Reflection Open-loop Tracking

The software receiver from [Borre, Akos, et al. 2006] already performs the acquisition, tracking and navigation solution. These steps are additionally documented in GNSS texts such as [Kaplan 2006]. The following describes the approach for determining the time delay and frequency of the reflected signal from the outputs of the navigation receiver. These are then used to perform the geometric-tracking of the reflection for which we need the code phase, code frequency and carrier frequency.

The receiver clock will typically have a time offset from the GPS system time, and the GNSS transmitter’s atomic clock will additionally be offset from the system time of the GNSS. The navigation correlators tracking the direct signal are therefore tracking this time-offset signal.

At any measurement instant the correlator code phase will be a measure of the pseudorange, so called because it is the range determined by multiplying the signal propagation velocity, 𝑐, by the time difference between the two, non-synchronised, clocks. The measurement contains (1) the geometric range, (2) the offset due to the difference in time between system time and

the receiver clock and (3) an offset between system time and the transmitter clock. Due to the different geometric ranges over the direct and reflected paths, the reflected signal at the measurement epoch, necessarily departed the transmitter before that of the direct signal.

During this time the satellites will have moved and so the transmitter has two relevant

locations, one for each of the two paths to the receiver. This is as shown in Figure 4.2 against the system time of the GNSS.

Figure 4.2 Timing of signal propagation for direct and reflected rays

The timing relationships are,

𝑇_𝑇𝑥^𝐷 = System time at which the signal left the satellite (for direct path) 𝑇_𝑇𝑥^𝑅 = System time at which the signal left the satellite (for reflected path) 𝑇_𝑆 = System time at which the signal reflected from the Earth’s surface 𝑇_𝑅𝑥 = System time at which both signals reached the receiver

𝛿𝑡 = Advance of the transmitter clock from system time 𝑡_𝑅𝑥 = Offset of the receiver clock from system time 𝑹 = Position of receiver at time of measurement

𝑺 = Position of specular point at time of measured signal’s reflection, 𝑇_𝑆 𝑻^𝑫, 𝑻^𝑹 = Position of transmitter at time of transmission for the direct and reflected

rays respectively 𝒄 = speed of light

For the direct signal the geometric range is,

|𝑹𝑻⃗⃗⃗⃗⃗⃗⃗⃗ | = 𝑐(𝑇^𝑫 _𝑅𝑥− 𝑇_𝑇𝑥^𝐷 ) _(4.1)

And the pseudorange is formed as in the conventional navigation receiver,

𝜌_𝐷 = 𝑐((𝑇_𝑅𝑥+ 𝑡_𝑅𝑥) − (𝑇_𝑇𝑥^𝐷 + 𝛿𝑡))

= 𝑐(𝑇_𝑅𝑥+ 𝑇_𝑇𝑥^𝐷) + 𝑐(𝑡_𝑅𝑥− 𝛿𝑡)

= |𝑹𝑻⃗⃗⃗⃗⃗⃗⃗⃗ | + 𝑐(𝑡^𝑫 _𝑅𝑥− 𝛿𝑡)

(4.2)

For the reflected signal the geometric range is,

|𝑹𝑺𝑻⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ | = 𝑐((𝑇^𝑹 _𝑆− 𝑇_𝑇𝑥^𝑅 ) + (𝑇_𝑅𝑥− 𝑇_𝑆))

= 𝑐(𝑇_𝑅𝑥− 𝑇_𝑇𝑥^𝑅 ) ^(4.3)

And the pseudorange of the reflected ray is,

𝜌_𝑅 = 𝑐(𝑇_𝑆− (𝑇_𝑇𝑥^𝑅 + 𝛿𝑡) + (𝑇_𝑅𝑥+ 𝑡_𝑅𝑥) − 𝑇_𝑆)

= 𝑐(𝑇_𝑅𝑥+ 𝑇_𝑇𝑥^𝑅 ) + 𝑐(𝑡_𝑅𝑥− 𝛿𝑡)

= |𝑹𝑺𝑻⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ | + 𝑐(𝑡^𝑹 _𝑅𝑥− 𝛿𝑡)

(4.4)

A simplification can be made in the case of the transmission time difference (𝑇_𝑇𝑥^𝐷 − 𝑇_𝑇𝑥^𝑅 ) being small. To test this we take a scenario of receiver and transmitter in circular orbits at 700 km and 20,200 km altitude respectively. The largest range difference (|𝑹𝑺𝑻⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ | − |𝑹𝑻^𝑹 ⃗⃗⃗⃗⃗⃗⃗⃗ |) ^𝑫 occurs when the receiver is directly below the transmitter. The maximum specular point to receiver time (𝑇_𝑆− 𝑇_𝑅𝑥) is the time for propagation over the receiver altitude, which is 2.3 ms. The maximum transmitter, specular point to receiver time is twice this, 4.7 ms.

The LEO receiver has speed 7.5 km/s which will cause the specular point to move at a speed of about 6.5 km/s so will move along the surface by 15 m between the reflection and

measurement time. The GNSS transmitter will have speed 3.8 km/s and travel 18 m between the transmission and reflection time.

These would be significant if the receiver application were surface altimetry, where the aim is to measure the surface height to centimetre accuracy. However as the focus of this research is on scatterometry, these distances are a fraction of a chip length and so we can treat 𝑺 to be at the measurement time and the transmission times to be the same, 𝑻^𝑫= 𝑻^𝑹.

It is therefore possible to set the delay of the reflected signal by applying an offset to the code phase of the navigation correlator as follows,

𝜌_𝑅− 𝜌_𝐷 = (|𝑹𝑺𝑻⃗⃗⃗⃗⃗⃗⃗⃗ | + 𝑐(𝑡_𝑅𝑥− 𝛿𝑡)) − (|𝑹𝑻⃗⃗⃗⃗⃗ | + 𝑐(𝑡_𝑅𝑥− 𝛿𝑡))

= |𝑹𝑺𝑻⃗⃗⃗⃗⃗⃗⃗⃗ | − |𝑹𝑻⃗⃗⃗⃗⃗ | ^(4.5)

where the geometric ranges are determined in the navigation solution and through calculating the specular point position.

To set the carrier frequency for the reflection, this was calculated in the software receiver by numerically differentiating path length with respect to time,

𝑣_𝑅𝑆𝑇 =𝛿|𝑹𝑺𝑻⃗⃗⃗⃗⃗⃗⃗⃗ |

𝛿𝑡 ^(4.6)

and keeping the time interval 𝛿𝑡 relatively small. The Doppler shifted carrier frequency, 𝑓_𝐿1,𝑅, of the reflected signal is then,

𝑓_𝐿1,𝑅 = (1 −𝑣_𝑅𝑆𝑇 𝑐 ) 𝑓_𝐿1

(4.7)

where 𝑓_𝐿1 is the nominal GNSS carrier frequency (marked as being the GPS L1 frequency here).

The numerical differentiation provides acceptable error as the coherent integration time is relatively short. If 𝑇_𝑐𝑜ℎ = 1 ms, then the signal AF is relatively wide at 2000 Hz null-to-null, so a 100 Hz error would be considered small, which corresponds to a relaxed requirement of 20 m/s velocity accuracy, which this method achieves.

Following calculation of the Doppler shifted carrier frequency of the specular point, this now needs to be offset due to the receiver clock drift rate, 𝑡_𝑅𝑥̇ , which is the rate at which the receiver clock is running fast or slow relative to system time. This is an output from the velocity part of the navigation solution and has the units seconds / second. The carrier frequency is therefore set to be,

𝑓^′= 𝑓_𝐿1,𝑅+ 𝑓_𝐿1𝑡_𝑅𝑥̇

(4.8)

The code rate of the reflected signal, 𝑓_𝑐′, is then scaled by the ratio of the nominal code frequency, 𝑓_𝑐, to carrier frequency.

𝑓_𝑐′ = 𝑓^′⋅ 𝑓_𝑐

𝑓_𝐿1 ^(4.9)

These code phase, code and carrier frequencies are calculated strictly from the geometry of receiver, transmitter, and the current clock drift and drift rate in as open-loop tracking and are calculated periodically steer the DDM processing. The DDM of the reflected signal was then calculated in the software receiver equivalently to that described in Section 2.3.1.