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I'm trying to make use of the ETOPO1 dataset as the bathymetry input of a wave model. I came upon this discussion of grid registration: Registration of Structured Square-Cell Grids.

I read this material several times and I'm still confused about some things. From what I understand of the diagram on that page: the blue dots represent the data values. We know from the dataset what the latitude and longitude of each of the blue dots is. And for each blue dot, it is always in the center of the cell, which is the area it represents. If I'm right so far, what is the difference or significance of these two schemes? What is the purpose of the red lines? Why might there be an ambiguity, since we already know the locations of the blue dots and the data values at those locations? Does it affect how we interpret the data?

If anyone can help me with this, or point me to relevant resources, it would be much appreciated.

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Yes, it makes a difference. In this case the grid-registered dataset is higher fidelity. However, the cell-registered is easier to handle.

As you see, both types of data are available on the dataset's page. The page states:

The grid-registered is the authoritative registration.

The page also points out:

The cell-registered is derived from the grid-registered, and the conversion produces slightly flattened relief.

So what's the difference? The grid-registered version puts the data points at the grid corners (the red lines, not the grey squares, define the grid boundaries. You have to imagine the grid cells in the left-hand image). So if there are $n \times m$ grid cells, there are $(n+1) \times (m+1)$ data points in the grid-registered data.

Since there are fewer data points in the cell-registered version, making it involves some smoothing (averaging) of the values in the grid-registered version. There's an illustration of this effect on the other page you linked to; notice how the centre cell of value 90 ends up being smoothed away:

Flattening grids

So if accuracy is paramount, you should use the grid-registered version.

But...

You might find cell-registered data more convenient to work with (for me at least it's more intuitively analogous to an image).

The catch is that if you have grid-registered but accidentally treat it as cell-registered, then the data will shift by half a cell. So you should keep careful track of what kind of data you have. As the NOAA page states:

Most grid applications recognize both types and avoid [the grid shift] problem but care should be taken.

Indeed, one popular open source GIS tool, GRASS, apparently only handles cell-registered data, as explained in the GRASS wiki. So after loading (to wherever) you should check where the pixels of the raster actually end up.

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Disclaimer: I have no special knowledge of this data set. But from reading the linked page, which I agree is not very clear, here's my understanding:

As you have stated, the data points actually represent cells with a finite area rather than points. The difference between the two approaches is whether the location of the cell is identified by giving the coordinates of the cell centre (shown by the blue dot), or of one corner of the cell.

To put it another way - the lat/lon coordinates that you have may not correspond with the locations of the blue dots, but be half a cell offset from them.

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I got in touch with the author of the dataset with my remaining concerns, who responded surprisingly quickly. Here is the essence of everything I learned from people I talked to, including kwinkunks, Simon W, and the author of the dataset, and is intended to augment the existing answers.

For both the grid registration and the cell registration, the dataset provides 3 pieces of info: Item #1: lat for each blue dot, Item #2: lon for each blue dot, Item #3: data (elevation/depth) at each blue dot. In addition, for both registrations, each blue dot is at the center of a cell area, and the blue dot represents the average elevation/depth of that cell area.

The difference between the two registrations is solely a question of where the outer boundaries (demarcated by the red lines) lie.

The cell registered version was created because it was more appropriate for creating maps and images. The grid-registered version presents difficulties for plotting near the poles, which "rolls over" to the other side.

As my application is wave modeling and only a subset of the dataset is required, the grid-registered version, being the authoritative and non-flattened version, is the most appropriate for me.

I am grateful for everyone who helped.

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The different schemes where to define grid-points has nothing to do with fidelity or easiness to handle.

The keyword here is a 'staggered grid'. In simulations of physics that compute conservation laws it makes sense to define scalar quantities (like density, temperature) at cell centers, while directed or vectorial quantities (momentum, energy flux) are defined at cell interfaces.

So depending which variable you're interested in looking at, the data for this variable can be shifted by 1/2 cell relative to the others. You have to take that into account when creating maps.

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