COSMIC & GNSS

Common-Volume Observations

Brian Breitsch

Geometry

Method

Diurnal results

Long-term results

Future work

Assumptions

  • model signal propagation paths as straight line segments

 

 

  • use SGP4 orbit calculation model with NORAD TLE
    • accuracy generally much better than 3km

SGP4 =

"Simplified General Perturbation Model"

version 4

Geometry

common observation volume

impact parameter

region of interest (ROI)

COSMIC satellites

  • 72 degree inclination
  • 780km (ish) altitude

Effects of Satellite Motion on ROI

r_{LEO}
rLEOr_{LEO}
r_{GPS}
rGPSr_{GPS}
r_{IP}
rIPr_{IP}
\sqrt{r_{LEO}^2 - r_{IP}^2}
rLEO2−rIP2\sqrt{r_{LEO}^2 - r_{IP}^2}
\sqrt{r_{GPS}^2 - r_{IP}^2}
rGPS2−rIP2\sqrt{r_{GPS}^2 - r_{IP}^2}
\alpha
α\alpha
\beta
β\beta
||v_{LEO}||\cos\theta_{LEO} \frac{\beta}{\alpha + \beta}
∣∣vLEO∣∣cosθLEOβα+β||v_{LEO}||\cos\theta_{LEO} \frac{\beta}{\alpha + \beta}
||v_{GPS}||\cos\theta_{GPS} \frac{\alpha}{\alpha + \beta}
∣∣vGPS∣∣cosθGPSαα+β||v_{GPS}||\cos\theta_{GPS} \frac{\alpha}{\alpha + \beta}

max vertical ROI movement

exercise

\theta_{LEO}
θLEO\theta_{LEO}
\theta_{GPS}
θGPS\theta_{GPS}

Effects of Satellite Motion on ROI

\alpha
α\alpha
\beta
β\beta

max horizontal ROI movement

exercise

||v_{LEO}||\frac{\beta}{\alpha + \beta}
∣∣vLEO∣∣βα+β||v_{LEO}||\frac{\beta}{\alpha + \beta}
||v_{GPS}||\frac{\alpha}{\alpha + \beta}
∣∣vGPS∣∣αα+β||v_{GPS}||\frac{\alpha}{\alpha + \beta}

3D Line Intersection

3D "intersection"

x_{RX}, x_{LEO}, x_{GPS_1}, x_{GPS_2}
xRX,xLEO,xGPS1,xGPS2x_{RX}, x_{LEO}, x_{GPS_1}, x_{GPS_2}
\delta_{GND} = x_{RX} - x_{GPS_!}
δGND=xRX−xGPS!\delta_{GND} = x_{RX} - x_{GPS_!}
\delta_{LEO} = x_{LEO} - x_{GPS_2}
δLEO=xLEO−xGPS2\delta_{LEO} = x_{LEO} - x_{GPS_2}
\delta_{GPS} = x_{GPS_2} - x_{GPS_1}
δGPS=xGPS2−xGPS1\delta_{GPS} = x_{GPS_2} - x_{GPS_1}
x^* = x_{GPS} + \mu \delta
x∗=xGPS+μδx^* = x_{GPS} + \mu \delta
\delta_{GND}
δGND\delta_{GND}
\delta_{LEO}
δLEO\delta_{LEO}
\delta_{GPS}
δGPS\delta_{GPS}

3D Line Intersection

3D "intersection"

\mu_{GND} = \frac{ \delta_{GPS} \cdot \delta_{LEO} * \delta_{LEO} \cdot \delta_{GND} - \delta_{GPS} \cdot \delta_{GND} * \delta_{LEO} \cdot \delta_{LEO} } { \delta_{GND} \cdot \delta_{GND} * \delta_{LEO} \cdot \delta_{LEO} - \delta_{LEO} \cdot \delta_{GND} * \delta_{LEO} \cdot \delta_{GND} }
μGND=δGPS⋅δLEO∗δLEO⋅δGND−δGPS⋅δGND∗δLEO⋅δLEOδGND⋅δGND∗δLEO⋅δLEO−δLEO⋅δGND∗δLEO⋅δGND\mu_{GND} = \frac{ \delta_{GPS} \cdot \delta_{LEO} * \delta_{LEO} \cdot \delta_{GND} - \delta_{GPS} \cdot \delta_{GND} * \delta_{LEO} \cdot \delta_{LEO} } { \delta_{GND} \cdot \delta_{GND} * \delta_{LEO} \cdot \delta_{LEO} - \delta_{LEO} \cdot \delta_{GND} * \delta_{LEO} \cdot \delta_{GND} }
\mu_{LEO} = \frac{ \delta_{GPS} \cdot \delta_{LEO} + \mu_{GND} \delta_{LEO} \cdot \delta_{GND}} {\delta_{LEO} \cdot \delta_{LEO}}
μLEO=δGPS⋅δLEO+μGNDδLEO⋅δGNDδLEO⋅δLEO\mu_{LEO} = \frac{ \delta_{GPS} \cdot \delta_{LEO} + \mu_{GND} \delta_{LEO} \cdot \delta_{GND}} {\delta_{LEO} \cdot \delta_{LEO}}
||x^*_{GND} - x^*_{LEO}|| = ||\delta_{GPS}||^2 + 2\delta_{GPS} \cdot (\mu_{GND} \delta_{GND} - \mu_{LEO} \delta_{LEO}) + ||\mu_{GND} \delta_{GND} - \mu_{LEO} \delta_{LEO} ||^2
∣∣xGND∗−xLEO∗∣∣=∣∣δGPS∣∣2+2δGPS⋅(μGNDδGND−μLEOδLEO)+∣∣μGNDδGND−μLEOδLEO∣∣2||x^*_{GND} - x^*_{LEO}|| = ||\delta_{GPS}||^2 + 2\delta_{GPS} \cdot (\mu_{GND} \delta_{GND} - \mu_{LEO} \delta_{LEO}) + ||\mu_{GND} \delta_{GND} - \mu_{LEO} \delta_{LEO} ||^2
\frac{\partial}{\partial\mu_{GND}} \rightarrow 2 \delta_{GPS} \cdot \delta_{GND} + 2 \delta_{GND} \cdot \delta_{GND} - 2 \mu_{LEO} \delta_{LEO} \cdot \delta_{GND}
∂∂μGND→2δGPS⋅δGND+2δGND⋅δGND−2μLEOδLEO⋅δGND\frac{\partial}{\partial\mu_{GND}} \rightarrow 2 \delta_{GPS} \cdot \delta_{GND} + 2 \delta_{GND} \cdot \delta_{GND} - 2 \mu_{LEO} \delta_{LEO} \cdot \delta_{GND}
\frac{\partial}{\partial\mu_{LEO}} \rightarrow 2 \delta_{GPS} \cdot \delta_{LEO} + 2 \delta_{LEO} \cdot \delta_{LEO} - 2 \mu_{GND} \delta_{LEO} \cdot \delta_{GND}
∂∂μLEO→2δGPS⋅δLEO+2δLEO⋅δLEO−2μGNDδLEO⋅δGND\frac{\partial}{\partial\mu_{LEO}} \rightarrow 2 \delta_{GPS} \cdot \delta_{LEO} + 2 \delta_{LEO} \cdot \delta_{LEO} - 2 \mu_{GND} \delta_{LEO} \cdot \delta_{GND}

Line-Segment Intersection

\mu_{LEO} = \max(\min(\mu_{LEO}, 1), 0)
μLEO=max(min(μLEO,1),0)\mu_{LEO} = \max(\min(\mu_{LEO}, 1), 0)
\mu_{GND} = \max(\min(\mu_{GND}, 1), 0)
μGND=max(min(μGND,1),0)\mu_{GND} = \max(\min(\mu_{GND}, 1), 0)
\mu_{1} > 1
μ1>1\mu_{1} > 1
\mu_{2} < 0
μ2<0\mu_{2} < 0

line-segment intersect

line intersect

This case is useful when the common-volume ROI is close to the COSMIC satellite.

x^*_{GND} = x_{GPS_1} + \mu_{GND} \delta_{GND}
xGND∗=xGPS1+μGNDδGNDx^*_{GND} = x_{GPS_1} + \mu_{GND} \delta_{GND}
x^*_{LEO} = x_{GPS_2} + \mu_{LEO} \delta_{LEO}
xLEO∗=xGPS2+μLEOδLEOx^*_{LEO} = x_{GPS_2} + \mu_{LEO} \delta_{LEO}

Common volume observation metric

We want a sense of how much "common-volume activity" is going on at a time...

line segment proximity

||x^*_{GND} - x^*_{LEO}||
∣∣xGND∗−xLEO∗∣∣||x^*_{GND} - x^*_{LEO}||

intersection point of interest (POI)

\frac{x^*_{GND} - x^*_{LEO}}{2}
xGND∗−xLEO∗2\frac{x^*_{GND} - x^*_{LEO}}{2}

Total Number of Geometries

1 receiver

6 COSMIC satellites

32 GPS satellites

1 \cdot 32 \cdot 6 \cdot 32 = 6144
1⋅32⋅6⋅32=61441 \cdot 32 \cdot 6 \cdot 32 = 6144

(but most of them we don't care about...)

possible common-volume geometries

at every moment

Mask/Filters

  • GPS 1 (for RX) elevation > threshold (5 degrees)
  • GPS 2 (for LEO) elevation > threshold (-20 degrees from LEO)

"We don't count geometries with line segments through the Earth!"

  • proximity < threshold (1500km)
  • POI-to-receiver distance < threshold (3000km)

"We don't count geometries that are not common or are too far out in space!"

Histogram

  • bin total # of valid common-volume geometries according to proximity metric
[50km, 100km, ..., 550km]
[50km,100km,...,550km][50km, 100km, ..., 550km]

bin cutoffs

Common volume observation

correlation with COSMIC elevation

CDAAC Data:

  • podPhs   (15-30min)
    • occultation and pre-occultation phase observables
    • LEO/GPS ECF coordinates
  • podTec  (15-30min)
    • calibrated TEC
    • LEO/GPS DCB values
    • LEO/GPS ECF coordinates
  • ionObs, opnGps, goxBin  (always)
    • ionosphere observations, open GPS format, GOX Binary format
    • phase observables, some for > 0 GPS elevation angle

 

we'll just use this

Common volume observation

Common volume observation

Peru

Alaska

Note: there is a lot of "missing" data, partially due to the fact that COSMIC 2 and 3 have essentially been out of commission since 2009.

Common volume observation

Peru

Alaska

"Event" detection

  • look at single COSMIC satellite
  • check against proximity threshold
    • i.e. 2 common-volumes at < 100km proximity
  • event window threshold to combine "short" events

"Event" detection

Peru

Alaska

Conclusions

  • definitely more common-volume observations (and occultations) at high latitude
  • nicer/more periodic common-volume at high latitude

Future Work

(future as in...tonight...)

  • extract long-term event results
    • periodicity of high-latitude
    • dominant azimuthal angle of ROI
    • CDAAC data availability statistics

References

  • SGP4:
    • http://utias-sfl.net/wp-content/uploads/SSC09-X-7.pdf
    • http://elib.dlr.de/87081/1/5a_P6_aida.pdf

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By Brian Breitsch

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