Has anyone ever measured the one-way speed of light perpendicular to the Earth at the Earth's surface? 1 - Has anyone ever measured the one way speed of photons traveling perpendicular to the Earth at the Earth's surface?
2 - Given our current understanding of Physics is there any way both the upward and downward speed would not be $c$?
3 - If the measurement were made and the downward speed were found to be considerably faster than $c$, would there be any plausible explanation given our current understanding of Physics?
4 - If it hasn't been done in the past, how would one do so and how difficult would it be to make such a measurement in both the upward and downward directions with less than 1 km/s error bars?
EDIT:
Someone helped me find my error on another thread.
 A: GPS satellites orbit at an altitude of around $15\times10^6 m$. Sometimes they are directly overhead, sometimes they are nearer the horizon. An error of 11187 m/s in the speed of light, varying with direction, would show up as an enormous error in estimating position, rendering the currently used GPS computations completely useless.
A: Beside the method proposed above, we can actually observe the lack of discrepancy between the one-way and reflected speeds by observing distant sources of light that we receive as one-way and reflected light.
For example, There was a supernova (SN1987A) which exploded 168,000 lightyears from Earth in 1987.
Although no neutron star has been detected, the center of the supernova remnant regularly brightens and fades. About 2/3 of a lightyear away from the star is a ring of debris, which means it takes roughly 8 months for the light from the star to reach it.
As we observe direct light from the supernova remnant, the rings ebb and flow roughly 8 months later.
If there is a discrepancy between the two directions of light, it is not detectable.
A: Everyone seems to be misunderstanding the question. The one-way speed of light cannot be measured even in principle. Einstein knew about this and even proclaimed that the one-way speed of light is not a feature of nature but rather a human preference. What we know as the speed of light c is actually 1/2 of the "Two-way speed of light." The one-way speed of light is simply DEFINED to be c for the sake of simplicity but this is not necessarily so. The one-way speed of light can be DEFINED to be ANY constant number within a range of values as long as the opposite direction of light compensates so that the Two-way speed of light is c. Many gets confused by this because this is not mentioned in basic relativity textbooks. This, I believe, for the sake of simplicity but at the expense of knowing a very peculiar and interesting part of Relativity.
To directly answer the questions:
1) No, the one-way of speed of light cannot be measured whether downward, upward, or any direction. (for a general reference see: http://en.wikipedia.org/wiki/One-way_speed_of_light)
2) Yes. Special Relativity allows you to DEFINE a constant, one-way speed of light as long as the two-way speed of light remains c. For instance, you can define the downward speed of light to be greater than c but you would then have the upward speed of light to be less than c such that you will still get c as the two-way speed of light.
3) Our current understanding of Physics (Special Relativity) allows you to define the downward speed of light to be greater than c. Infinity is also possible (this means it takes 0 time for distant light to arrive to an observer) and have the upward speed of light to be 1/2 c. (The total time of travel of light back and forth would be 2t corresponding to two-way speed of light to be c). 
3) See answer 1.
A: I think we can measure one way speed of light.
we need to redefine simultaneity. My proposition is as follows:
if a rigid body AB of length l is moving without any acceleration parallel to X axis and at time t0 its point A is at location x, then simultaneously its point B is at location x+l
Let's now design the experiment to synchronize distant clocks and measure one way speed of light:
Imagine four spaceships flying as perfect square EFGH towards (or away from) not moving (at least relative to each other) points ABCD, where AD is parallel to EF and distance EF equals AD. Points EG should be collinear with points AB and points FH collinear with CD. Making sure that ABCD (and EFGH) is a square is relatively easy, since 2-way speed of light is constant:
we can measure (and correct, if necessary) distances BD and CA by sending light signals from B to D (and from C to A) and back
Now at certain time (clocks at A and D can be pre-synchronized using Einstein convention, but it is not absolutely necessary) we can measure distance from A to G (L) and from D to H (L’) using light (laser) signal send from A to G (and reflected back to A) as well as distance from D to H. If the distance AG (L) equals DH (L’) signals from A and D had been sent simultaneously; if not, it would be easy to adjust the clocks so they are synchronized.
Please let me know if I made any wrong assumption
Of course, theoretically it would be sufficient to have only the lines AD parallel to GH, but practically it could be difficult to make sure they are parallel to each other

