for Titan is inadequate. The data do not support it. In a
letter that I sent to Elliot in the summer of 1979, I pointed out
that the optimum value for k found from the data depends very
much on the model of background light. That, in fact, reasonable
variations in the background light model could push the derived
diameter for Titan to values much greater than 5800km, which
Elliot in fact preferred at the time ^{2,3}, or to values
much less than 5800km, which it turns out is correct.
Geophysicists refer to inadequacy of a model as representational
error. A person has to avoid it in the analysis a priori, because
when there are many parameters in the model, correlated with one
another to some extent, a person cannot detected it from an
objective analysis of the data. Representational error is
completely unlike random error.

Second, Elliot used only the central portion of the light curve to find k, which makes k and diameter highly correlated. Figure 3 shows there is almost perfect anticorrelation between k and duration using only the central half of the light curve; but, using the entire curve decorrelates the two almost completely. Thus, an improvement was possible if Elliot had used a method which considered the entire occulation light curve. Exhaustive search had something to offer in this instance.

Third, Titan has a visible atmosphere that rises well above
its limb. Images from Voyager missions showed that the visible
image of Titan is 400km in diameter larger than its solid surface
^{6}. Thus, an estimate made from an occultation light
curve ought to be about 5550km, which is very close to the size I
determined. Titan's atmosphere actually extends 300 km above its
surface. It is tenuous at this height but loaded with enough
aerosol so that its light scattering may affect the occultation
light curve. Add 600km to the true solid diameter of Titan, and
the result is practically within the uncertainties of both
Elliot's estimate and mine. The occultation light curve provides
the diameter of the visible image, not necessarily that of the
solid surface.

The smaller satellites are another story. They have no
atmospheres to complicate the issue of their diameters, but they
are very complex bodies. They have highly cratered surfaces that
vary in brightness. Some of the brightness variation results from
eruptive products being distributed unevenly on the surface.
Additional brightness variation results from the way the
satellites orbit Saturn. Some of these satellites present the
same face to Saturn as they orbit. The side constantly facing
into the orbit picks up debris, which gives it different optical
properties than the trailing face. Iapetus is so extreme in this
regard that the albedo of the forward face is about one-tenth
that of the trailing face. Cassini claimed that he could see
Iapetus only when it was to one side of Saturn
^{7,8}.

Despite this complexity, Elliot's estimates of the diameters of each satellite except Dione were extremely good. Dione presents a very strange example. The leading hemisphere of Dione is heavily cratered and uniformly bright. The trailing hemisphere is very dark except for wispy material that brightens the center of the body. The image on the frontispiece of this paper shows the complexity of the trailing hemisphere. This bright wispiness is too subtle to resolve through the Moana Kea telescopes, but it was apparent in the occultation light curve. A bright center and dark periphery is precisely a highly limb darkened object. Thus, if Elliot had been willing to accept a value for k of 2.5, his estimate of Dione's diameter would have been very accurate. I have not performed the analysis myself, but exhaustive search may have had something to offer here, as well.

- Chisholm, T. and Chapman, D.S., 1992. Climate change inferred from borehole temperatures. JGR. 97, 14,155-14,176.
- K. Kilty. The Analysis of Occulation Light Curves. 1978. A paper for GG 628, University of Utah, Dept of Geology and Geophysics.
- J. L. Elliot et al. 1975. Lunar Occultation of Saturn: I. The Diameters of Tethys, Dione, Rhea, Titan, and Iapetus. Icarus, 26, 387-407.
- J. L. Elliot et al. 1978. Lunar Occultation of Saturn: II. The Normal Reflectances of Rhea, Titan, and Iapetus. Icarus, 35, 237-246.
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