[Python] How to interpolate a multidimensional xarray DataArray?

I am using the xarray library and I have some doubts/questions.

I have this dataset::

ds
<xarray.Dataset> Size: 2GB
Dimensions: (Latitude: 364, Longitude: 246, Lon_u: 247, Lat_v: 364,
Time: 1087)
Coordinates:
Longitude (Latitude, Longitude)
Latitude (Latitude, Longitude)
Lon_u (Latitude, Lon_u)
Lat_v (Lat_v, Longitude)
* Time (Time) datetime64[ns]
Data variables:
Lat_u (Latitude, Lon_u)
Lon_v (Lat_v, Longitude)
uSurf (Time, Latitude, Lon_u)
vSurf (Time, Lat_v, Longitude)


I want to interpolate the data (Lat_v > Latitude and Lon_u > Longitude). The result must be like something like this:

ds
<xarray.Dataset> Size: 2GB
Dimensions: (Latitude: 364, Longitude: 246, Time: 1087)
Coordinates:
Longitude (Longitude)
Latitude (Latitude)
* Time (Time) datetime64[ns]
Data variables:
uSurf (Time, Latitude, Longitude)
vSurf (Time, Latitude, Longitude)


Or actually I don't know if converting 2D coordinates to 1D makes sense. I read that if they are 2D coordinates, they are logical coordinates (Longitude, Latitude), working as a irregular matrix. But I think the example above, it's also a irregular matrix just because the dimensions are different: {Latitude: 364, Longitude: 246}.

So I don't know if I should preserve the coordinates like this:

ds
<xarray.Dataset> Size: 2GB
Dimensions: (Latitude: 364, Longitude: 246, Lon_u: 247, Lat_v: 364,
Time: 1087)
Coordinates:
Longitude (Latitude, Longitude)
Latitude (Latitude, Longitude)
* Time (Time) datetime64[ns]
Data variables:
uSurf (Time, Latitude, Longitude)
vSurf (Time, Latitude, Longitude)


I don't understand this notation though. Because I find it recursive. Imagine I want to get all the possible Latitude values: I can do this:

ds.Latitude.values[:, 0]


But Latitude is also a coordinate of Latitude itself. So I could do something like this

ds.Latitude.Latitude.Latitude.Latitude.values[:, 0]


Is that a normal behaviour? Or it's something wrong with my DataSet and I need to flatten it?

I have tried this code:

interp_coords = {'Latitude': ds.Latitude.values[:, 0], 'Longitude': ds.Longitude.values[0, :]}

uSurf_interp = ds.uSurf.interp(
Latitude=interp_coords['Latitude'],
Lon_u=interp_coords['Longitude'],
method='linear'
)

vSurf_interp = ds.vSurf.interp(
Lat_v=interp_coords['Latitude'],
Longitude=interp_coords['Longitude'],
method='linear'
)

ds_interp = xr.Dataset(
{
'uSurf': uSurf_interp,
'vSurf': vSurf_interp
},
coords={
'Latitude': interp_coords['Latitude'],
'Longitude': interp_coords['Longitude'],
'Time': ds.Time
}
)


I get this error in the first interp instruction:

ValueError: Input DataArray is not 1-D.


As interp accepts only 1-D coordinates. So I need to flatten the coordinates. It's one thing I wanted to avoid.

I have read the documentation. But I cannot find a solution:

• Multi-dimensional interpolation
• Working with Multidimensional Coordinates

I found this diagram quite ilustrative. But for 1D coordinates



UPDATE


I'm trying now to do it manually with np.meshgrid and scipy.interpolate.griddata. Also, I'm trying to combine them with xarray.apply_ufunc

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