Questions about the number of parameters for the Earth-specific positional bias

[15b3]

Thank you for your excellent research.
I have two questions about the number of parameters for the Earth-specific positional bias.

1. formula

In the preprint, the formula is given as $(2W_\mathrm{pl}-1)\times2(W_\mathrm{lat}-1)\times(2W_\mathrm{lon}-1)$.
However, I believe the correct formula should be $(2W_\mathrm{pl}-1)\times(2W_\mathrm{lat}-1)\times(2W_\mathrm{lon}-1)$.
Could you please confirm if this is an error or if there is a specific reason for the given formula?

2. magnification factor

In the first block,

• $(N_\mathrm{pl}, N_\mathrm{lat}, N_\mathrm{lon}) = (8, 186, 360)$ (186 includes padding).
• $(W_\mathrm{pl}, W_\mathrm{lat}, W_\mathrm{lon}) = (2, 6, 12)$.
• Therefore, $(M_\mathrm{pl}, M_\mathrm{lat}, M_\mathrm{lon}) = (4, 31, 30)$.

According to these values, the number of parameters for the former would be:
$(2\times2-1)\times(2\times6-1)\times(2\times12-1) = 759$.
And for the latter:
$4\times31\times2^2\times6^2\times(2\times12-1) = 410,688$.
This results in a ratio of:
$\frac{410,688}{759} \approx 541$.

However, the paper mentions a magnification factor of "$527\times$".
Could you please clarify this discrepancy?

[JNR000]

The only explanation I can think is the paper estimated the value approximately using 400000/759.