Atmospheric delay. X, Y, Z : satellite cartesian coordinates. Z : receiver cartesian coordinates. In the vacuum the signal speed c is constant

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1 Atmospheric delay In the vacuum the signal speed c is constant c τ = ρ = ( X X ) + ( Y Y ) + ( Z Z ) S S S 2 S 2 S 2 X, Y, Z : receiver cartesian coordinates S S S X, Y, Z : satellite cartesian coordinates In the atmosphere, the signal propagation velocity varies according to the physical state of the medium. It is the so called atmospheric effect and ranges from 5 m to 200 m.

2 The signal traveling time depends on its propagation velocity v. Outside the atmosphere v is equal to c. In the atmosphere it varies from point to point: τ S = dx S v( x ) ρ ρ is the geometric trajectory of the signal, x is the integration point, v( x ) is the propagation speed.

3 Therefore cdx cτ = = n( x) dx= dx+ ( n( x) 1) dx= ρ +Δ S v( x) S S S S S S ρ ρ ρ ρ n( x ): refraction index in x, n( x) = c/ v( x ), S ρ : distance between satellite and receiver, S S S Δ = cτ ρ = ( n( x ) 1) dx: atmospheric delay, in length units. ρ S

4 Ionospheric delay Due to the free electrons presence between 100 and 1000 km altitude: A NE ( ) 1 ( x n x = ± ) 2 f 3 2 A= 40.3m s f : frequency (Hz), either of L 1 or L 2 N ( ) E x : local electron density (n/m3) Iono delay is positive (1+) for codes, negative (1-) for the carriers.

5 Final delay (order of magnitude) I S f (t) = Iono (n(x) 1)dx = ± A TEC S (t) f 2 Elevation ( ) Low (m) Medium (m) High (m) Low: night time. Medium: day time, common solar activity. High: day-time, intense solar activity.

6 Satellites send predicted models (Klobuchar model) to compute TEC along the signals trajectories. Ionospheric error is not equal on L1/P1 and L2/P2: particularly between I1 and I2 the following holds TEC () t f TEC () t f TEC () t I2 ( t) A A A S 2 S 2 S S 1 1 = = = f2 f1 f2 f2 f1 = f f S I1() t

7 One example of Ionospheric delay From the Astronomisches Institut, Universität Bern (CH)

8 Tropospheric delay Due to air and water vapor content of the atmosphere from 0 to 40 Km of height. The local refraction index is given by: P( x) e( x) n( x) = 1+ k1 + k2 2 T( x) T ( x) P: pressure (mbar) T : temperature ( K) e : water vapor pressure (mbar) k 1, k 2 : constants experimentally determined.

9 The first term is called dry component: about 90% The second integral is called wet component. Elevation Low (m) Medium High (m) ( ) (m) Low: dry, cold climate. Medium: temperate climate. High: hot-humid climate

10 Standard models to estimate the tropospheric delay exist, that are based on some ideal hypothesis on the atmosphere. Saastamoinen model Tropospheric models do never completely represent the real meteorological state at the surveying place, during the survey. Any modeling of T leaves a residual error, whose order of magnitude is about 5% of the total delay.

11 The final code observation equation S S S P () t = cτ () t + c( dt () t dt ()) t S S S S = ρ () t + cdt ( () t dt () t ) + I () t + T () t S S S Satellite position [ X, Y, Z ] (in ρ ): known from the ephemerides receiver position [ X, Y, Z ] (in ρ ): unknown, S satellite clock offset dt, known from the navigation message, receiver clock offset dt unknown, ionospheric and tropospheric delays IT:, computable by standard models. The equation contains 4 receiver related unknowns [,,, ] X Y Z dt. 4 satellites in view guarantee their estimation in single epoch!

12 The final carrier phase observation equation S S S S S S L () t = ρ () t + c( dt () t dt ()) t I () t + T () t + λη () t The observation equation is similar to the code one, but for the ambiguity λη S () t. S Each new satellite adds a new λη term! No estimation is possible in single epoch

13 Configuration: one receiver - satellite couple, two consecutive epochs, t 1 and t 2 L ( t ) = ρ ( t ) + c( dt ( t ) dt ( t )) I ( t ) + T ( t ) + λη ( t ) S S S S S S S S S S S S 2 = ρ λη 2 L ( t ) ( t ) c( dt ( t ) dt ( t )) I ( t ) T ( t ) ( t ) The integer part of the ambiguity at t 2 is given by N ( t ) = N ( t ) + n ( t, t ) S S S S ( 1, 2) n t t : integer number of cycles passed during ( t1, t 2). The observation equation at epoch t 2 is given by S S S S S L ( t ) = ρ ( t ) + c( dt ( t ) dt ( t )) I ( t ) + T ( t ) λ( N ( t ) + n ( t, t )) + λ( φ φ ) S S S

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15 If no loss of lock happens during ( t1, t 2), the receiver can count and S record n ( t1, t 2) λη ( t ) = N ( t ) + ϕ ϕ = N ( t ) + ϕ ϕ + n ( t, t ) = λη ( t ) + n ( t, t ) S S S S S S S S For a satellite-receiver couple, a unique unknown initial ambiguity η S ( t ) exists, related to the first observation epoch. 1 The initial ambiguity is unknown but constant in time However, what happens in case of a loss of lock between satellite and receiver? Cycle slip

16 Cycle slip Usually a cycle slip (CS in the following) occurs when an obstacle (such as a wall, a building..) blocks the signal going from satellite to receiver, anomalous phenomena happen in the ionosphere, causing the so-called signal scintillations.

17 During a cycle slip the counting is interrupted. When the signal is again tracked, a new unknown ambiguity has to be introduced. Imagine to have a clock with seconds, but not minutes. By looking at the clock, you never know the absolute time, but you have access to the seconds: this information contains an initial ambiguity with respect to the absolute time. If you continuously look at the clock, you can always know the elapsed minutes from the first observation: the only ambiguity is the absolute minute of the first observation. On the contrary, if for a certain time span you do not look at the clock, you are not able to know how many minutes are passed. This new ambiguity is the equivalent of the GPS cycle slip.

18 Observations available at one epoch for each receiver-satellite couple Code observations PY on L 1, PY ( ) on L 2. C/ A and ( ) If military codes can be read, less precise civilian codes are not used. Phase observations All receivers: L 1 and L 2.