Klobuchar Ionospheric Correction

Klobuchar ionospheric correction, in brief

The Klobuchar model estimates an L1 ionospheric group delay from broadcast coefficients, receiver location, satellite direction and time. Its intermediate angular quantities use semicircles. A correction sign follows the observation equation: code group delay and carrier-phase advance have opposite signs.

Prepared by OpenGNSSLabUpdated 30 September 2026Sources, reproducibility and corrections

Learning objectives

Locate the ionospheric pierce point.
Evaluate broadcast alpha/beta polynomials.
Convert the L1 time delay into meters.

Model idea

Klobuchar is the standard broadcast ionospheric model for single-frequency GPS. It approximates the vertical ionospheric delay at an ionospheric pierce point and maps that value to the slant path.

The model uses receiver location, satellite elevation and azimuth, GPS time, and eight navigation-message coefficients.

Input parameters

Parameter	Value
Elevation E	45.23456789012 deg
Azimuth A	138.67890123456 deg
User latitude	37.43678912345 deg
User longitude	51.32456789012 deg
GPS time	259400 s
alpha	[0.2335e-7, 0, 0.5960e-7, 0.1192e-6]
beta	[0.1464e6, 0.1966e6, 0, 0.1966e6]

Step 1

Earth-centered angle

Compute the angle between the receiver radius vector and the ionospheric pierce point radius vector.

\psi = \frac{\pi}{2}-E-\arcsin\left(\frac{R_E}{R_E+h}\cos E\right)

\psi = 2.888407^\circ

Step 2

Ionospheric pierce point

Propagate from receiver latitude/longitude along the satellite azimuth by the earth-centered angle.

\varphi_I=\arcsin(\sin\varphi_u\cos\psi+\cos\varphi_u\sin\psi\cos A)=35.244303^\circ

\lambda_I=\lambda_u+\frac{\psi\sin A}{\cos\varphi_I}=53.659764^\circ

Step 3

Geomagnetic latitude and local time

The daily ionospheric model is evaluated using geomagnetic latitude and local solar time at the pierce point.

\varphi_m=28.404910^\circ

t = 43200\frac{\lambda_I}{\pi}+t_{GPS}=13078.34\ \text{s}

Step 4

Amplitude, period, and phase

Evaluate the broadcast coefficient polynomials and the phase of the daily delay curve.

A_I=\sum_{n=0}^{3}\alpha_n\left(\frac{\varphi_m}{\pi}\right)^n=2.530261\times 10^{-8}\ \text{s}

P_I=\sum_{n=0}^{3}\beta_n\left(\frac{\varphi_m}{\pi}\right)^n=178197.06\ \text{s}

X_I=\frac{2\pi(t-50400)}{P_I}=-75.398531^\circ

Step 5

Slant factor and L1 delay

The mapping factor converts the modeled vertical delay to the slant path. The result is a time delay on L1.

F=\left[1-\left(\frac{R_E}{R_E+h}\cos E\right)^2\right]^{-1/2}=1.343040

I_1=(5\times10^{-9}+A_I\cos X_I)F=1.528197\times10^{-8}\ \text{s}

I_{meters}=cI_1=4.584592\ \text{m}

Legacy-source note

The retained legacy exercise uses a thin-shell slant factor. The standard IS-GPS-200 Klobuchar algorithm uses F = 1 + 16(0.53 − E)^3, with E in semicircles; these mapping functions must not be substituted silently. See the specification below when implementing the broadcast model.

The original HTML page computes 4.584592 m in the final step but its summary says 2.01 m. The numerical chain supports 4.584592 m, so the migrated tutorial uses the computed value and avoids the contradictory summary.

Result

Computed result

4.584592 m

For this geometry and coefficient set, the GPS L1 ionospheric slant delay is about 4.58 m.

What you should understand now

The broadcast coefficients are not delays by themselves; they shape a daily delay curve.
Elevation controls the slant factor strongly.
The correction enters the SPP code observation equation as a positive range delay.

Help improve this tutorial

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

Klobuchar Ionospheric Correction