Those puzzles may be explained by Mars' other long-term cycle of climate change: the rhythmic changes-- thanks mostly to the long-distance gravitational tuggings of Jupiter -- that it undergoes every hundred thousand years or so, as its "obliquity" (its axial tilt) gradually rocks from zero degrees up to levels far higher than Earth's tilt, and then back again. (By contrast, Earth's tilt oscillates only four degrees, thanks to the stabilizing tidal tuggings of our big Moon -- although even that small shift is enough to bring on our ice ages.)

First, let me give you a crash course in gamma-ray and neutron spectrometry.

Odyssey's gamma-ray spectrometer counts the gamma-ray photons given off by the natural radioactive decay of the traces of uranium, thorium and a radioisotope of potassium in Mars' crust. But it can also measure a dozen or so non-radioactive elements through a different technique.

Cosmic rays -- those extremely energetic protons coming from outer space -- are continually crashing into Mars' upper surface and plowing down through its upper few meters. When they crash into a nucleus of one of the atoms of Mars' surface material, they cause that nucleus to undergo various kinds of nuclear reactions which often cause it to emit one or more high-energy ("fast") neutrons.

Some of these fast neutrons escape into space, but others ricochet off other atomic nuclei in the surface soil and rock. Every time they do, they naturally lose some of their rebound speed, and thus their energy, by shoving the nucleus they hit backwards. In the case of some elements, they also interact with the nucleus in other ways when they bounce off it, transferring still more of their energy of motion to the nucleus, which then re-emits it as a gamma-ray photon of specific energy.

Thus the original fast neutrons slow down, and are then called less energetic "epithermal" neutrons. Those epithermal neutrons that don't eventually escape upwards back into space continue to bounce around and lose energy this way until they're moving no faster than the speed at which the surface material's atoms are vibrating because of its surface temperature -- at which point they become known instead as "thermal" neutrons.

And those thermal neutrons that don't first escape back up into space eventually run into a nucleus of one of several different kinds of elements which captures and absorbs them permanently, in the process emitting still more gamma-ray photons.

The gamma-ray photons emitted by these three processes -- natural radioactive decay, "inelastic" or sticky neutron rebound off a nucleus, and permanent neutron capture by a nucleus -- all exist at specific energy levels, depending on the element whose nucleus is emitting them. Odyssey's gamma ray spectrometer counts them, and can thus measure the amounts of these different elements in the upper 30 cm or so of Mars' surface.

These gamma-ray photons are, however, sparse -- after all, the level of radioactivity from cosmic rays isn't very high -- and so it takes months or even years for Odyssey to count enough of them to get an estimate of the amount of such an element at one spot on Mars' surface that isn't hopelessly fuzzy.

(For some elements, the number of gamma rays emitted is so sparse that even if Odyssey makes its measurements for years, there won't be enough of them to get a useful estimate of the amount of that element unless the total number of photons from the element over a very wide area of the surface is added together -- so Odyssey must make a necessary tradeoff between the sharpness of its spatial mapping of such elements and the accuracy of its actual measurement of their amounts.)

But Odyssey also measures the energy of all those neutrons of various energies that have escaped upwards back into space at some point -- and its purpose for this is to measure hydrogen in the surface.

A hydrogen nucleus is vastly lighter than the nucleus of any other element existing in significant quantities in a planet's surface minerals -- after all, it's only a single proton, weighing slightly less than a neutron. And so a hydrogen nucleus is especially effective at soaking up part of the momentum of any neutron that hits it: the nucleus goes flying backwards at half the speed with which the neutron was originally moving, while the neutron rebounds in the other direction with only half its original speed -- just as a billiard ball rebounds much more slowly after hitting another billiard ball than it would if it hit a bowling ball.

So hydrogen is far more effective than any other element at reducing the energy of the original "fast" neutrons produced by cosmic rays, and thus converting them into epithermal and then thermal neutrons. However, a hydrogen nucleus is also pretty efficient at permanently absorbing a thermal neutron that hits it at a slow enough speed.

So, by measuring the relative amounts of fast, epithermal and thermal neutrons that escape from Mars' surface, Odyssey's two neutron spectrometers can provide an estimate of the amount of hydrogen in the surface. And since hydrogen exists in a planet's surface in significant amounts only in water, or in minerals that have been chemically "hydrated" by past exposure to water, measuring hydrogen is equivalent to measuring surface water.

Neutrons penetrate material more easily than gamma rays, so Odyssey's neutron spectrometers can estimate the total water content of the upper meter or so of Mars' surface -- whereas the gamma rays given off by hydrogen when it actually captures neutrons provide a separate measurement of the water in just the upper third of a meter of the surface.

And thanks to the multi-step complexity of the way in which Mars' surface interacts with neutrons, measuring the ratios of fast, epithermal and thermal neutrons and gamma rays can provide an estimate not only of how much water there is in the near-surface, but whether it's buried beneath a shallow layer of soil with less water, and how thick that layer is.