Relativistic location algorithm in curved spacetime

In this article, we describe and numerically implement a method for relativistic location in slightly curved but otherwise generic spacetimes. For terrestrial positioning in the context of Global Navigation Satellite Systems, our algorithm incorporates gravitational as well as tropospheric and ionospheric effects modeled by the Gordon metric. The algorithm is implemented in the squirrel.jl code, which employs a quasi-Newton Broyden algorithm in conjunction with automatic differentiation of numerical geodesics. Our work provides a practical solution to the relativistic location problem in a generic spacetime and consolidates relativistic and atmospheric effects in a single framework. Though optimization is not our primary focus, our implementation is already fast enough for practical use, establishing a position from five emission points in $<1\mathrm{s}$ on a desktop computer for reasonably simple spacetime geometries. In vacuum, our implementation can achieve submillimeter accuracy considering the Kerr metric with terrestrial parameters and submeter accuracy including tropospheric and ionospheric effects.

Figure 1

$2+1$ illustration for the relativistic location algorithm in the case of a spacelike configuration hyperplane $\mathrm{\Sigma }$ in an adapted frame. If the configuration hyperplane $\mathrm{\Sigma }$ is spacelike, then one can perform a Lorentz transformation so that the emission points lie on a surface of constant time coordinate $t=X^{\prime 0}$ (the corresponding $t$ axis is vertical) with a value $t=t_0$. In this frame, the emission points lie on a sphere $\mathcal{S}$ (represented here as a circle) in the configuration hyperplane $\mathrm{\Sigma }$. The problem consists of first finding the circumcenter ${\mathbf{x}}_{\mathrm{c}}$ and the circumradius $r_{\mathrm{c}}$. In this frame, the time it takes for light to travel the distance $r_{\mathrm{c}}$ is $\mathrm{\Delta }t=r_{\mathrm{c}}$ in units where the speed of light is unity. The intersection point is then given by $X_{\mathrm{c}}=(t_0+\mathrm{\Delta }t,{\mathbf{x}}_{\mathrm{c}})$.

Figure 2

$2+1$ illustration for the relativistic location algorithm in the case of a timelike configuration hyperplane $\mathrm{\Sigma }$ in an adapted frame (as in Fig. 1, the $t$ axis is vertical). If $\mathrm{\Sigma }$ is timelike, then one can perform a Lorentz transformation so that the emission points lie on a surface of constant coordinate $z=X^{\prime 3}$, with the value $z=z_0$. In this frame, the emission points lie on a hyperboloid $\mathcal{H}$ (represented here as a hyperbola). The problem consists of first finding the coordinates of the vertex ${\mathbf{v}}_{\mathrm{c}}$ in this frame and the distance $R$, the latter being the distance between the vertex ${\mathbf{v}}_{\mathrm{c}}$ and a point on $\mathcal{H}$. The distance between the $z=z_0$ plane and the intersection point $X_{\mathrm{c}}$ is given by $R=\mathrm{\Delta }z$. With these quantities in hand, one may obtain the intersection point $X_{\mathrm{c}}=({\mathbf{v}}_{\mathrm{c}},z_0\pm \mathrm{\Delta }z)$.

Figure 3

Horizontal positioning errors for our implementation of the five emission point algorithm of [21] relative to the Kerr-Schild geometry for ${10}^5$ test cases. The closely spaced vertical lines correspond to the rms values and 95% confidence level for the errors, with respective values of 0.0226 and 0.0227 cm. Out of ${10}^5$ samples, two samples (0.002%) have an error $>2\mathrm{cm}$ with the largest error being 3.3 cm.

Figure 4

Vertical positioning errors for our implementation of the five emission point algorithm of [21] relative to the Kerr-Schild geometry for ${10}^5$ test cases. The vertical lines correspond to the rms values and 95% confidence level for the errors, with respective values of 1.56 and 2.00 cm. Out of ${10}^5$ samples, 11 samples (0.011%) have an error $>5\mathrm{cm}$ with the largest error being 15.0 cm.

Figure 5

Horizontal positioning errors ($n=5$ emission points) in the Kerr geometry for ${10}^5$ test cases. The vertical lines correspond to the 95% confidence level and rms values for the errors, with respective values of 0.0608 and 0.277 mm. Note that the results here are expressed in units of millimeters. Out of ${10}^5$ samples, four samples (0.004%) have an error $>2\mathrm{cm}$ with the largest error being 3.39 cm.

Figure 6

Vertical positioning errors ($n=5$ emission points) in the Kerr geometry for ${10}^5$ test cases. The vertical lines correspond to the 95% confidence level and rms values for the errors, with respective values of 0.0862 and 0.286 mm. Note that the results here are expressed in units of millimeters. Out of ${10}^5$ samples, 1 sample (0.001%) has an error $>2\mathrm{cm}$ with the largest error being 2.40 cm.

Figure 7

Horizontal positioning errors ($n=5$ emission points) in the analog geometry incorporating tropospheric and ionospheric effects (${10}^5$ test cases). The vertical lines correspond to the 95% confidence level and rms values for the errors, with respective values of 0.418 and 0.693 mm. Out of ${10}^5$ samples, 13 samples (0.013%) have an error $>2\mathrm{cm}$ with the largest error being 3.20 cm.

Figure 8

Vertical positioning errors ($n=5$ emission points) in the analog geometry incorporating tropospheric and ionospheric effects (${10}^5$ test cases). The vertical lines correspond to the 95% confidence level and rms values for the errors, with respective values of 0.618 and 1.02 mm. Out of ${10}^5$ samples, 58 samples (0.058%) have an error $>2\mathrm{cm}$ with the largest error being 3.59 cm.

Figure 9

Horizontal positioning errors ($n=5$ emission points) in the perturbed analog geometry corresponding to a fractional uncertainty of 0.1% (${\delta }_1={10}^{-3}$) in the tropospheric index of refraction and 1% (${\delta }_2=0.01$) in the ionospheric electron density (${10}^5$ test cases). The vertical lines correspond to the rms and 95% confidence level values for the errors, with respective values of 6.27 and 9.70 cm. Out of ${10}^5$ samples, seven samples (0.007%) have an error $>2\mathrm{m}$ with the largest error being 3.42 m.

Figure 10

Vertical positioning errors ($n=5$ emission points) in the perturbed analog geometry corresponding to a fractional uncertainty of 0.1% (${\delta }_1={10}^{-3}$) in the tropospheric index of refraction and 1% (${\delta }_2=0.01$) in the ionospheric electron density (${10}^5$ test cases). The vertical lines correspond to the rms and 95% confidence level values for the errors, with respective values of 25.6 and 39.0 cm. Out of ${10}^5$ samples, 10 samples (0.01%) have an error $>2\mathrm{m}$ with the largest error being 4.56 m.

Figure 11

Horizontal positioning errors ($n=5$ emission points) in the perturbed analog geometry corresponding to a fractional uncertainty of 0.1% (${\delta }_1={10}^{-3}$) in the tropospheric index of refraction and 10% (${\delta }_2=0.1$) in the ionospheric electron density (${10}^5$ test cases). The vertical lines correspond to the rms and 95% confidence level values for the errors, with respective values of 59.4 and 93.1 cm. Out of ${10}^5$ samples, six samples (0.006%) have an error $>20\mathrm{m}$ with the largest error being 31.9 m.

Figure 12

Vertical positioning errors ($n=5$ emission points) in the perturbed analog geometry corresponding to a fractional uncertainty of 0.1% (${\delta }_1={10}^{-3}$) in the tropospheric index of refraction and 10% (${\delta }_2=0.1$) in the ionospheric electron density (${10}^5$ test cases). The vertical lines correspond to the rms and 95% confidence level values for the errors, with respective values of 2.44 and 3.70 m. Out of ${10}^5$ samples, 10 samples (0.01%) have an error $>20\mathrm{m}$ with the largest error being 42.5 m.

Figure 13

Horizontal positioning errors ($n=6$ emission points) in the perturbed analog geometry corresponding to a fractional uncertainty of 0.1% (${\delta }_1={10}^{-3}$) in the tropospheric index of refraction and 1% (${\delta }_2=0.01$) in the ionospheric electron density (${10}^5$ test cases). The vertical lines correspond to the rms and 95% confidence level values for the errors, with respective values of 3.27 and 5.81 cm. Out of ${10}^5$ samples, 1 sample (0.001%) has an error $>1\mathrm{m}$ (none greater than 2 m) with the largest error being 1.56 m.

Figure 14

Vertical positioning errors ($n=6$ emission points) in the perturbed analog geometry corresponding to a fractional uncertainty of 0.1% (${\delta }_1={10}^{-3}$) in the tropospheric index of refraction and 1% (${\delta }_2=0.01$) in the ionospheric electron density (${10}^5$ test cases). The vertical lines correspond to the rms and 95% confidence level values for the errors, with respective values of 24.2 and 35.0 cm. Out of ${10}^5$ samples, 1 sample (0.001%) has an error $>1\mathrm{m}$ (none greater than 2 m) with the largest error being 1.10 m.

Figure 15

Horizontal positioning errors ($n=6$ emission points) in the perturbed analog geometry corresponding to a fractional uncertainty of 0.1% (${\delta }_1={10}^{-3}$) in the tropospheric index of refraction and 10% (${\delta }_2=0.1$) in the ionospheric electron density (${10}^5$ test cases). The vertical lines correspond to the rms and 95% confidence level values for the errors, with respective values of 33.5 and 61.1 cm. Out of ${10}^5$ samples, seven samples (0.007%) have an error $>5\mathrm{m}$ with the largest error being 14.6 m.

Figure 16

Vertical positioning errors ($n=6$ emission points) in the perturbed analog geometry corresponding to a fractional uncertainty of 0.1% (${\delta }_1={10}^{-3}$) in the tropospheric index of refraction and 10% (${\delta }_2=0.1$) in the ionospheric electron density (${10}^5$ test cases). The vertical lines correspond to the rms and 95% confidence level values for the errors, with respective values of 2.32 and 3.24 m. Out of ${10}^5$ samples, five samples (0.005%) have an error $>7\mathrm{m}$ with the largest error being 10.4 m.