*** THE CACHE IS NOT AT THE POSTED COORDINATES ****
Global Positioning System
The GPS consists of three parts: the space segment, the control segment, and the user segment. The U.S. Air Force develops, maintains, and operates the space and control segments. GPS satellites broadcast signals from space, which each GPS receiver uses to calculate its three-dimensional location (latitude, longitude, and altitude) plus the current time.
The space segment is composed of 24 to 32 satellites in medium Earth orbit and also includes the boosters required to launch them into orbit. The control segment is composed of a master control station, an alternate master control station, and a host of dedicated and shared ground antennas and monitor stations. The user segment is composed of hundreds of thousands of U.S. and allied military users of the secure GPS Precise Positioning Service, and tens of millions of civil, commercial, and scientific users.
A GPS receiver calculates its position by precisely timing the signals sent by the GPS satellites high above the Earth. Each satellite continually transmits messages which include
• the time the message was transmitted
• precise orbital information (the ephemeris)
• the general system health and rough orbits of all GPS satellites (the almanac).
The receiver utilizes the messages it receives to determine the transit time of each message and computes the distances to each satellite. These distances along with the satellites' locations are used with the possible aid of trilateration, depending on which algorithm is used, to compute the position of the receiver. This position is then displayed, perhaps with a moving map display or latitude and longitude; elevation information may be included.
Three satellites might seem enough to solve for position, since space has three dimensions and a position near the Earth's surface can be assumed. However, even a very small clock error multiplied by the very large speed of light—the speed at which satellite signals propagate—results in a large positional error. Therefore receivers use four or more satellites to solve for the receiver's location and time. Since the GPS receiver cannot handle precise times in nano-seconds, it will always calculate the time by correcting the clock with the 4th satellite. If the calculated distance to the 4th satellite from the calculated 3-satellite position calculation is more than the calculated distance to the 4th satellite from its time information, then clock is delayed. If it is less, then the clock is advanced. This 4th satellite requires that the GPS receiver clock be advanced and delayed several times until all 4 satellites converge into one point and get a more accurate reading. The clock correction requires some iteration but it is one-variable iteration because it solves for the position from the 3 satellites directly with tiraleration and the receiver time with iteration. When using more than 4 satellites most receivers use a Newton-Rapson multi-variable iterative method to find the most accurate location and clock correction. So when you have 9 satellites (9 equations) and you are solving for four variables (3 for position and 1 for time) you have more equations than variables and your position will be more accurately calculated even if one of the readings is off.
To find this cache, you are going to become a GPS mathematician. For the purpose of the simplicity and accuracy of this puzzle, we are going to assume that the GPS receiver’s time has already been corrected by the fourth satellite so there will be no time-based errors in calculating the distance from the cache location to the satellites to 5 decimal places of a mile. So no iterations are needed here. With these precise distances here, there is no need for more than 3 satellites to calculate the position. However, there are 4 satellites posted here so that you can verify your math and select any 3 of the 4 satellites to compute the cache location.
The satellites follow specific orbits, but since this is my puzzle cache, I placed the positions of the 4 satellites wherever I wanted to. I decided to place them to be exactly above islands that are part of the United States.
Satellite 1:
Flying directly above the Statue of Liberty in Liberty Island, NY
Location = N 40° 4.354’ W 74° 2.677’
Orbital radius = 16,400 miles
Distance to cache = 12,477.53873 miles
Satellite 2:
Flying directly above the little Alaskan Diomede Island in the Bering Strait.
Location = N 64° 45.223’ W 168° 55.317’
Orbital radius = 16,700 miles
Distance to cache = 14,512.46282 miles
Satellite 3:
Flying directly above a crater on Mt. Kilauea in the Island of Hawaii.
Location = N 19° 24.4’ W 155° 17.004’
Orbital radius = 16,600 miles
Distance to cache = 15,293.42569 miles
Satellite 4:
Flying directly above the building where I grew up as a kid (my mom’s residence) in the US Island of Puerto Rico.
Location = N 18° 24.996’ W 66° 6.45’
Orbital radius = 16,500 miles
Distance to cache = 13,137.03211 miles
By selecting 3 of the four satellites above you can calculate the exact location of the cache by finding the intersection point of the three spheres with centers on each satellite and radius of their distance to cache.
The first two spheres intersect at a circle. The third sphere intersects that circle at two points, one of which is the location of the cache (the one closest to Earth)
Latitude, longitude and distance from center of earth are spherical coordinates with the origin at the center of the Earth.
You need to convert latitude and longitude to decimal degrees and then to radians.
Then Transform the spherical coordinates of all satellites to Global Cartesian x, y, z coordinates in miles with (0,0,0) at the center of the Earth; x-axis from the center of the earth and both latitude and longitude of zero values; z-axis is from the center of the Earth to the North Pole.
Then you can solve for the position of the cache using TRILATERATION. Search for the term to know how to use it. Trilateration is what most GPS System uses to calculate your position.
To use trilateration effectively you need to transform all the coordinates from global to local. The new local coordinates will be with the 0,0,0 at one of the satellites, its x-axis in the direction of a second satellite and the z-axis perpendicular to the plane that contains the 3 points of the 3 satellites. To find this local coordinate system you must know how to use vectors and cross product calculations and to create a rotational matrix from the three unit vectors that define the local coordinate system. Search on how to transform from global coordinates to local coordinates. The reason for which we need a local coordinate system is because the equations needed for trilateration are based on three spheres all with centers in an x-y plane with one center at 0,0,0; another center along the x-axis and the 3rd center anywhere in the x-y plane. This is done so that no iterations are needed to get x,y,z in local coordinates.
Once you transform all points to that local coordinate system, you use trilateration formulas to calculate the cache position in x,y,z at this local coordinate system (not at the global)
Then you transform the cache position from the local back to the global Cartesian coordinate system.
Then you convert from global Cartesian to spherical radians, then to latitude and longitude decimal and then to latitude and longitude in degrees and minutes.
The math for this puzzle is intensive but fun. You will learn some of what your GPS unit does.
I verified this puzzle using the trilateration method for all different permutations of 3 of the 4 satellites and it works exactly for the cache coordinates in the geochecker to the 3rd decimal place of the minute.
There are other methods to solve this but are not necessarily the ones used by your GPS unit.
Have fun with my puzzle and learning about the math behind GPS.
Leave cache exactly as found.
FTF will get TWO kart races.
You can check your answers for this puzzle on Geochecker.com.
Congratulations to Band of Hawks and Uncle Mike 2 for this very well done FTF!