The Rainy Lake Experiment was designed to show, how we can figure out the shape of the earth, Flat or a Globe, by observing and measuring a clever arrangement of targets over a distance of 10 km, taking terrestrial refraction into account and using modern equipment. The experiment is an advanced version of the Bedford Level experiment executed in 1838. The Experiment leads to the conclusion that the earth must be a Globe with a radius of 6371 km.

Overview

To come to a Conclusion the following requirements have been set:

Execution

Although the whole Experiment was mainly George Hnatiuk's work, Jesse Kozlowski and Soundly joined him in early April 2018 to make GPS measurements (Jesse), images and videos (Soundly) of the Experiment. Later Walter Bislin created a Computer Model of the experiment, visualized the data in his GNSS Data Viewer and wrote this reported.

Location

Rainy Lake is a relatively large freshwater lake (930 km²) straddling the border between the United States and Canada. Rainy Lake is part of an extremely large system of lakes forming the Hudson Bay drainage basin that stretches from west of Lake Superior north to the Arctic Ocean.

Img 1: Rainy Lake panorama on winter 2018 with Georges dog Khan at the snowmobile

During the winter months until May, the lake is covered by a meter-thick layer of ice. The lake was choosen for this experiment because George Hnatiuk lives right at the lake and is equipped with the tools and vehicles like a snowmobile necessary for the experiment. The lake provides an unobstructed path of about 10 km from George's home to a small island called Home Island, ideal for this experiment.

Img 2: Rainy Lake target locations

See Location Graph and Data for data assigned to the numbers.

Choosing Target Heights

The target center heights were chosen in such a way, that on each earth model one of two rows of targets will be at eye level of an observer while the other row curves up (Flat Earth) or down (Globe). The lower targets are called Bedford targets, the upper targets are called Tangent targets.

Img 3: Rainy Lake Experiment Design: Flat Earth side-view

The lower Bedford targets consist of a row of 6 targets, all 1.85 m above water level. If the earth is flat, all this targets will appear at eye level for an observer at 1.85 m height. The upper Tangent targets will curve up for an observer at 3.91 m height.

Img 4: Rainy Lake Experiment Design: Globe-Earth side-view

The upper Tangent targets consist of a row of 4 targets, increasing in height with increasing distance. If the earth is a Globe, all Tangent targets will appear in line with eye level for an observer at 3.91 m height. The lower Bedford targets will curve down for an observer at 1.85 m height.

To see how the targets were planted and measured, see Planting the Targets and Measuring the Targets.

Calculating Tangent Target Heights

The height h of the Tangent targets is the sum of the observer height h_o and the curvature drop x_i at the i-th target. On distances as small as 10 km the leaning can be neglegted. The target center height can be computed as follows:

Method used to Mount the Tangent Targets

The Tangent targets were first placed roughly at the pre-calculated target center heights. Then they were adjusted to the eye level height as seen through the auto level at the observer. Because the targets were adjusted at different times and days, refraction was slightly different for each target so the targets did finally not align perfectly. But the height deviations are within the calculated variations of common low refraction.

Calculated and Measured Tangent Target Heights

The Calculated Heights are calculated from the curvature drop of a sphere with radius R = 6371 km with a standard refraction coefficient of k = 0.17062 applied according to (1). The expected lift due to refraction for each target is shown in row Refraction Lift. The Lift depends on the Distance to the target.

Refraction_Lift = k * Distance² / 2R

In row Measured Height is the target center height with respect to water level at the observer, measured using GPS. I use the height with respect to water level at the observer rather than to water level at the target, because the targets were adjusted visually to the eye level at the observer. Measuring the Targets explains how the GPS Vectors to the target centers were measured.

Note: The water level elevation of the lake decreases about linearly with increasing distance from the observer. At the last target (7) the water level elevation is about 25 cm lower than at the observer with respect to the Reference Ellipsoid, due to variations in the gravitational field of the earth. This means that mean sea level is also 25 cm lower with respect to the Ellipsoid. The Geoid defines mean sea level with respect to the Reference Ellipsoid for each location on earth. See Obtaining Elevations for how the Geoid height is obtained.

The row Refraction at Adjustment shows the calculated prevailing refraction at the time each target was adjusted to eye level, using the measurements of the target center heights:

Refraction_at_Adjustment = 0.17 * (1 - Difference / Refraction_Lift)

The different prevailing refractions at the adjustment times plus the current deviation from k = 0.17 at an observation results in the apparent height variations.

The Computer Model uses the measured GPS Vectors for the predictions after transforming them into a local coordinate system at the observer, see Target Positions and Sizes relative to the Observer. Therefore if the refraction in the Computer Model is set to the same refraction that was present when a photo was shot, the photo and the Computer Model image should match exactly.

Size and Placement of the Targets

To optimize the visibility of the targets and to help better distinguish the targets, they are arranged in the following way:

Targets of the same size appear smaller with increasing distance (left image). To counter this perspective effect, the targets were built in such a manner, that their angular size at the observer is about the same. This way they appear the same size no matter how far away from the observer they are (center image). The target (7) is about 4 times larger than target (2), see Calculating Target Size. Target (1) and (6) are different for better recognition.

The targets are not placed in a straight row, so that they do not overlap each other as viewed from the observer (right image).

Img 5: Rainy Lake Experiment target design

The targets have two horizontal black or orange bars with a clearly visible gap between. A target is perfectly at eye level, if the crosshair of the theodolite or auto level is dead center in the gap.

Note: the slant of the orange target (6) is not intentional. It was damaged by strong winds.

Calculating Target Size

To achieve that all targets appear the same size, they must have the same angular size $\theta$ at the observer. The target size s then depends on the distance d from the observer according to the following equation:

See Target Positions and Sizes relative to the Observer for the calculated and choosen target sizes.

Data and Tools

Follow the links below for all data gathered and processed and the tools used:

Videos

See the following video documentations for the planing, execution and discussion of the Rainy Lake Experiment:

[1]

Rainy Lake Survey Progress Report
George Hnatiuk, Soundly, and Jesse Kozlowski get together for a hangout live stream to discuss the data and observations collected on Rainy Lake and plans for completing the work.
http://www.youtube.com/watch?v=kOqlAir8cdI

[2]

Survey Overview Rainy Lake - is the earth curved?
George Hnatiuk: This is the first part of a series of videos that will follow an ongoing set of measurements on a frozen lake in northern Minnesota to determine the contour of the earth's surface.
http://www.youtube.com/watch?v=GUO-mbxybVA

[3]

Laser with corner reflector prism
George Hnatiuk: This was a quick test to see how difficult it is to aim my laser to a corner reflector placed 1.36 miles away. The corner reflector effectively reflected the laser light back to me. The laser did not have a beam collimator attached.
http://www.youtube.com/watch?v=Y9CRO50F8Kk

[4]

Survey Update 1 --- Rainy Lake Minneosta
George Hnatiuk: This is the first of weekly updates for the survey on Rainy Lake in northern Minnesota which discusses the point of observation and placement of the reference poles.
http://www.youtube.com/watch?v=VxyGQyLaD-k

[5]

Survey Update 2 --- Rainy Lake Minnesota
George Hnatiuk: This update details the deployment of marker targets and reflectors for a laser to shoot and return light back.
http://www.youtube.com/watch?v=TcbgNDJanmw

[6]

Survey Update 3 - Rainy Lake Minnesota
George Hnatiuk: This update shows the deployment of the six mile target and discusses why a laser is not the correct tool for aligning targets separated by long distances.
http://www.youtube.com/watch?v=LwZuXuiMhOg

[7]

Survey Update 4 --- Rainy Lake Minnesota
George Hnatiuk: This video details the deployment of target poles and targets at 4 miles and 6 miles from the observation station which are used to survey the contour of the water surface of Rainy Lake in northern Minnesota.
http://www.youtube.com/watch?v=To6IscXbO3k

[8]

Survey Update 5 -- Rainy Lake Minnesota
George Hnatiuk: On a previous day, the target heights were adjusted to be centered about a tangent line set by an auto level. Some preliminary measurements are made on the 1.36 mile target (time 5:02) and Fransen Island target at 2.72 mile.
http://www.youtube.com/watch?v=WCL6SpsBzow

[9]

Survey Update 6 -- Rainy Lake Minnesota
George Hnatiuk: Setting the lower Bedford targets for a Bedford level type of test. All Bedford targets are set to 73" above the water surface, a constant height with distance from the spotting scope.
http://www.youtube.com/watch?v=NuIpxiqrqyg

[10]

George, Soundly & Jesse Talk About The Work Planned On Rainy Lake
Soundly and Jesse Kozlowski are traveling on 3/30 to meet George Hnatiuk. In this live hangout, the three will discuss their plans for the work to be performed conducting and recording surveying observations on the frozen Rainy Lake.
http://www.youtube.com/watch?v=lL3693c0RxI

[11]

GPS Signal Test
George Hnatiuk: This video is a simple demonstration that GPS receivers capture signals that originate from the sky above rather than ground based towers.
http://www.youtube.com/watch?v=YTOA5RrkBdE