…And Standing Up Straight
In this post we cover New Zealand’s new gravity-based vertical datum, NZVD2016. Along the way we’ll get familiar with the weirdly wonderful geoid and WGS84. This post follows on from Keeping Our Heads Above Water, hence the title, and has benefited greatly from comments by LINZ’s Geodetic team – thanks!
In a previous post we looked at sea level and saw that it’s not quite as straightforward as most of us would like to think. This time around I’m afraid things are only going to get worse, so buckle up and let’s talk about gravity. The two are related, really, though gravity definitely gets a lot stranger.
Maybe we should start with the shape of the earth:
From space, the home planet looks nice and spherical. Our globes are nice, perfect spheres, too:
As you might have already guessed, that’s not quite the reality. Due to gravity and the earth’s spin, it tends to look a bit more like an orange; flattened at the poles and bulging at the equator:
We need a model of the earth’s shape to do things like create flat two-dimensional maps, and carry out topographic surveys and anticipate where water will flow. Geodesy, the science (and mathematics) of measuring the shape of the earth (and other planets) has developed a range of models of the earth. Some are roughly spherical, others more like squashed spheres – these latter models are called ellipsoids and are better models of the earth than spheres.
A Global Ellipsoid: WGS84
Let’s talk about one particular ellipsoid that’s become quite important. If you’ve ever used GPS to measure elevation, you’ve already had direct contact with this one (now go and wash your hands!) GPS, the global positioning system, uses a constellation (yes, that’s the proper collective noun) of satellites to determine positions on the surface of the earth.
(okay, this image is not to scale.) Exactly how it works is the topic for another post, but for now, bear in mind that it’s actually mapping locations in a three-dimensional space. Built in to every receiver (and satellite) is an ellipsoid model of the shape of the earth. When you look at your receiver screen and it displays your latitude and longitude, those are the coordinates on the ellipsoid. And your elevation is how high above or below the ellipsoid surface you are. The ellipsoid surface usually doesn’t coincide with the land surface. In New Zealand, the differences between the two may be as much as 35 m in Northland, and roughly zero near Stewart Island/Rakiura. Here’s an image of my GPS showing details about a point on the lawn in front of Ivey Hall:
(Apologies for image quality) The GPS receiver has calculated my horizontal position as well as an elevation. Note the elevation shown above is 20.5 m – that’s my calculated height above the GPS ellipsoid, not mean sea level. If I look on the topo map, my elevation at that same location is roughly 10 m but that value is with respect to mean sea level. (Note: if you reset your receiver to a different coordinate system like NZTM, it will probably correct your elevations to match closer with reality – geez – that simple comment cries out for a whole ‘nother post…) This ellipsoid has a very grand title: the World Geodetic System 1984 (affectionately known as WGS84). It’s a geocentric ellipsoid, meaning it shares its centre with the centre of mass of the earth, and has very much become the standard global model of the earth’s shape.
Models like WGS84 are quite useful but they can’t accurately represent the rough, convoluted topography that we’ve built our lives upon – they’re not designed to represent topography but do give us a simple model of the earth that makes some computations easier.
Gravity and Mean Sea Level
All well and good, but let’s get back to mean sea level and something more meaningful. Tide level measurements are relatively easy to make. Average them out over a long enough period and you can estimate mean sea level (pay no attention to the elephant in the room – we know it’s changing). We’ve already seen that mean sea level is different in different places: the driver behind this is gravity. And gravity varies as the mass beneath our feet varies. Happily, it is measurable,
No, not by dropping apples are various places and measuring their acceleration (my sources at LINZ tell me this is actually pretty close to how it’s really done), with gravimeters. Increasingly, gravity measurements are remotely sensed by instruments in satellites and airplanes, so it’s become much easier to measure and map gravity around the globe. Here’s a map of how gravity varies in our neighbourhood:
If you’re familiar with New Zealand bathymetry, some of this map should look a bit familiar. The Kermadec Trench trends northeast from the east coast of the North Island and coincides with the Pacific plate subducting beneath the Australian plate. To the southwest, another subduction zone can be seen, the Puysegur Trench – notice these are areas of higher gravity (reds) right next to areas of lower gravity (blues). The line of irregular red dots to the north east are the remains of a migrating hot spot (well, really it’s the plate that’s been migrating), the same type that would have created the Hawaiian Islands. They are areas of denser volcanic magma and so have higher gravity.
A Gravity Model: the Geoid
We must now talk about another three-dimensional surface: the geoid. Using gravimeters to map out the local gravitational field, we could create a surface where every point has the same arbitrary value of gravitational potential (which is closely related to gravity, although not quite the same) – this is called an equipotential surface. Since gravity determines where sea level is, if we picked our arbitrary value of gravitational potential such that it matched sea level, we then have a (very) special surface – the geoid. At sea, the geoid essentially matches what sea level would be without the effect of tides and winds and currents. The geoid continues under the land surface and if you start reading up on this you’ll come across notions of canals being cut through the land and filling up with sea water – the level the water would settle at would be the geoid surface (try getting a resources consent for that!). Because this is a surface of gravity, objects dropped from above the geoid fall towards the surface at a right angle, the plumb line, and spirit levels would be horizontal to it. Here’s the thing – rather than being essentially flat, it undulates, varying from place to place depending on local mass. The geoid surface is quite irregular but relatively smooth, a bit like our orange peel. Where land surface elevations may range from +8,848 meters above sea level (there’s that pesky phrase again) at Mt Everest to -427 m below sea level at the Dead Sea, vertical distances between high and low points on the geoid may only differ by about 200 m. Here’s an image of how the height of the geoid varies globally:
What does zero mean here? (Good question!) On this figure, zero is where the geoid and WGS84 meet. Positive values are above WGS84, negative values are below. (Note that if you travelled from Papua New Guinea to Sri Lanka, you would never notice the difference.) Below is an animated image of the geoid that shows the variations in its shape:
The reds are areas of higher gravity while the blues are lower (not the same colours as the other image). These differences are really very slight and have been exaggerated by quite a bit here but you get the idea.
The geoid in our neighbourhood is called NZGeoid2016 – here’s a map of it from LINZ:
The units here are in metres; the zero question arises again – here, zero is where the geoid and (yet) another ellipsoid, NZGD2000, coincide. For most practical purposes, NZGD2000 is the same as WGS84 (and my list of posts that follow on from this one is growing steadily…) and is the reference ellipsoid that NZ’s national coordinate system is based on. Latitude explains some of what we’re seeing here but you can also see some echoes from the NZ gravity map shown above.
How Are These All Related?
We’ve now talked about several different surfaces – how are they all related? The image below shows a cross-section through the earth with the geoid, a reference ellipsoid (such as WGS84), surface topography and how they’re all related:
Now if you’re still with me (and I’d understand if you weren’t) this does bring us back to the problem of having multiple sea levels. As we’ve noted in another post, we currently have 13 different standard values for mean sea level across New Zealand. What’s the practical effect of this? Imagine two teams of surveyors setting off from two different places to measure the height of Mt Cook, one from Lyttelton and the other from Bluff using traditional surveying methods. They would use their local measure of sea level as their zero. When they meet on the High Peak of Mt Cook their elevation values would probably be different. (I shouldn’t say this now, but I will…if they used GPS their measurements would be the same.)
With better mapping of the geoid, LINZ has addressed this problem by recently introducing a new vertical datum based on the NZ geoid – a single zero level across the country that not so much replaces the previous 13 as compliments them. This is called NZVD2016 (New Zealand Vertical Datum 2016).
Hurrah! All of our elevation problems have been solved! Well…mostly. As we can see in the geoid map above, mean sea level varies from place to place, so if our surveyors use the geoid as their datum, they will all calculate the same elevation at Mt Cook no matter where they start from (for the same reason that GPS measurements would agree). So that we can still work with our 13 values of mean sea level, LINZ has done the hard yards of developing spatial data that allows surveyors and engineers to more consistently calculate elevations across the country more easily. (See for yourself: look in J:\Data\NZVD2016 for layers of the NZ geoid and conversion grids. Data sourced from the LINZ Data Service. We’ll have to talk more about these conversions in another post.) While this new vertical datum is big news in the world of geodesy, it’s not going to have any effect on the vast majority of us. But you can now sleep more easily knowing that there’s now one source of vertical truth in the country.
Understanding gravity is pretty fundamental – our internal gravimeter senses local gravity and literally allows us to stand up straight. To correctly model water flow, engineers and designers must have a good sense of how local gravity behaves. Happily, it’s almost indistinguishable from local topography, but the differences are real and measurable.
How does this all relate to GIS? Coordinate systems, which include vertical measurements, are fundamental to GIS working. In order for our data to play nicely on a map, their locations need to be unambiguous. It’s sort of like a language. If we moved to a new country and wanted to conduct our day to day lives, we’re best off using the local language. With coordinate systems, that language is geographic and allows our data to map correctly. With the new vertical datum, we can now all speak the same vertical language. There are a few more complications we need to discuss, but I suspect we’ve covered enough ground in this post.
Much of the reason for talking about gravity has been to get a better understanding of what mean sea level means. As a parting note – would you be shocked if I told you that Mars has a “sea level”? It may not have any (liquid) water, but it does have a mean sea level-like datum that’s used to measure elevation across the planet: it’s been set as the mean Martian radius (3382.9 km).
Crazy stuff – and there’s more to come…
Also have a look at Catching Fish with Gravity from New Zealand Geographic (it’s worth tracking down the hard copy – there’s a nice map of NZ gravity)
Watch Graeme Blick, LINZ’s chief geodesist, talk about understanding the shape of the earth.