Start with a spinning sphere with a known axis / rate of rotation and some bit of mass on its surface that you can move. You can move it in three directions, let's call them latitude, longitude and altitude [1]. The classic examples I learned in physics class are about altitude (e.g. figure skater spinning faster when she brings her arms inward) which alters the magnitude of angular velocity (assuming conservation of angular momentum).

My intuition is that, if changing the altitude only affects the magnitude of the angular velocity, the other 2 degrees of freedom (longitude and altitude) must determine the direction of the angular velocity.

You start with a model of mass distribution of Earth over time, let us say M(x, y, z, t). Let us call w(t) the Earth's angular velocity at time t. If you know w(t_0) for some time t_0, you can calculate what the model says w(t) will be. The givens are: M(x, y, z, t_0), M(x, y, z, t), w(t_0), and conservation of angular momentum. You want to calculate w(t) from the model, then compare that calculation to the measurement to test the model. Your hypothesis is the model is accurate; your experiment is comparing the model's prediction of w(t) against the measured w(t).

I'm immensely curious to know how M(x, y, z, t) is calculated. They show some satellite images but it seems like they would only measure lakes, rivers, and maybe surface level soil saturation. But to me "groundwater" implies things like aquifers and underground storage, how do they measure that? You'd need to not only know the amount of water but also its change in latitude and longitude. Do you assume that if groundwater is used it ends up in the oceans? That seems a bit presumptuous, wouldn't a lot of it soak into the ground, get taken up by plants, find its way back down into an aquifer, etc.? Having water soak into the soil and become integrated into a plant is literally the point of watering crops, if we assume agricultural water ends up in the ocean doesn't that mean farmers are using too much water, which would be economically irrational because water is not free?

For that matter, why focus so much on water? Solid matter also has mass and we change its latitude, longitude and altitude when we mine it and turn it into products that we ship all over the world. For that matter, people and cars and ships and airplanes and wild animals all have mass and move around every day.

I'm a bit lost trying to follow the paper, it says "Changes in terrestrial water and oceanic mass loads were converted to spherical harmonic (SH) coefficients of the geoid..." but I only have a vague notion of what spherical harmonics are, and I don't really understand the given formulas.

[1] Latitude: Along the surface in the direction it's spinning. Longitude: Along the surface parallel to the axis of rotation. Altitude: Toward or away from the center of the sphere.