Comparison of Calculated and Measured View Parameters From Vetter Mountain

Description of the Osborne Firefinder
Data and Analysis
   Data-Taking and Preliminary Analysis
   Detailed Analysis


I have compared theoretical predictions for azimuth and elevation angles for peaks as seen from Vetter Mountain Lookout Tower to observed values obtained with an Osborne Firefinder. After correction of the data to remove instrument calibration errors, there is close agreement between the predicted and observed angles, with a one sigma scatter of 0.14° separately for both the azimuth and elevation comparisons. This close agreement makes the identification of peaks unambiguous, revealing one previously-accepted peak identification that was erroneous.

Other items revealed by this analysis:


Several years ago, I began calculating the Distances, Azimuths and Elevation Angles Of Peaks as seen from any given location. This was in response to the notoriously difficult problem of answering the question: "What peak is that?" when one is hiking.

Although the calculations are quite straightforward, there are always a near-infinite number of ways that one can make a mistake. Thus verification is an important step for any theoretical prediction. Experimental confirmation of predictions ensures that the foundation of the theoretical calculation is correct and that no errors have been made in carrying out the calculation.

Unfortunately, it is quite difficult without surveying instruments to measure the predicted elevation and azimuth angles accurately enough to check the predictions. In order to provide a useful check, these angles must be measured to tenths of a degree, far beyond the ability of most consumer-grade instruments.

Fortunately, the San Bernardino National Forest Association has begun a vigorous program to rebuild, restore and reopen the Fire Lookout Towers in both the San Bernardino and Angeles National Forests. The restoration includes a wonderful piece of equipment called an Osborne Firefinder, an instrument that measures the azimuth and elevation angles to a given point. That instrument is perfect for verifying the predictions for the view parameters of azimuth and elevation angle made here. The Association kindly allows visitors to observe with the Osborne, making the work reported here possible.

This analysis was performed primarily in order to verify whether the theoretical predictions were correct. In order to do so, it is necessary to calibrate the Osborne instrument, and this calibration is presented here. The analysis also gives the accuracy of the comparison, and compares it to the expected accuracy. As always, a number of other interesting things appeared in the course of the analysis.

Description of the Osborne Firefinder

Some of the parameters for the Osborne instrument itself reported below are estimates, which need to be verified with a repeat visit to Vetter.

The main body of the Osborne consists of a metal horizontal ring about 85 cm (33.5") in diameter which rotates about a fixed base. Two items are mounted vertically at opposite end of the rings: a double cross-hair and a calibrated sighting pinhole. The double cross-hair has guides to precisely hold two horizontal horse-hairs and one vertical horse-hair, forming two cross-hairs. The upper cross-hair is used to site positive elevations, and the lower one is for negative elevations.

The fixed base is marked with the azimuthal angle in degrees and half-degrees. A vernier scale allows the azimuthal angle to be read with TBD precision. Since 0.5° corresponds to ~0.4 cm (0.14"), it is easy to read the angle to a precision of ~±0.05°, one tenth of the smallest marked division, even without the vernier scale provided. For comparison, the angular width of the horse hair is ~0.014° (0.02 cm at a distance of 85 cm). Thus the Osborne is a very precise instrument, easily measuring an angle to within about three times the width of a horse hair seen at 85 cm!

For this initial investigation, I did not use the vernier scale and simply took quick readings accurate to ~±0.05°.

The sighting pinhole is in the middle of a metal slider mounted on two vertical posts marked with the elevation angle in steps of 10'. Although it is relatively easy to read the position of the metal slider to about one tenth of the smallest division, ~1', or ~0.02°, the pinhole severely restricts both the field of view and the amount of light transmitted to the eye. Hence it is likely that the fundamental limitation for elevation is not the instrument itself, but on the ability to gauge whether the cross-hair and the peak are aligned. Thus the error in the observed elevation angle depends on the atmospheric transparency at the time of observation, as well as the contrast of the peak being measured against its surroundings. In the two times I have used an Osborne, there were a handful of cases where I could clearly discern the location of a peak with my naked eye, but could not do so through the pinhole.

The fixed base has two attachments that allow it to mount on parallel rails on a cabinet at the center of the Lookout. There are three parallel rails, which allow the Osborne to be mounted on either the two northernmost rails or the two southernmost rails. It was noted which data were taken in which configuration.

In either mounting, the Osborne can slide along the rails. The two mounts and the ability to slide allow the Osborne to observe without the obstructions caused by the walls between the windows. If the rails are not perfectly parallel, there will be a bias in the azimuth and elevation angle that depends on the position along the rails. However, for this initial investigation, I did not keep track of where along the rails the observation occurred.

Data and Analysis

Data-Taking and Preliminary Analysis

On 6/22/00, with the assistance of Jane Strong, the volunteer Lookout Host, I used the Osborne Firefinder at the Vetter Mountain Lookout Tower to measure a number of peaks visible from Vetter. The opacity of the atmosphere was fairly high since the monsoon had just begun, rendering a number of possible peaks invisible or hard to measure. In addition, there were clouds building to the east, making it impossible to measure the elevation of Mt. Baldy, for example. Nonetheless, I was able to measure a large number of peaks and produce a rich data set.

Before taking data, Jane and I installed a new vertical horse-hair and tightened it to provide good cross-hair positions.

Beginning in the north and continuing east, I attempted to measure every peak that stood out as a feature on the horizon. The only notable peak I missed was Josephine, a literal oversight on my part.

The initial comparison of the raw observed azimuth angles with the predicted ones immediately revealed that the Osborne is a quite precise tool. There was no ambiguity at all about whether an identification of a measurement with a given peak was correct. Every measurement except one agreed within the predicted azimuth for each peak to within ~0.6°. The exception was "Winston Peak", where the difference was 4.55°, which clearly ruled out this identification. The measurement was easy to associate instead with Pallett Peak, whose theoretical azimuth agreed with the observed azimuth to within 0.24°.

The mean of all the azimuth differences was about 0.1°. Since it was expected that there would be some error in the alignment of the Osborne with true North, I subtracted that mean from the raw observed azimuth angles to produce corrected observed azimuth angles. In the following, I define the delta azimuth angle to be the corrected observed angle minus the predicted angle.

I then examined the outliers (those points with the largest delta azimuth angles). Much to my surprise, in every case the source of the error was a poor position for those peaks used in the theoretical prediction. The position I used for the peaks came from the USGS Geographic Names Information System, which gives positions quoted to the nearest second of arc. Using the position derived by using the Topo! software significantly decreased the error in every case.

I verified that this was not due to mixing coordinates from different Datums by comparing the USGS coordinates to two sets derived from Topo!, one using the NAD27 and one using the WGS84. (A Datum is the reference system to define the origin of the latitude and longitude system. The two main standards are the North American Datum defined in 1927 and the World Geodetic Standard defined in 1984. The average difference between them for these peaks in the San Gabriel Mountains is ~20 feet in latitude and 300 feet in longitude, with NAD27 having the higher latitude and WGS84 having the higher longitude value.) The analysis showed that the USGS coordinates use the NAD1927 Datum, since only in that system was there no bias between the USGS and the Topo! coordinates.

Although there was no bias in the USGS coordinates, there were errors of up to ±5" in both latitude (~500 feet) and longitude (~400 feet). This error is much larger than the error in locating the peak using Topo!, which was at most 0.00005°, equal to 0.2", since I was careful to use only peaks whose positions were marked on the USGS 7.5' Topographic Maps.

It turns out that this error of ±5" is indeed stated as the accuracy of the USGS Name Server coordinates deep within their documentation. However, I, like many other users, expected that the maximum error would apply to peaks that are not precisely located on the USGS topographic maps. It is surprising that the USGS, producers of very accurate maps, would report such low-accuracy coordinates without a clear up-front statement about the accuracy.

Hence for an accurate comparison, I rederived coordinates for all the visible peaks, and for the Vetter Mountain Fire Lookout, by measuring their locations by hand using Topo! with the NAD27 Datum (the same as the 7.5' USGS Topographic maps). The Predicted Azimuth and Elevation Angles were derived as explained previously.

In addition to the misidentification of Winston, there were three peaks whose true identifications were not known to us before this work. One of these was at a slightly higher azimuth angle than Twin Peaks, which turned out to be 6834, a peak southeast of Twin Peaks. The other two were neighboring peaks of Condor Peak, which turned out to be Fox and 5260, a peak west of Condor Peak.

With the new theoretical angles and a complete list of measurements for identified peaks, I then performed the detailed analysis below.

Detailed Analysis

The Table of Raw and Corrected Observed Data shows the raw data, the observations corrected as specified below, and the theoretical calculations just for the measured peaks.

The peaks in the data table that are within 26 miles of Vetter are plotted, with the peaks separately shown for those with positive and negative elevations. (This plot shows the actual elevation angle, not the residual error in it.) A few peaks that are not visible from Vetter (Mt. Williamson and Mt. Hawkins) are also shown to help deduce what foreground peaks block their observation. This plot shows that the nearest peaks to Vetter Mountain are ~five miles away, with mostly higher peaks to the north and east and mostly lower peaks to the south and west.

Consider first the azimuthal values. The measured azimuths agree well with the predicted azimuths, as mentioned above. Since most of the data were taken using the northernmost set of rails, the average difference between the observed azimuth and the predicted azimuth for that subset of the data was taken to be the error in orienting the Osborne finder to true North. That average difference was 0.12°. Hence I corrected all measured azimuths (including the southernmost data) by subtracting 0.12° from them, producing the "corrected azimuth angles".

The value of 0.12° corresponds to 0.2 cm (~0.1") along the circumference of the Osborne, assuming a radius of 85 cm. This error is a rotation of only the diameter of nine horse hairs (18 human hairs) at the circumference of the wheel! It is very impressive that the Osborne was lined up to true North to that accuracy.

After correction, the differences between the observed and predicted azimuths are quite small, falling in the range of ±0.2° except for a single measurement of the westernmost part of Twin Peaks. Since the data taken using the southernmost set of rails fit well with the northernmost data, the bias of 0.12° apparently applies to the southernmost set as well.

The 0.14° scatter of the delta azimuth angles is much larger than the value of 0.04° expected from the root sum square combination of input coordinate errors (~0.02° for azimuth) and the precision I used in reading the azimuth angles from the Osborne (±0.05°, equivalent to a sigma of 0.03°). (The scatter is defined as one sigma, the average value of the squared delta angles.) This larger scatter is caused by a trend in the residuals seen clearly in the plot. Nearly all are positive for azimuths of 50-220° and nearly all are negative for azimuths of 230-360°. This pattern is not correlated with elevation angle, as seen from examining the peak location plot mentioned earlier. Investigation into the cause of this pattern will be done next.

Revision and additions still needed below here.

Consider next the elevation angles. A plot of the measured vs. predicted elevation angles reveals that the raw relationship is not as good as it was for the azimuths. Plotting the differences between the observed and predicted elevation angles vs. either the predicted elevation angle or the observed elevation angle immediately reveals that the difference linearly depends on the elevation angle. (The three points at the extreme right makes it look like the relationship becomes more constant at high elevation angles. Although this may be the case, the three points are consistent with a linear trend within the accuracy of the data.)

This linear trend implies that the Osborne has a calibration error, with the scale marks for the elevation angle having a ~16% error. Since the marks are spaced by roughly a couple of mm, it is not surprising that their spacing might be in error by 0.3 mm (0.01"). This error should be relatively easy to verify with measurements made observing a yardstick at the Lookout.

The correction applied is:

elevation anglecorrected = 1.16 * elevation angleraw - 0.089°

Thus the Osborne reads correctly at an angle of 0.56° ( = 0.089/0.16), with larger or smaller angles being compressed to values closer to 0.56°, needing the above correction to make them accurate.

Using the corrected elevation angles then produces the following residuals vs. predicted elevation angle. (The residual here is the delta elevation angle, equal to the corrected elevation angle minus the predicted angle.) The residuals are now unbiased, and quite good, with most falling in the range of ±0.2°.

The 0.14° scatter of the residual elevation angles is also very much larger than the value of 0.01° expected from the combination of input coordinate errors (~0.001° for elevation) and the precision in reading the elevation angles from the Osborne (±0.02°, equivalent to a sigma of <0.007°). This is probably simply due to the difficulty in precisely aligning the cross-hair with the position of the peak being sighted.

Finally, I can check to see if there is any correlation between the azimuth error and the elevation error. This might happen, for example, if the vertical cross-hair piece is not exactly vertical. A plot of the elevation vs. azimuth residuals reveals a satisfying scatter plot showing no observable trend. The histogram of the residuals also looks good.

(more to come)

A curious effect still remains in the data that may indicate that the observation position is slightly but significantly different than derived from the USGS topographic maps.


Future work: measure scale error directly, take repeated observations to determine the consistency errors, determine effect of moving the instrument off the mountain (thermal effects?).

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Copyright © 2000 by Tom Chester.
Permission is freely granted to reproduce any or all of this page as long as credit is given to me at this source:
Comments and feedback: Tom Chester
Updated 28 June 2000.