The major range error for GPS measurements is mainly due the deviation of the speed of the signal from its actual light speed because of the presence of free electrons in the ionosphere medium. This medium is extended from 50 km to about 1000 km above the earth surface. The variations of the ionospheric effects are mainly governed by the ionization processes, which is caused by the solar radiation. Hence there is a direct relationship and the state of the ionosphere can be realized by observing the intensity of the solar activity. The physical characteristics of the ionosphere have noticeable diurnal (day and night) variations. During the sun rise, the electron density starts to build up due to the ultraviolet radiations which help to break up gas molecules into ions and free electrons (Leick, 2004; Hofmann-Wellenhof et al., 2008).

Single-frequency receivers can access eight ionospheric coefficients, located in the GPS navigation message to estimate the ionospheric delay based on the Klobuchar model. These coefficients are generated at least once per 6 days but no more than once a day and they are updated by the 5 GPS Ground Control Segment stations. The Klobuchar algorithm is a physical model that considers the changes in latitude, season, solar flux and magnetic activity representing the amplitude change along with the associated diurnal period change of the ionospheric delay. The Klobuchar model permits to correct about 50% of the ionospheric error for mid-latitudes location and average ionospheric environment. Therefore, the ionospheric modeling has been investigated for last few decades to develop an accurate ionospheric correction for single-frequency receiver applications. To develop a regional ionospheric model, the ionospheric error is estimated using dual frequency receivers distributed in the area under consideration as discussed below.

GPS signals are transmitted in two *L* frequency bands (*L1* and *L2*) and a dual frequency receiver can be used to provide code measurements (*P*
_{
f1} and *P*
_{
f2}) and carrier-phase measurements (*ϕ*
_{
f1} and *ϕ*
_{
f1}). The receiver is able to assess the ionospheric delay on *f1* using code measurements combination *I*
_{
P − f1} or using carrier-phase measurements combination *I*
_{
ϕ − f1} of identical amplitudes as follows (Hofmann-Wellenhof et al., 2008):

$$ {I}_{P- f1}=\frac{\left({P}_{f1}-{P}_{f2}\right).{f}_2^2}{f_2^2-{f}_1^2} $$

(3)

$$ {I}_{\phi - f1}=\frac{\left({\phi}_{f2}-{\phi}_{f1}\right).{f}_2^2}{f_2^2-{f}_1^2} $$

(4)

The integration of the above code measurements combination and carrier-phase measurements combination along with the ionospheric delay from equation (1) can produce the smoothed value of the TEC as follows:

$$ T E C=\frac{1}{40.3}{\left(\frac{1}{{f_1}^2}-\frac{1}{{f_2}^2}\right)}^{-1}\left(\left({\phi}_{f2}-{\phi}_{f1}\right)-{\left({\phi}_{f2}-{\phi}_{f1}\right)}_0+{\left({P}_1-{P}_2\right)}_0\right)\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\left( el/{m}^2\right) $$

(5)

where (*ϕ*
_{
f2} − *ϕ*
_{
f1})_{0} and (*P*
_{1} − *P*
_{2})_{0} are the initial values of the carrier-phase measurements combination and code measurements combination, respectively, and *el* is the number of electrons. Fig. 1 shows an example of a noisy and smoothed TEC for one satellite.

The ionospheric delay is estimated at the ionosphere pierce point (IPP) that is defined as the intersection between the constant ionosphere ellipsoid (350 km above the WGS84 ellipsoid) and the line in view from the receiver antenna reference point to the satellite antenna reference point. Fig. 2 shows he cross-section of main components; Earth, receiver, ionosphere shell, satellite and their spatial relationship. If the elevation angle of a satellite (*α*) is estimated, the Zenith angle at receiver (*z* ') and the Zenith angle at IPP (*z*) can be estimated as follows:

$$ z\hbox{'}={90}^{\circ }-\alpha $$

(6)

$$ z={ \sin}^{-1}\left(\frac{R. \sin \left({z}^{\prime}\right)}{R+ h}\right) $$

(7)

It is worth noting that an appropriate cut-off angel is used to ignore those satellites with bad geometry and under horizon level (Hofmann-Wellenhof et al., 2008).

In practice, the ionospheric model is developed using the vertical total electron content (VTEC) that should be estimated at the IPP. Using a simple geometric function, vertical Total Electron Content can be estimated as:

$$ VTEC= T E C\times C o s(z) $$

(8)

where *z* is the zenith angle at pierce point.

In this paper, the wavelet network model is proposed to model the estimated VTEC form a number of GPS receivers in a regional area and the model was validated by the Global Ionospheric Model (GIM) developed by the CODE analysis center.