A drop of water caught in the gravitational field of the Earth
What the hell is water?
There’s an old Delta blues song called “High Water Everywhere” about the Great Mississippi Flood of 1927. This title could also be a perfect nickname for our planet. Indeed, water is everywhere. 72% of Earth’s surface is covered with water. 60% of your body is made of water. Try living one week without water, and see what happens. But what is water? It is a very simple molecule. Hydrogen and oxygen had a baby and they called it water. Water is in the ground, in the sea, in the ice, in the human body, in your food and in your glass. Fish swim into it. There is a very humorous story about fish and water, we can’t resist the pleasure to share it with you. It goes like this. Two young fishes are swimming in the ocean. An older fish comes along and says: “Good morning boys! How’s the water?”. The two young fishes pause for a second, look at each other, puzzled, and say: “What the hell is water?”. We love this story. It reminds us about the influence of environment, and about everything we take for granted that we’re not even aware of. When you hear the popular mantra of “Thinking out of the box”, the first step is to be aware that you are in a box. What the hell is a box? Think about it. So, let’s reflect wisely about fishes and boxes, and for now come back to our main topic. Here, on our blue planet, we are surrounded by water.
Photons don’t have a mass but water does
If you are interested about water on Earth and more generally about Physics, there is a very good book by Richard Feynman called “Light and Matter”. Actually, all books by Feynman are excellent. You should read them all, including “Surely you’re joking Mr. Feynman!”, and if you’re a serious aficionado, the Feynman lectures on Physics from Caltech. In those books, you will learn that photons don’t have a mass, but that water does. Fill your glass with water, and you’ll feel the weight in your hand. Fill a whole barrel with water and you’ll need a horse to carry it. Fill an ocean with water and you’ll get enough mass to change the path of a satellite. Shocking! Isn’t it? Masses attracts masses. You know that from Newton. Because the mass of water is so important on Earth, it modifies the gravitational attraction of the Earth, and it impacts the orbits of satellites. Every time a satellite flies over an area, such as the Amazon basin for example, or Greenland, the mass of water below the satellite is different, and it impacts the orbit differently. By measuring precisely the deviation of the orbit, and by applying sophisticated reverse calculations, scientists can determine the quantity of water below the satellite’s path. Deviation is first converted into a value of gravitational attraction which is then converted into a quantity of water. This is the reason why gravity missions (past, current and future), are in fact a very innovative tool to monitor climate and the water cycle: draughts, floods, ice melt, and so on.
Observing Earth’s gravity field from space
Space geodesy made a huge leap forward in the 2000’s with the CHAMP, GRACE, GOCE and today GRACE Follow-on missions. The GRACE mission (as well as its successor GRACE-FO) is made out of two satellites, a polar pair orbiting between 400 and 500 km of altitude and following each other at a 200 km intersatellite distance. The main instrument is the KBR (for K-Band Ranging), which measures the distance and velocity between the two satellites. GRACE Follow-On additionally has an LRI (Laser Ranging Interferometer), which is basically a better version of the instrument fulfilling the same purpose. Along with other instruments such as GPS, accelerometers, and satellite laser ranging, those measurements allow us to compute the average gravity field model over a given period of time. With 10 days of data, we have a dense enough coverage to compute a map of the gravity field that represents the state of the Earth during those 10 days. 30 days is the standard.
GRACE mission illustration (Credits: Astrium/GFZ)
From telemetry to maps of equivalent water heights
So, how does it work? We receive the GRACE data by telemetry, basically stacks of zeroes and ones sent from the satellite to Earth. We process all this data and we produce monthly gravity field models. But what is this? Mathematically, Earth’s gravity field can be described by different manners. The most straightforward is the gravitational potential, which we can describe as an infinite sum of spherical harmonics (read about it here). So for every month, we produce a set of spherical harmonics (our set is not infinite though). Once we have the potential, we can easily derived it into several other quantities : gravity anomalies, geoid heights, or equivalent water heights (EWH). In our case, the most interesting is equivalent water heights. Of course, if you think rigorously, you will immediately object that all variations in gravity field are not 100% related to water. You are absolutely right. But first, this transformation is only an abstract change of units, and second, the approximation is good enough to give us a good overview of what happens at the Earth’s surface: the evolution of the level of the lakes, the content of water in tropical regions, the evolution of the polar caps… In science, we have the right, and sometimes the duty, to make approximations. Galileo’s approximation was perfectly fine until Einstein improved it. In practical terms, the products we deliver are monthly 1°x1° grids of equivalent water heights differences between the monthly gravity field model and an average gravity field model over the long-term. Typically our graphic scale goes from -35 cm to 35 cm. Let’s show you two images, one for the month of April 2009 and the other one for the month of October 2009. Those maps are computed 6 months apart, which means they are at the opposite of each other in terms of seasons. Watch the differences.
Equivalent water heights for April 2009 (difference with long-term average)
Equivalent water heights for October 2009 (difference with long-term average)
The maps in rectangular and polar projections represent the gridded values. The red color represents an excess in water, the blue color represents a default in water. This is a comparison to a long-term average. You can clearly see the influence of seasons: the colors over the Amazon, Africa, India and Alaska are inverted between April and October. For specialists, the triangle on the top right represents the amplitudes of the spherical harmonics coefficients and the curves at the bottom right represent the spectrae. It’s nothing but another vision of the same signal.
Once we have produced these monthly grids, it is easy to compile them over time, and to extract time series in one given place: a single grid point, a square, a rectangle, a specific shape, you name it… We can then clearly see water cycles on dedicated areas, such as hydrological basins (Amazon, Congo, etc.), place where monsoons are important (India), and in the polar caps (Antarctic, Greenland, etc.). Here is an image over the Amazon basin. You can clearly see the seasonal amplitude (more or less 30 cm).
Time series of equivalent water heights over the Amazon from the GRACE mission (2002-2017)
Last time we talked about spherical harmonics and Joseph Fourier. Today we made a brief introduction about how we can monitor the water cycle from space with gravity missions. In the next article, we are going to skip a few steps and jump both feet into highly specialized matters. Mathematics. More precisely: how we use the diagonalization of matrices to improve the quality of the gravity maps we produce. How we get rid of the noisy vertical stripes that come from the GRACE polar geometry, and the lack of measurements in the East-West directions. We are going to talk about diagonalization, subspaces, eigenvalues and eigenvectors. Only recommended if you love linear algebra. Put on your seatbelts.