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My users would like to be able to see the ocean contour lines as they zoom in. But mapbox outdoors style only seems to provide contours for land.

I'm not a GIS pro but a web dev, so please keep that in mind in any answers. I do have experience in publishing my own tilesets to mapbox with points, shapes, and labels. But a tileset with all of the contours for the entire water area of the earth sounds impractical and very large for me to host on my little portion of mapbox. But I don't see that they have such a layers available.

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You could have a look at the Mapbox custom map style "North Star" which is using 1:10m bathymetry data from Natural Earth.

Here is a blog post from the style designer with some more information on this:

https://blog.mapbox.com/designing-north-star-c8574e299c94

It is possible to use the Natural Earth bathymetry data also with other styles.

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