SP3 Interpolation

SP3 interpolation, in brief

SP3 files provide sampled precise satellite positions and, when available, clock corrections. Interpolation estimates a state between those samples. File units, time system, interpolation interval and missing-value markers must be respected; interpolating the orbit is only one input to a positioning solution.

Prepared by OpenGNSSLabUpdated 30 September 2026Sources, reproducibility and corrections

Learning objectives

Identify SP3 header and position records.
Understand coordinate and clock units.
Interpolate a satellite coordinate at a non-SP3 epoch.

Why SP3 exists

Broadcast navigation files give Keplerian parameters and meter-level orbit quality. SP3 files provide precise Cartesian satellite coordinates and clocks estimated by analysis centers.

SP3 is essential for PPP and precise geodesy because the satellite position is already expressed in an ECEF reference frame such as IGS20/ITRF.

Navigation file vs SP3

Aspect	Broadcast navigation	SP3
Content	Keplerian parameters	Cartesian coordinates and clocks
Typical orbit accuracy	Meters	Centimeters
Main use	Routine SPP	PPP, POD, precise geodesy
Computation	Solve broadcast orbit model	Read and interpolate coordinates

Raw data excerpt

Annotated SP3 coordinate excerpt

example.sp3

hover or focus fields

1*Epoch markerAn asterisk starts an SP3 epoch record.  2024 01 01 00 00 00.00000000Epoch timeCoordinate epoch in the SP3 time scale.2PPosition recordP lines contain satellite position and clock.G01SatelliteGPS PRN 01.  12908.438647XX coordinate in kilometers. -10025.115815YY coordinate in kilometers.  20508.373980ZZ coordinate in kilometers.    164.997276ClockSatellite clock in microseconds; 999999.999999 marks missing data.

SP3 coordinates are Cartesian positions in kilometers; most positioning engines convert them to meters.

SP3 position line

PG01 12908.438647 -10025.115815 20508.373980 164.997276

Line interpretation

`P` means position record, `G01` means GPS PRN 01, the next three columns are X/Y/Z in kilometers, and the last column is satellite clock in microseconds.

A clock value of `999999.999999` marks missing or invalid clock data.

Lagrange interpolation

X(t)=\sum_{i=0}^{n}X_iL_i(t),\qquad L_i(t)=\prod_{j\ne i}\frac{t-t_j}{t_i-t_j}

Apply the same expression independently to Y(t), Z(t), and clock values when appropriate.

Five G01 epochs

Epoch	tk (s)	X (km)	Y (km)	Z (km)
00:00:00	0	12908.438647	-10025.115815	20508.373980
00:05:00	300	12988.440556	-9216.822128	20838.836691
00:10:00	600	13084.429049	-8398.467123	21127.872753
00:15:00	900	13196.669424	-7571.999376	21374.927788
00:20:00	1200	13325.318847	-6739.388472	21579.534446

Step 1

Choose the target time

Assume the receiver observation occurred at 00:07:30 GPST, or 450 seconds after the first SP3 epoch.

t=450\ \text{s}

Step 2

Build basis polynomials

The first two basis polynomials show the pattern; the remaining bases follow the same product structure.

L_0(t)=\frac{(t-300)(t-600)(t-900)(t-1200)}{(0-300)(0-600)(0-900)(0-1200)}

L_1(t)=\frac{(t-0)(t-600)(t-900)(t-1200)}{(300-0)(300-600)(300-900)(300-1200)}

Result

Interpolated coordinate

(13034.417, -8808.781, 20988.570) km

Evaluating the Lagrange polynomial at 450 s gives X = 13034.417 km, Y = -8808.781 km, and Z = 20988.570 km.

Convert to meters before feeding most positioning software: approximately (1.3034e7, -8.8088e6, 2.0989e7) m.

Migration note

The legacy `interpolation_example.html` is a broadcast ephemeris satellite-position calculation, not an SP3 interpolation page. It is useful as a contrast between broadcast orbit propagation and SP3 interpolation, but it should not be merged blindly.

What you should understand now

SP3 coordinates are already Cartesian and precise.
The file sampling interval rarely matches observation epochs exactly.
Interpolation must be performed independently for each coordinate component.

Help improve this tutorial

Point out numerical ambiguity, missing prerequisites, or a step that needs a fuller derivation.

SP3 Interpolation