GPS World, October 2009

INNOVATION Algorithms Methods 3 But how is multipath frequency related to quantities measured by our GNSS receivers the code range carrier phase and signal to noise ratio SNR To answer that question we must introduce another set of multipath quantities which describe the dominant signal strength factors TABLE 2 for the direct and multipath signals we ignore thermal noise cable losses etc The amplitude of the direct signal A d is equivalent to the GNSS signal strength as it is received and is affected by the antenna gain pattern Figure 1b The multipath signal comes through the antenna gain pattern at a different angle by design most GNSS antennas will apply less gain at angles consistent with common multipath geometries such as below the antenna horizon The multipath signal will also experience some amount of attenuation upon refl ection the combination of attenuation and antenna gain yields the amplitude of the multipath signal A m Note that the broadcast GNSS signals are right hand circularly polarized RHCP which are largely converted to left hand polarization upon refl ection Thus the simplifi ed gain pattern introduced here must incorporate both RHCP and LHCP patterns Under the simplifi ed model of GNSS receiver response to tracking direct plus short delay smaller than 15 code chips refl ected signals the multipath relative phase and signal amplitudes describe both the code and carrier phase multipath errors respectively denoted and 4 5 These equations are derived from code and carrier tracking behavior in the presence of multipath Look in Further Reading for precise derivations and additional background material In addition to carrier phase and code observables GNSS receivers routinely record SNR or the related carrier to noisedensity ratio C N 0 for each satellite As the term indicates SNR is a ratio of signal power to the noise fl oor of the GNSS observation and has conventionally been used only for comparison of signal strengths between channels and satellites or to assess interference Like code and carrier phase multipath errors SNR is a function of multipath phase and signal strengths 6 If we remove the effects of the direct signal the remaining SNR is due only to multipath and is reduced to a simple function of multipath signal amplitude relative phase and a timeinvariant phase offset 7 Note that the equations for code multipath carrier phase multipath and SNR contain the cosine or sine of the multipath relative phase Therefore all three GNSS observables will have quasi sinusoidal behavior driven by To illustrate this FIGURE 2 gives an example for a rising satellite refl ecting off horizontal ground 10 meters below the antenna All three GNSS observables oscillate at the same frequency however pseudorange error and SNR are in phase while carrier phase error is 90 degrees out of phase In this article we use SNR observations to understand and quantify multipath effects We choose SNR over the other observable types because multipath effects on SNR have the most unambiguous relationship to multipath Typical levels of pseudorange noise will swamp all but the most extreme of multipath errors carrier phase data are more precise but extracting multipath from these data requires fi rst modeling clocks orbits and atmospheric delays SNR data are directly related to carrier phase multipath are largely independent of the above effects and are determined independently for individual satellites Unfortunately not all GNSS receivers provide SNR data with the requisite precision and accuracy to clearly observe the multipath relationships see Scientifi c Utility of the Signal to Noise Ratio SNR Reported by Geodetic GPS Receivers in Further Reading for information on high utility SNR When SNR data are of suffi cient quality they can provide a unique and direct window on the multipath errors affecting the code and carrier observations SNR Multipath Applications A number of new scientifi c applications of SNR data are evolving to exploit the above multipath relationships In the following sections we describe three different SNR multipath applications and provide relevant although not exhaustive references All of these applications draw upon the above relationships and require precise and accurate SNR data that conform to the simplifi ed multipath model described above Multipath Corrections Recall that the multipath errors in GNSS observables are simply a function of signal amplitudes and the relative phase between direct and multipath signals It stands to reason that if these amplitudes and phases can be estimated we can model and remove multipath errors from our code and carrier observations SNR data allow us to do just that After extracting the direct signal A d to reveal the SNR due only to multipath SNR MP this remaining time series depends only on A m and As shown in Figure 2 and Equation 7 SNR due to multipath oscillates with a constituent frequency which is the time derivative of and has an amplitude envelope equivalent to A m Therefore from SNR due to multipath we can estimate multipath relative phase and multipath amplitude as a function of time This idea of modeling SNR data to estimate multipath parameters as time varying quantities was fi rst explored in a GPS World October 2009 www gpsworld com 34