FMT grid search doubles the number of point per axis in the delta coordinate (iso component)

[SeismoFelix]

Hi all,

In the MTUQ module FullMomentTensorGridSemiregular
when the number of points per axis is defined to determine how coarse the grid search will be, I found that that number is doubled for the W variable (i.e., delta in the lune, related to the ISO component):

v, w = _semiregular(npts_per_axis, 2*npts_per_axis + 1, tightness, uniformity).

Therefore, if npts_per_axis = 21, for a FMT search the grid size will be:

grid shape: (21, 43, 21, 21, 21)

The 43 corresponds to the w (delta,iso) coordinate: 2*npts_per_axis+1.

Is there any reason behind this hard-coded condition?

I wonder whether this may affect the uncertainty analysis method proposed by Tape and Tape, 2016..

When the grid search is double couple, I plot the derivative of the volume fraction as a function of the angular distance from a reference moment tensor, and I could obtain roughly the expected curve (mesa shape curve).

On the other hand, when I plot the same curve for the FMT search, the fractional volume gradient curve shows an unusual increment in the fractional volume at 90° from the reference moment tensor, deviating from the theoretical curve of Tape and Tape, 2016:

I am trying to figure out if such deviation is because the oversampling in the isotropic component of the moment tensor, compared to the other variables.

Thanks for your kind attention.

[rmodrak]

Hi Felix, The current full moment tensor grid aims for a tradeoff between good plotting and low computational expense (e.g. sampling near the edges of the domain but not oversampling, especially near the poles)

[rmodrak]

The aim is not to simplify the integration over angular distance. In principle, I think you could still do the integration, but you would need quadrature that accounts for uneven sampling and a Jacobian that accounts for the change of variables (outside of special cases considered by Tape & Tape, these terms no longer drop out)

[rmodrak]

With regard to the hardcoded factor of 2-- wasn't it your suggestion to always sample the center of the domain?

[rmodrak]

In v w domain the integration becomes simpler. The ratio between the two sides of the v w domain is some factor of pi. But for the grid, obviously we need an integer number of samples along each side. I suspect (npts, 2*npts + 1) is probably a good choice?

[carltape]

@SeismoFelix and @rmodrak, thanks for pursuing this. Here are comments on three issues raised. First, the goal in the vw plane is to have uniform patches, since our parameterization will then have approximately the same number of moment tensors in each patch, which means same number of MTs in equal volumes of MT space. (This also assumes a fixed magnitude.) This allows the vw rectangle to serve as a convenient method for displaying probability density of source type, with a uniform value representing a homogeneous distribution of moment tensors. If the rectangle is displayed one-to-one, then it's 2/3 units wide [-1/3, 1/3] and 3pi/4 tall [-3pi/8, +3pi/8], giving an aspect ratio of 3.53 (see Alvizuri et al., 2018, Figure 2b). So if we want approximately square patches for plotting purposes, then we would want 3-4 times the number of w values as v values. (This supports Ryan's point that we'd want more points in w than in v; probably a factor of 2 is fine, but 4 is also possible.) Second, the 2*npts+1 ensures that we have values for w=0, the deviatoric points, which is a good thing. The third issue is the key one: Your distribution of omega angles for the uniform search does not match the homogeneous distribution, which is sin^4(omega). All my testing of various analytical representations of a homogeneous distribution are in this Matlab script, which will generate the mesa and sin4 plots, among others.

[SeismoFelix]

Thanks all for your comments and suggestions.
@rmodrak: Indeed, we suggested selecting an appropriate number of points per axis (odd number) to guarantee they pass through the center of the lune, and hence, sampling a double couple source type. The 2nppa+1 guarantees that for the latitude, not so for the longitude. So I assume we are still relying on the user to choose an odd number for sampling the center of the lune, then the 2nppa + 1 factor for w doubles the points of v, allowing a roughly equal volume sampling as @carltape pointed out. and +1 guarantees that w will always be odd.

@carltape: doubling the number of points of w for having the same moment tensors in a volume unit, it is what I thought after reading this from Alvizuri et al., 2018:

I read that after posting the issue; sorry. Thank you also for clarifying that we would need 3 or 4 points per axis to have uniformity in the number of tensors per volume unit. In Figure 5 of Alvizuri et al., 2018, I noticed v = 13 and w = 35, around a 2.7 ratio, not exactly 2, so I hesitated to understand this part.

By the way, in the e) inset of the figure, I suspect that the curve labels are swapped. It looks like the green curve is V'(ω), not the blue one, and vice versa. Could you confirm that?

Thanks all again for your feedback. I will keep you posted about the progress made in the uncertainty analysis following Tape and Tape, 2016.

[carltape]

The key point is that the patch size on the rectangle needs to be the same everywhere, since we're summing probabilities at the points inside each patch. It can be a square patch or any-shaped rectangle (or perhaps another shape), as long as all patches are the same size and not overlapping. But for plotting purposes, aiming for a square-like patch makes sense.

Yes, the colors are swapped in (e) -- I never noticed this mistake. V'(omega) should be the sin^4(omega) function. Thanks for pointing this out.