choclo.prism.gravity_u#

choclo.prism.gravity_u(easting, northing, upward, prism_west, prism_east, prism_south, prism_north, prism_bottom, prism_top, density)[source]#

Upward component of the gravitational acceleration due to a prism

Returns the upward component of the gravitational acceleration produced by a single rectangular prism on a single computation point.

Parameters
  • easting (float) – Easting coordinate of the observation point. Must be in meters.

  • northing (float) – Northing coordinate of the observation point. Must be in meters.

  • upward (float) – Upward coordinate of the observation point. Must be in meters.

  • prism_west (float) – The West boundary of the prism. Must be in meters.

  • prism_east (float) – The East boundary of the prism. Must be in meters.

  • prism_south (float) – The South boundary of the prism. Must be in meters.

  • prism_north (float) – The North boundary of the prism. Must be in meters.

  • prism_bottom (float) – The bottom boundary of the prism. Must be in meters.

  • prism_top (float) – The top boundary of the prism. Must be in meters.

  • density (float) – Density of the rectangular prism in kilograms per cubic meter.

Returns

g_u (float) – Upward component of the gravitational acceleration generated by the rectangular prism on the observation point in m/s2.

Notes

Returns the upward component gz(p) of the gravitational acceleration g on the observation point p=(xp,yp,zp) generated by a single rectangular prism defined by its boundaries x1,x2,y1,y2,z1,z2 and with a density ρ:

gz(p)=Gρ|||kz(x,y,z)|X1X2|Y1Y2|Z1Z2

where

kz(x,y,z)=[xsafe-ln(y,r)+ysafe-ln(x,r)zsafe-arctan(xy,zr)]
r=x2+y2+z2,

and

X1=x1xpX2=x2xpY1=y1ypY2=y2ypZ1=z1zpZ2=z2zp

are the shifted coordinates of the prism boundaries and G is the Universal Gravitational Constant.

The safe-ln and safe-arctan functions are defined as follows:

safe-ln(x,r)={0x=0,r=0ln(x+r)x0ln((r2x2)/(rx))x<0,rxln(2x)x<0,r=x
safe-arctan(y,x)={arctan(yx)x0π2x=0andy>0π2x=0andy<00x=0andy=0

These were defined after [Fukushima2020] and guarantee a good accuracy on any observation point.

References