[34] In tracking the transmitted GPS code signal, the received signal is correlated with a locally generated replica of the code. The receiver computes the correlation function between the received signal and the internally generated signal at three different modeled delays called “prompt”, “early”, and “late”. The “early” and “late” delays are different from the “prompt” delay by a receiver sampling interval +*S* and −*S**ns*, respectively. The receiver effectively fits a equilateral triangle with base length of two chip period, 2*T*, on these three points and declares the location of the peak to be the true delay. In the absence of any multipath, the correlation between the received signal and the receiver generated code can be approximated by a equilateral triangle with a peak value of *A* as shown in Figure 2 (thin solid line triangle) and a phase exp[*i*(ϕ_{m} − ϕ)]. τ_{m} and ϕ_{m} are the modeled time delay and phase. In the presence of a single multipath signal with an additional time delay of Δτ_{1}, amplitude of *A*_{1}, and phase shift ϕ_{1}, the correlation function can be modeled as the sum of the two triangles corresponding to the direct and the multipath signal as shown in Figure 2. The presence of multipath signals corrupt the triangular shape.

##### 3.2.1. Wide Sampling Interval

[36] In the case of the sampling intervals longer than half of the chip length (*S* > *T*/2), the resulting error in the code measurement, Δτ^{g}, induced by the presence of a single multipath is given by one of the following formulas [*Young and Meehan*, 1988]: Region 1 which applies when Δτ_{1} < *T* − *S* + Δτ^{g}:

Region 2 which applies when *T* − *S* + Δτ^{g} < Δτ_{1} < *S* + Δτ^{g}:

Region 3 which applies when *S* + Δτ^{g} < Δτ_{1} < *T* + *S* + Δτ^{g}:

Region 4 which applies when Δτ_{1} > *T* + *S*:

[37] Figure 3 shows the P1 pseudorange multipath induced error Δτ^{g} as a function of the multipath delay Δτ_{1}. Values of *T* = 98 *ns*, *S* = 60 *ns*, and *A*_{1}/*A* = 0.1 are used. The envelope of the multipath error can be readily seen in the figure. The upper envelope corresponds to the multipath error which is in-phase with the direct signal, while the lower envelope corresponds to the out-of-phase case. The three different slopes of the upper or lower envelopes correspond to the three different regimes of equations 15–17 and they are separated by two thick black lines in the figure for clarity. The asymmetry of the envelope is amplified for higher values of *A*_{1}/*A*.

[38] The C/A pseudorange multipath induced error is also given by equations 15–18 as long as *S* > *T*/2, where *T* is a C/A code chip period. This causes the different regimes of equations 15–17 to trigger at different values of Δτ_{1}. Specifically, the C/A code multipath induced error can grow to 10 times larger than the P-code and does not vanish until Δτ_{1} > *T* + *S*. On the other hand, for various reasons it is the region of small multipath delays that is most important for most GPS applications, and in this region the P1 and C/A multipath errors are the same.

##### 3.2.2. Narrow Sampling Interval

[39] When *S* < *T*/2, Δτ^{g} is given by one of the following formulas: Region 1 which applies when Δτ_{1} < *S* + Δτ^{g}:

Region 2 which applies when *S* + Δτ^{g} < Δτ_{1} < *T* − *S* + Δτ^{g}:

Region 3 which applies when *T* − *S* + Δτ^{g} < Δτ_{1} < *T* + *S*:

Region 4 which applies when Δτ_{1} > *T* + *S*:

Note that equation 20 is not a function of *T*, and the boundary values of each region are different.

[40] Figure 4 shows the P1 pseudorange multipath induced error Δτ^{g} for the *S* < *T*/2 case. Values of *T* = 98 *ns*, *S* = 48 *ns*, and *A*_{1}/*A* = 0.1 are used. A main distinction between the narrow and wide sampling interval is that in the former, the error exhibit a constant peak value in region 2. This becomes clear as we consider the corresponding figure for the C/A code. Figure 5 was generated using *T* = 980 *ns*, *S* = 48 *ns*, and *A*_{1/}*A* = 0.1. Note that the C/A code multipath induced error exhibits a very wide region 2 and does not vanish until *T* + *S* = 1028 *ns*.

##### 3.2.3. Multipath Induced Bias

[41] When *A*_{1}/*A* ≪ 1, region 1 translates to Δτ_{1} < *T* − *S* (wide sampling) and to Δτ_{1} < *S* (narrow sampling). For *S* = 60 *ns* (wide sampling) and 48 *ns* (narrow sampling) this translate to *c*Δτ_{g} < 11 *m* and *c*Δτ_{g} < 14 *m*, respectively. These are above the multipath distance that we will deal with in our discussion later. Therefore, equation 15 (which is identical to 19) becomes valid for P1, P2, and C/A code.

[42] In order to understand the effect of the denominator in equation 15, we plot

versus *x* for several values of *A*_{1}/*A*. Figure 6 shows that the effect of the denominator is to change the amplitude asymmetrically for in-phase and out-of-phase multipath. For example, if *A*_{1}/*A* = 0.5 then the positive peak is reduced to 2/3 and the negative one to −2. Note that this will introduce a negative bias when integrating over a complete cycle. Averaging equation 23 over a complete cycle gives

By averaging over *N* multipath cycles, we get *N* times the one cycle average. This implies that multipath does not average out and could cause a significant bias when *A*_{1}/*A* is not small. Figure 7 shows a magnified version of region 1 for *A*_{1}/*A* = 0.5.When *A*_{1}/*A* ≪ 1, Δτ^{g} varies more or less sinusoidally as a function of the multipath delay, and the multipath bias becomes second-order in *A*_{1}/*A*. For small ratios of *A*_{1}/*A*, the multipath bias is approximately, −1/2 Δτ_{1}(*A*_{1}/*A*)^{2}.

[43] Setting aside the amplitude scaling effect of the denominator in equation 15, we can concentrate on the numerator part of the equation and use the approximation

In order to understand more the implication of equation 25, we consider an example of a plane reflector as shown in Figure 8. In this example, we have

where *L*_{m} is the extra traveled distance due to multipath, *L* is the antenna height, and θ is the satellite elevation angle. Let θ = ω_{s}*t*, where ω_{s} is the orbital angular frequency of the transmitting satellite as seen by the receiver. By substituting equation 26 in equation 25, we get:

where λ is wavelength of the carrier signal, and Δ*P* is *c*Δτ^{g}. Based on equation 27, Δ*P* is governed by two oscillations: a slow one, ω_{s}, and a fast one governed by the *cos* term in equation 27. In order to quantify this fast oscillation, we expand the *sin* term around a reference angle ϕ_{r} as follows:

Using this approximation in equation 27, dropping constant phase terms, and keeping the time varying terms, we can get:

Since ω_{s} = 2π/τ_{s} where τ_{s} is the orbital period of the satellite, we can derive from equation 29 that the period for high frequency multipath oscillation is

As is evident from equation 30, the multipath oscillation period is proportional to the wavelength. This implies that P1 multipath is oscillating faster than P2.

[44] Using the values of λ_{1} and λ_{2}, and the average value of cos (ϕ_{r}) from 0 to π/2 which is 2/π in equation 30, we arrive at the simple “rule of thumb”: In about one quarter of a GPS satellite's revolution (i.e. tracking from horizon to zenith), we should count 10*L* fast oscillations for P1 code multipath error and 8*L* fast oscillations for P2 code, respectively, where *L* is measured in meters. When the receiver is on a low earth orbiter (LEO), ω_{s} will be mainly determined by the LEO orbital frequency.

[45] Considering the situation of Figure 8 with equation 15, we can plot the P1 and P2-code range error due to one multipath source. Figures 9a and 9b show multipath error for P1 and P2 for *L* = 1 *m* case, respectively. These figures clearly show fast 10 and 8 oscillations, respectively.

[46] When the ionospheric free linear combination is considered (equations 7 and 8), then the multipath frequencies cause a beating phenomenon with two main frequencies: (*f*_{1} + *f*_{2})/2 and (*f*_{1} − *f*_{2})/2. These frequencies correspond to a long (λ_{L}) and short (λ_{S}) wavelengths given by

and

Note that these wavelengths are not to be confused with the widely known wide lane and narrow lane wavelengths.

[47] Using λ_{1} and λ_{2} equal to 0.19 *m* and 0.24 *m*, we get λ_{L} and λ_{S} equal to 1.8 *m* and 0.21 *m*, respectively. We can arrive at a similar rule of thumb for PC measurement: During the time the GPS satellite goes from horizon to overhead, we should count 10*L* fast oscillations and *L* slow oscillations where *L* is measured in meters. Figure 9c shows multipath error for ionospheric-free PC measurement for *L* = 1 *m*, and demonstrates 10 fast oscillations and 1 slow oscillation. All three figures show that averaging over one satellite pass does not average to zero.

[48] Equation 30 can also be directly arrived at by noticing that

and by using equation 28. Equation 33 is intuitively obvious from the following argument. If in one second the extra multipath distance changes by *N* carrier wavelength, we should expect *N* oscillations in the P-code delay or the carrier phase measurements. This is true irrespective of the type of the antenna or tracking loop. From this argument, we can conclude that equation 30 is true for flat surfaces for both the P-code and the carrier phase and for all types of receivers. For other types of surfaces, we should apply equation 33 with the appropriate value for *L*_{m}. The above description treats the special case of an antenna above a reflecting plane. There are corresponding rule-of-thumb derivations for reflectors and diffractive sources at various angles.