Genzod
Banned
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Determination of PERSONAL Minimum Required Flashlight Intensity to Identify an Appalachian Trail White Blaze on a Tree as a Function of Distance Utilizing only Stone Knives, Bearskins and a Nerdy Girl with Glasses
EDIT: At the time of this experiment, subject had 20/40 vision and amazingly wide 8.5 mm dark adapted pupils. That acuity has since been corrected. Also, it was later learned that the experimental curve derived here could be adapted to all pupils and degrees of acuity with a convenient two equation piece-wise function discussed here in post #76 of this thread.
PREPARE TO BE AMAZED! (Probably how it will go, too) :laughing:
Abstract
Fast-packing the Appalachian Trail at night over long distances between trail towns requires a reliable headlamp, preferably one with a large enough boost mode that is able to see an appreciable distance when identifying trail markers for navigation (a vertical blaze of white paint, 2 x 6 inches in size, placed about 2 meters high on a tree and spaced about every 140 feet/43 meters). When such a boost is unavailable on the primary lamp because the manufacturer of your favorite headlamp model lost their minds and nixed lithium ion support, procuring a compact, lightweight secondary flashlight that has enough throw to find trail markers is necessitated. (Nah, this experimenter really would have needed redundancy for his lighting needs, either way).
Throw performance of a flashlight is defined by intensity in lux at one meter (e.g., candela) and/or throw distance to 0.25 lux. A minimum requirement to identify white blazes at a distance of 140 ft /43 m is straightforward, but various factors like weather and foreground spill light can impede both flashlight performance and perception. So it was decided a minimum constraint for marker identification at 100 meters under ideal conditions would be expedient. This choice provides brighter, easier identification under normal conditions, while providing some reserve capacity for performance robbing conditions.
An experiment was devised to determine the experimenter's personal minimum useful intensity required to meet this goal. A flashlight of known intensity was set at various distances from target so as to create intensity at marker of 0.5, 1, 2 and 4 lux. At each intensity, the experimenter attempted to identify target marker at 100 meters. The marker was discernible at 4, 2, and marginally discernible at 1 lux. At 0.5 lux, the marker was not discernible at 100 meters but was marginally discernible at 75 meters.
Experiment establishes that 1 lux provided the most practical minimum cutoff intensity for target identification at 100 m. 1 lux at 100 m is 10,000 lux/candela at 1 meter or 200 meters throw to 0.25 lux. A flashlight meeting those minimum requirements should provide the experimenter useful throw for identifying white blazes on the Appalachian Trail.
At a later date, a more useful set of data was collected at equally spaced intervals of 30 meters from target up to 180 meters. This new data allowed investigation of a suspected acuity limitation that began to deviate the required minimum intensity vs distance curve from the square law curve after about 85 meters. The equal spacing of data also allowed for easier handling of the regression curve fit.
The regressed curve becomes a useful tool for predicting target identification with various sized scopes at distances beyond the 180 meter range of this investigation. Although not discussed in this post, later posts in this thread demonstrate that feature. It was determined that as a rule of thumb, 0.5 lux is a practical minimum intensity for identifying the given target at any virtual sighting distance from an observation distance of 200 meters.
Methods and instruments
A makeshift white blaze, 2 x 6 inches in size was carefully cut from the poster-board rigid page of a spent spiral notebook (Did I not tell you this was stone knives & bearskins?). The surface color was white having a cool white tint, not flat and not entirely satin in sheen, either, providing a highly contrasted and highly reflective target for emulating white blazes. Duct tape was looped and placed at three points covering the entire backside of the marker, and placed on a pine tree of about 8-10 inches diameter at a bend in a forested section of a very wide bike path. (I'm taking the 5th on where exactly this was).
THAT TICKET WILL
COST YOU..............
1 MILLION DOLLARS!
A Princeton Tec Quad of current make was utilized for illumination, having the following characteristics and performance:
Three modes: low, medium and high.
Max output: 78 lm.
Range (high): 50 m @ 0.25 lux.
Range (medium): 24m
Range (low): 17m
(Note: Specifications in manual are no longer valid for this lamp. Correct data was derived from product package labeling.)
Maximum intensity of 625 cd was determined from maximum throw (I=(R^2)/4).
The low mode was used to generate predetermined target intensities for this experiment. The low mode intensity of 72.25 cd at one meter is determined from the 17 m throw and the maximum throw (I=Imax*(R/Rmax)^2).
The headlamp was secured to a tripod and aimed at the marker so the most intense part of the spot illuminated the center of the marker. The following table provides corresponding distances for the four tested target intensities:
Table 1 Target intensities and corresponding distances of headlamp
0.5 lux....12.00 m
1.0 lux......8.50 m
2.0 lux......6.00 m
4.0 lux......4.25 m
1.0 lux......8.50 m
2.0 lux......6.00 m
4.0 lux......4.25 m
A set of three fresh AAA alkaline Duracell batteries were used for this test. The lamp on low mode has 10 hours of regulation, and the lamp was allowed to stabilize output before attempting to identify target.
A time of night was selected providing no moon and minor starlight due to high altitude haze (June 13, 2017 between 10:50 and 11:50 PM EDT). Temperature was about 80F and humidity was high at 71%. No rain, no fog, NO SKITTLES. Location had little ambient light and even less in the area where the tree with the marker was isolated by forest. Closest street lights impacting viewing zone while discerning target were 620 meters to the right of viewer. A hand was used as an effective visor to eliminate glare in peripheral vision due to the distant street lamps. The target marker was isolated by trees and bushes on all but the immediate viewing side and was not directly impacted by street lighting or residential light. The marker could be marginally discerned with the naked eye at 30.4 meters in this ambient light. The fox that passed by the target tree did not run off with the blaze marker, although, he may be the reason for the absence of ducks at the pond this year.
mmmmmm.....SNACKS!
The lamp on the tripod was set at a predetermined distance, turned on to low mode, aimed and locked, then the experimenter walked quickly to the 100 meter viewing line to identify target, starting with 4 lux and working down to 2, 1 and 0.5 lux. Identification at 100 m was successful for 4, 2 and 1 lux and unsuccessful at 0.5 lux. Identification with 0.5 lux was restored at 75 meters. The identification at 1 lux was considered marginal.
A 50 something year old male with corrected vision was utilized as a testing instrument. This instrument was ideal in that the investigation was conducted for the purpose of determining a useful minimum intensity at 100 meters as a constraint in procuring a magical flashlight for his own personal use. (And no, you still can't borrow it.)
Taken under advisement, a 40 something year old woman with corrected vision (allegedly superior to the geriatric male experimenter's vision) was asked to view the 0.5 lux case at 50 meters and 75 meters to render moot the assertions of love seat rending, irritable puppies once and for all. Identification was successfully made at 50 meters. At 75 meters, identification was successfully made but described by the woman as "marginal", just as it also was for the male experimenter--proving once and for all, that a man with a woman need not go blind. :huh: (This experimenter sees extremely well.)
No scopes or binoculars were used in this target identification. But please, do keep digging for a way to render this effort nebulous and pointless, I'm sure you'll come up with something. You did get an early start, after all.
Discussion and Conclusions
An intensity of approximately 1 lux was determined to be a boundary constraint for target identification at 100 m for the eyes of the experimenter. (Your results may differ.) Required intensity at target appears to closely approximate a quadratic function of the distance within the range of 0-88 meters. The farther the eyes are from the target, the higher the intensity at target is required in accordance to the square law.
After 88 meters the intensity as a function of range to target curve deviates noticeably from the square law primarily due to the acuity limitations of the observer (20/40--max range to acuity limit for this target is 88 m). Although suspected at first, there is no asymptote the required intensity approaches implying a limit to distance where no amount of light will resolve target.
Although one might imagine that looking for a white blaze marker is like hunting for chipmunks that move in and out of holes and blend in with their surroundings, the white blaze target is highly contrasted, easily anticipated, found typically on stationary trees (when they aren't being chased by a pack of marauding termites) spaced every 140 feet or so at an elevation of about 2 meters and is found more often than not along an easily identifiable, well worn path. But, if you still fancy the marker being an elusive chipmunk, don't worry--I won't interrupt your delusion. 1 lux as a limit will be fine for my purposes. (Your purposes may be different).
RUN, FORREST, RUUUUNNNN!
Ambient light: On June 11th in the same test location, phase angle of moon was about 27 degrees and Z = 39. Moonlight intensity was calculated at 0.11 lux The ambient light impinging on the isolated marker was substantially darker in comparison. On the following two test nights without moon, the marker was isolated in even less light. Distance for marginal marker visibility was measured on two moonless nights as 30.4 meters. Whatever the intensity of the ambient light impinging on the marker, it was perceived to be substantially less than the moonlight at 0.11 lux plus any background ambient light from the city. So ambient light impinging the marker was much, much smaller than the 0.5- 4 lux added by the lamp.
Perhaps a lumen stud could enlighten us on what that value is based on the distance needed to marginally identify the target without lamp. Go, Speed Racer, go! (In the mean time, see below for how I finally backed it out of the collected data.)
A new data set was collected on July 25th, 2017, 2-3:30am EDT. Crescent moon on opposite side of earth, no rain, very little cloud cover, no haze or pollen, no foxes, no Clint Eastwood running cattle between observer and target, no Skittles. A rectangular box visor was used this time to block the effect of ambient light above tree level, residences and street lights in the distance. Nevertheless, the new data was very similar to the old data with the exception that investigated range to target was extended from 100 meters out to 180 meters. The equally spaced data points made for better regression curve fit determination. Differences in regressed curves between old and new data were primarily due to the spacing of data points and initially imposing incorrect boundary conditions on the data during the regression. Those issues have been resolved and updated.
Table 2 Marginal target identification intensities* (less ambient light) and corresponding distances
..............DATA...............
Ia + 0.000 lux........30 m
Ia + 0.271 lux........60 m
Ia + 0.739 lux... ... 90 m
Ia + 1.640 lux .....120 m
Ia + 3.770 lux......150 m
Ia + 8.800 lux......180 m
Ia + 0.271 lux........60 m
Ia + 0.739 lux... ... 90 m
Ia + 1.640 lux .....120 m
Ia + 3.770 lux......150 m
Ia + 8.800 lux......180 m
*Ambient intensity at target, "Ia" is unknown at this point.
ACHTUNG! The data above represents the intensity from the lamp ONLY. It is not total intensity. Ia was later placed as a variable in the above data due to the fact that this forum is PREGNANT with looking for any false appearance of loose threads to pull to blow off neurotic steam. Future assaults by neurotic combatants vill face military tribunal und be shot.. den zent to zee Russian Front!
Ambient light intensity impinging target was determined by imposing the boundary condition I'(0)=0 and I(0)=0 on the data set and regressed curve. These constraints simply mean the minimum required light to see target approaches zero as the distance to target approaches zero. Since there is no other light but ambient in the interval between zero and 30 meters, and ambient light is the minimum required light to identify target from 30 m, less light than ambient is required at distances less than 30 m. The trick now is to guess Ia and bend the regressed curve with guesses for Ia until the shape of the curve at range equals zero has met the condition I'(0)=0
It is important to differentiate between ambient intensity and minimum required intensity at distances less than 30 meters, as ambient light is constant in the interval (lamp is off and not needed here), but minimum required intensity drops off as the target is approached from 30 m. The minimum required intensity curve from 0-30 m should theoretically follow a square law with approximately zero intensity at zero meters and ambient intensity at 30 meters--And indeed, as shall be seen, it does.
To find ambient intensity, a guess for ambient is provided as I(0)=-Ia. The data set of table 2 and this guess are regressed in a quintic polynomial regression (5th order polynomial), and the derivative is checked to see if I'(0) = 0. This involves monitoring the shrinking of the constant of the 1st order term of the polynomial with each progressive guess until it approaches approximately zero. Ia=0.085 lux when that requirement is accomplished. Once the curve has the proper shape with I'(0)=0, the constant term of the polynomial is truncated from the function, vertically moving the curve up the value of Ia so that the curve intersects the origin (0,0). The resulting curve represents all values for minimum required intensity to identify target at distances from 0 to 180 meters. Then we add the ambient intensity of 0.085 lux to each of the 6 collected data points and superimpose the curve over the data points.
The value of Ia makes sense as it is a small fraction of moonlight intensity plus background intensity (there was no moonlight during this data collection, therefore Ia is background from suburban light pollution and anything else in the atmosphere at the time--stars, light reflecting off clouds, etc--approximated as 0.39 lux*). Ambient light impinging on the target blaze in the shaded portion of the test area is 0.085 lux which is about 22% of the local suburban light pollution.
*Two nights prior to the first data collection, the moon was 39 degrees above horizon and estimated from a chart to be about 0.11 lux. Max sighting distance with only ambient light was found to be 34 m. On two test nights that followed much later with no moon, max sighting distance with only ambient light was 30 m. Unshaded suburban light pollution in the test area, P, can be backed out of the intensity square law equation (0.11+P)/P=(34/30)^2. P becomes 0.39 lux.
Boundary conditions: I'(0)=0 and I(0)=0
Table 3 Marginal target identification intensities and corresponding distances (with ambient light added)
...........DATA...............
0.000 lux.........0 m
0.085 lux*......30 m
0.356 lux........60 m
0.824 lux... ... 90 m
1.725 lux .....120 m
3.855 lux......150 m
8.885 lux......180 m**
0.000 lux.........0 m
0.085 lux*......30 m
0.356 lux........60 m
0.824 lux... ... 90 m
1.725 lux .....120 m
3.855 lux......150 m
8.885 lux......180 m**
*Artificial, guessed value used as a "bending point" to force and maintain a horizontal tangent in the regressed curve at point (0,0), i.e. I'(0)=0 . True value of ambient light is Ia=0.085 lux, which comes from the red regressed quintic (5th order) polynomial curve fit.
**EDIT: This one data point was in error due to the fact the correct location down range for marginal identification for 8 lux+ambient on target was over the surface of the lake. I apologize for not only being unable to walk on water like Jesus, but for also fudging this point by making an estimation instead of tossing it out altogether--be assured, it caused me no little grief. I discuss this in greater detail in post #76 where I describe the physics equation governing requirement for increased intensity beyond the acuity limit.
**EDIT: This one data point was in error due to the fact the correct location down range for marginal identification for 8 lux+ambient on target was over the surface of the lake. I apologize for not only being unable to walk on water like Jesus, but for also fudging this point by making an estimation instead of tossing it out altogether--be assured, it caused me no little grief. I discuss this in greater detail in post #76 where I describe the physics equation governing requirement for increased intensity beyond the acuity limit.
Red Curve Regression:
Minimum Intensity to ID Target (lux) Vs. Observation Range to Target (meters)
I(x) = 7.3990e-05*x+6.5004e-05*x^2+1.4421e-06*x^3-2.1628e-08*x^4+1.1145e-10*x^5
(Intensity values valid only in the range from 0 to 180 meters. Notice I(0)=0 and that I'(0)<0.0001 which for our purposes is approximately 0.
PLOT*
We can determine from the plot, utilizing the point tool, that required intensity of the lamp at 100 meters is 1.0512-0.085 (ambient) = 0.9662 lux, which is consistent with 1 lux determined in the first data set, a deviation with relative error of only 3.4%.
I(x) = 7.3990e-05*x+6.5004e-05*x^2+1.4421e-06*x^3-2.1628e-08*x^4+1.1145e-10*x^5
(Intensity values valid only in the range from 0 to 180 meters. Notice I(0)=0 and that I'(0)<0.0001 which for our purposes is approximately 0.
PLOT*
We can determine from the plot, utilizing the point tool, that required intensity of the lamp at 100 meters is 1.0512-0.085 (ambient) = 0.9662 lux, which is consistent with 1 lux determined in the first data set, a deviation with relative error of only 3.4%.
*Note: The blue curve is the square law curve for intensity and distance. The square law appears to be valid up to about 80-85 meters, which in terms of visual acuity represents a focus of 12 one inch square pixels filling the 2x6 inch target for a person with 20/20 vision, three 2x2" pixels for 20/40 vision which I had at the time of the data collection. As the target shrinks smaller than the 3x1 pixel resolution and focus blurs, the regressed curve increasingly deviates from the square law curve. At increasing distance, the target blends into the tree and dims, becoming a point source of light, ever diminishing.
EDIT: There is no vertical asymptote at a distance that no amount of light will resolve the target. I had suspected that at first, but it is not the case. Once the target rectangle slips inside the pixel dimension of 2x2", distance progressively dims the pixel, which stays the same size relative to the viewer.
Compact Flashlight: The 0.966 lux at 100 m constraint requires the performance of a flashlight with at least 0.966x100x100 = 9660 candela. (See plot of Range vs. Intensity for constant target lux of .5, 0.71, 1, 1.41) Since foreground spill can constrict pupils and require more intensity at target to identify it, it is necessary to minimize spill with a deep reflector, narrow beam spot, or impeding the spill light in some way without altering the spot, such as using a tube extended off the head of the lamp or using a sighting tube to isolate the pupil.
The Manker T01 II seems well suited for the job, capable of 10,000 cd with a primary AA (same battery as my headlamp) and 20,000 cd with an IMR battery for even greater throw. The Manker T01 II has output settings that make it also useful as a back up running light. Plus, it's empty weight is only 55 grams, which is desirable when wanting to keep things light running up hills.
Manker T01 II
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