Can I change Free Air anomaly to gravity disturbance?

[prince-Ha]

Hi
I'm learning a lot from fatiando. thank you

I became curious after reading the "Should geophysicists use the gravity disturbance or the anomaly?" article.

The picture below is what I organized after reading the article. There was a difference in the gravitational effect between gravity disturbance and Free Air anomaly as much as the geoid height.

If I calibrate the gravitational effect corresponding to the height of the geoid to Free Air Anomaly, can I make a gravity disturbance and use it? The data I'm using is Free Air Anomaly from Geoid, so I'm asking you to use gravity disturbance

Thank you.

[santisoler]

Hi @prince-Ha. Thanks for opening this question.

Your derivations seems right, so using that last equation you could get from the free-air anomaly to an approximation of the gravity disturbance. I say approximation because the expression you have for the gravity disturbance in equation 1 is an approximation:

$$ \delta g_p = g_P - \gamma_p \simeq g_p - (\gamma_Q + \frac{\partial \gamma}{\partial h} h_p ) $$

The term between parenthesis is actually a first order Taylor approximation of the normal gravity on the point p. This approximation is actually not needed, Li and Gotze (2001) present a closed-form formula to compute the normal gravity on any external point. We have this formula implemented in Boule.

If you have details of how the free-air anomaly was applied to your data you can "undo" it and recover the observed gravity values $g_p$:

$$ g_p = \Delta g_F + (\gamma_Q + \frac{\partial \gamma}{\partial h} H_p) $$

And then you can compute the gravity disturbance using the analytical solution we have in Boule. Does it make sense?

[prince-Ha]

thanks to you, I was able to convert free air anomaly to gravity disturbance

However another problem was occured at interpolation

I.
Artifacts occur while interpolating EIGEN6C4 geoid(ensaio, gridstep: 0.167 deg) data with gebco elevation grid(15").
I thought that interpolation could not be done in the latitude and longitude coordinate system, so I projected it with Mercator coordinates and interpolated it

[code]
gebco

gebco_table = vd.grid_to_table(gebco)
geoid

projection = pyproj.Proj(proj = 'merc', lat_ts = gebco_table.lat.mean())
easting, northing = projection(gebco_table.lon.values, gebco_table.lat.values)

gebco_proj = vd.project_grid(gebco.elevation, projection, method="nearest")
geoid_proj = vd.project_grid(geoid_eigen, projection, method="nearest")
geoid_proj_table = vd.grid_to_table(geoid_proj)
interpolator = vd.ScipyGridder(method = 'linear')
interpolator.fit((geoid_proj_table.easting, geoid_proj_table.northing), geoid_proj_table.geoid)
gebco_table = gebco_table.assign(geoid=interpolator.predict((gebco_table.easting_m, gebco_table.northing_m)))
gebco_table

[at mercator interpolation]

cmap = jet

[geoid]

II.
Interpolation in latitude and longitude coordinates is artifact-free. Is it right to interpolate at longitude latitude coordinates?

[at longitude & latitude interpolation]

cmap = turbo
[geoid]

It's cold. Be careful not to catch a cold.
thank you

[santisoler]

Hi @prince-Ha. You're right: interpolating on geographic coordinates using the current interpolators we have in Verde is not the best solution. But in this case where the region is quite long, projecting the grid also introduces high errors. Instead of having a projected and interpolated grid to then interpolate on the observation points, I would just run the interpolation on longitude latitude directly. I think this is the best compromise for now.

I hope at some point we can have nice interpolators for geographic coordinates that measure distance using Haversine distances rather than Euclidean distances.

[prince-Ha]

Thank you @santisoler

Projections are bound to be accompanied by errors on a large scale....

I want to clarify the conventional terms. Help me.

In my country, gravity disturbance is still called free-air anomaly, so I'm really confused about the terminology.

So, the data provided by the site below is called Free Air Anomaly, but please ask if the actual concept is gravity disturbance.

https://www.generic-mapping-tools.org/remote-datasets/earth-faa.html

I think it's a disturbance because it's produced by satellite altimetry

[santisoler]

In my country, gravity disturbance is still called free-air anomaly, so I'm really confused about the terminology.

They are actually two different quantities. I think you mean that people still use free-air anomaly as if they were gravity disturbances. If that so, chill out, it's not your country only, it's a worldwide issue 😅

So, the data provided by the site below is called Free Air Anomaly, but please ask if the actual concept is gravity disturbance.

https://www.generic-mapping-tools.org/remote-datasets/earth-faa.html

I think it's a disturbance because it's produced by satellite altimetry

I'm pretty certain that that is free-air anomaly and not gravity disturbance. But if you need to get satellite gravity disturbance, you can use ICGEM Calculation Service. In there you can choose which model you want to use (you can choose EGM2008, which is the one that GMT provides in the url you shared, but you can also choose a more update one like EIGEN-6C4), then choose which field you want to download (gravity_disturbance_sa for example) and give details of the grid where you want it and also the height above the ellipsoid.

You can use Harmonica's load_icgem_gdf function to load the file you'll download into an Xarray DataArray.

[prince-Ha]

I think you're good at teaching easily.

Thank you so much!!!