INTRODUCTION OF CORRECTIONS INTO MEASURED PARAMETERS ACCOUNTING FOR SIGNAL DELAY IN TROPOSPHERE AND IONOSPHERE

Key Parameters

Notation	Description
\(\Delta_{trop}^j\)	Tropospheric correction in pseudorange for j-th navigation satellite (meters).
\(\Delta_c^j\)	Dry component of tropospheric signal delay for j-th navigation satellite (meters).
\(\Delta_{wet}^j\)	Wet component of tropospheric signal delay for j-th navigation satellite (meters).
\(\Delta_{ion}^j\)	Ionospheric correction in pseudorange for j-th navigation satellite (meters).
\(\varepsilon\)	Satellite elevation angle in local topocentric coordinate frame (radians).
\(t\)	Temperature in Celsius (°C).
\(T\)	Temperature in Kelvin (K), where \(T = t + 273.16\).
\(P\)	Atmospheric pressure (hPa).
\(f\)	Relative humidity (%).
\(e\)	Partial pressure of water vapor (hPa), where \(e = 0.01 \times f\).
\(T_0\)	Standard temperature, \(273.16\) K.
\(f_{L1}\)	GPS L1 carrier frequency, \(1575.42\) MHz.
\(f_{L2}\)	GPS L2 carrier frequency, \(1227.60\) MHz.
\(\tau_{GD}^j\)	Group Delay for j-th satellite (seconds), inter-frequency bias correction.
\(X_{j,\text{ENU}}, Y_{j,\text{ENU}}, Z_{j,\text{ENU}}\)	Satellite coordinates in East-North-Up local topocentric coordinate frame.
\(a, b\)	Atmospheric model coefficients for tropospheric delay calculation, dependent on meteorological conditions.
\(\delta P\)	Pressure deviation from standard value, \(\delta P = P - 1000\) hPa.
\(\delta t\)	Temperature deviation from standard value, \(\delta t = t - 20\) °C.
\(\delta \beta\)	Temperature gradient deviation, \(\delta \beta = \beta + 6.5\) °C/km.
\(\delta H_T\)	Tropopause height deviation, \(\delta H_T = H_T - 11.29\) km.
\(\beta\)	Temperature gradient, \(\beta = -5.930 - 0.0359 \cdot t\) °C/km.
\(H_T\)	Tropopause height, \(H_T = 7.508 + 2.421 \cdot \exp(t/22.9)\) km.
\(\dot{\Delta}_{trop}^j\)	Tropospheric correction in pseudovelocity for j-th navigation satellite (m/s).
\(\dot{\Delta}_c^j\)	Dry component of tropospheric pseudovelocity correction (m/s).
\(\dot{\Delta}_{wet}^j\)	Wet component of tropospheric pseudovelocity correction (m/s).
\(\dot{\Delta}_{ion}^j\)	Ionospheric correction in pseudovelocity for j-th navigation satellite (m/s).
\(\dot{\varepsilon}\)	Rate of change of satellite elevation angle (rad/s).
\(S_{L1}^j, S_{L2}^j\)	Pseudorange measurements on L1 and L2 frequencies for j-th satellite (meters).
\(\dot{S}_{L1}^j, \dot{S}_{L2}^j\)	Pseudovelocity measurements on L1 and L2 frequencies for j-th satellite (m/s).
\(S_{i,j,k,2}, S_{i,j,k,3}\)	Pseudorange before and after atmospheric corrections for i-th receiver, j-th satellite, k-th epoch.
\(\dot{S}_{i,j,k,2}, \dot{S}_{i,j,k,3}\)	Pseudovelocity before and after atmospheric corrections for i-th receiver, j-th satellite, k-th epoch.
\(c\)	Speed of light in vacuum, \(299,792,458\) m/s.
\(i\)	Receiver index in measurement processing sequence.
\(j\)	Satellite index (NSA - Navigation Satellite Apparatus).
\(k\)	Measurement epoch index for time-series processing.

Variable Definitions

Atmospheric Parameters:

• t — temperature in Celsius [°C]
• T — temperature in Kelvin [K], where T = t + 273.16
• P — atmospheric pressure [hPa]
• f — relative humidity [%]
• e — partial pressure of water vapor [hPa], where e = 0.01 × f
• T₀ = 273.16 — standard temperature [K]

GNSS Signal Parameters:

• f _L1 = 1575.42 MHz — GPS L1 carrier frequency
• f _L2 = 1227.60 MHz — GPS L2 carrier frequency
• τ _GD ^j — Group Delay for j-th satellite [seconds]

Measurement Indices:

• i — receiver index
• j — satellite index (NSA - Navigation Satellite Apparatus)
• k — measurement epoch index

1. Calculation of tropospheric correction in pseudorange for \( j \)-th NSA is performed by the formula:

\[ \Delta_ { trop } ^ { j } = - \left( \Delta_{c}^{j} + \Delta_{wet}^{j} \right), \]

Total Tropospheric Delay Correction: Sums dry and wet delays to adjust GPS signal timing. Critical for precise navigation through atmospheric slowdown.

where \( \Delta_ { c } ^ { j } \), \( \Delta_ { wet } ^ { j } \) — dry and wet components of signal delay in troposphere for \( j \)-th navigation satellite;

\[ \Delta_ { c } ^ { \,j } \;=\; 2.309 \times 10^ { -3 } \;\times\; \dfrac { \,T - 3.28\, } { \displaystyle \sin\varepsilon + \dfrac { \,a\, } { \displaystyle \operatorname { tg } \varepsilon + \dfrac { \,b\, } { \sin\varepsilon } } } \;\times\; \dfrac { P } { T } , \quad m \]

Dry Tropospheric Delay Correction: Computes dry atmosphere delay using pressure, temperature, and elevation angle. Important for accurate GPS positioning.

\[ \Delta_ { wet } ^ { \,j } \;=\; \dfrac { \displaystyle 1.094 \cdot 10^ { 3 } \left( 3.101 + \exp\!\left(\tfrac{t}{22.9}\right) \cdot \exp\!\left(\tfrac{19.83 \cdot t}{T}\right) \right) } { \displaystyle \sin\varepsilon + \dfrac { 0.00035 } { \displaystyle \tan\varepsilon + 0.0017 } } \times \dfrac { e } { T_0 ^2 } , \quad m \]

Wet Tropospheric Delay Correction: Adjusts for water vapor using humidity, temperature, and elevation angle. Fundamental to GPS signal accuracy.

\[ \varepsilon = { \rm arctg } \left( \frac{Y_{j,{\rm ENU}}}{\sqrt{X^2_{j,{\rm ENU}} + Z^2_{j,{\rm ENU}}}}\right); \]

Satellite Elevation Angle: Calculates satellite's angle above horizon in ENU coordinates. Critical for delay adjustments due to longer atmospheric paths at low angles.

where ENU — East-North-Up coordinate system (local topocentric coordinate frame):

• X _ENU — East component (positive towards East)
• Y _ENU — North component (positive towards North)
• Z _ENU — Up component (positive towards zenith)
• Origin at the receiver position

\[ a = 0 { , } 00118 \left( 1 + 6{,}071 \cdot 10^{-5} \delta P - 1{,}471 \cdot 10^{-4} e + 3{,}072 \cdot 10^{-3} \delta t + 1{,}965 \cdot 10^{-2} \delta \beta - 5{,}645 \cdot 10^{-3} \delta H_T \right); \]

Tropospheric Coefficient 'a': Tunes delay model with pressure, humidity, and temperature deviations. Necessary for weather-adjusted GPS corrections.

\[ T = t + 273 { , } 16; \]

Temperature Conversion to Kelvin: Converts Celsius to Kelvin for atmospheric calculations. Important for consistent GPS delay adjustments.

\[ b = 0 { , } 00114 \left( 1 + 1{,}164 \cdot 10^{-5} \delta P - 2{,}795 \cdot 10^{-4} e + 3{,}109 \cdot 10^{-3} \delta t + 3{,}038 \cdot 10^{-2} \delta \beta - 1{,}217 \cdot 10^{-3} \delta H_T \right); \]

Tropospheric Coefficient 'b': Refines delay model using pressure, humidity, and atmospheric deviations. Critical for precise GPS weather corrections.

\[ c = -0 { , } 0090.; \]

Tropospheric Constant 'c': Sets constant (-0.0090) for atmospheric models, supporting GPS delay calculations.

\[ \delta P = p [hPa] - 1000 [hPa]; \]

Pressure Deviation: Measures deviation from standard 1000 hPa for tropospheric modeling. Necessary for accurate GPS weather adjustments.

\[ \delta \beta = \beta \left[ \frac { ^\circ C } { km } \right] + 6 { , } 5 \left[ \frac { ^\circ C } { km } \right]; \]

Temperature Gradient Deviation: Adjusts temperature gradient for atmospheric modeling. Important for precise GPS signal corrections.

\[ \delta t = t [^\circ C] - 20 [^\circ C]; \]

Temperature Deviation: Measures deviation from standard 20°C for tropospheric modeling. Critical for GPS positioning accuracy.

\[ \delta H_T = H_T [km] - 11 { , } 29 [km]; \]

Tropopause Height Deviation: Computes deviation from standard 11.29 km for atmospheric modeling. Necessary for accurate GPS corrections.

\[ H_T = 7 { , } 508 [km] + 2 { , } 421 \cdot \exp \left( \frac{t}{22{,}9} \right), \, km; \]

Tropopause Height: Estimates tropopause height from temperature for atmospheric modeling. Important for precise GPS delay corrections.

\[ \beta = -5 { , } 930 \left[ \frac { ^\circ C } { km } \right] - 0 { , } 0359 [km^ { -1 } ] \cdot t; \]

Temperature Gradient: Calculates temperature change rate with altitude for tropospheric modeling. Critical for accurate GPS delay adjustments.

\[ e = 0 { , } 01 \cdot f, \, hPa. \]

Water Vapor Pressure: Derives water vapor pressure from humidity for wet delay modeling. Necessary for precise GPS signal corrections.

Correction in pseudorange for signal delay in ionosphere is calculated by the formula:

\[ \Delta_ { \text { ion } } ^j = \frac { S_ { L1 } ^j - S_ { L2 } ^j } { 1 - \left( \dfrac{f_{L1}}{f_{L2}} \right)^2 } - c \cdot \tau_ { GD } ^j, \]

Ionospheric Delay Correction (Dual-Frequency Method): Uses L1-L2 frequency differences to correct ionospheric delays, adjusted for group delay. Critical for precise GPS positioning.

where \( \tau_ { GD } ^ { j } \) is the Group Delay for the j -th navigation satellite.

GD stands for Group Delay and refers to the difference in signal delays between carrier frequencies (e.g., L1 and L2). It is used in GNSS (such as GPS) to correct ionospheric errors and improve the accuracy of pseudorange measurements.

Introduction of tropospheric and ionospheric correction into pseudorange is performed by the formula:

\[ S_ { i,j,k,3 } = S_ { i,j,k,2 } + \Delta_ { trop } ^ { j } + \Delta_ { ion } ^ { j } . \]

Corrected Pseudorange: Combines tropospheric and ionospheric corrections with raw pseudorange for accurate GPS positioning.

2. Introduction of corrections into pseudovelocity

2.1. Calculation of correction in pseudovelocity for j-th NSA, caused by troposphere influence, is performed by the formula:

\[ \dot { \Delta } _ { trop } ^ { j } = - \left( \dot{\Delta}_{c}^{j} + \dot{\Delta}_{wet}^{j} \right), \]

Tropospheric Pseudovelocity Correction: Sums dry and wet delays to adjust GPS signal velocity. Important for precise navigation in motion.

where

\[ \dot { \Delta } _ { c } ^ { j } = 2 { , } 309 \cdot 10^ { -3 } \cdot \frac { (T - 3{,}28) } { \sin \varepsilon \cdot \tan \varepsilon } \times \frac { P } { T } \cdot \dot { \varepsilon } , \quad \text { m /s } ; \]

Dry Tropospheric Pseudovelocity Correction: Corrects signal velocity for dry atmosphere using pressure, temperature, and angle rate. Critical for accurate GPS velocity.

\[ \dot { \Delta } _ { wet } ^j = \frac { 1 { , } 094 \cdot 10^3 \left( 3{,}101 + \exp \left( \frac{t}{22{,}9} \right) \cdot \exp \left( \frac{19{,}83 \cdot t}{T} \right) \right) } { \sin \varepsilon \cdot \tan \varepsilon } \times \frac { e } { T_0 ^2 } \cdot \dot { \varepsilon } , \, \text { m /s } ; \]

Wet Tropospheric Pseudovelocity Correction: Adjusts signal velocity for water vapor using humidity and angle rate. Necessary for precise GPS motion tracking.

\[ \dot { \varepsilon } = \frac { \dot { y } _ { j } \cdot \left( x_j^2 + z_j^2 \right) - y_ { j } \cdot x_j \cdot \dot { x } _j - y_ { j } \cdot z_j \cdot \dot { z } _j } { \left( x_j^2 + y_{j}^2 + z_j^2 \right) \cdot \sqrt { x_j ^2 + z_j^2 } } . \]

Elevation Angle Rate: Calculates satellite angle change rate in ENU system. Important for GPS velocity adjustments for moving satellites.

Correction in pseudovelocity for signal delay in ionosphere is calculated by the formula:

\[ \dot { \Delta } _ { ion } ^j = \frac { \dot { S } _ { L1 } ^j - \dot { S } _ { L2 } ^j } { 1 - \left( \frac{f_{L1}}{f_{L2}} \right)^2 } , \]

Ionospheric Pseudovelocity Correction (Dual-Frequency Method): Uses L1-L2 frequencies to correct ionospheric velocity delays. Critical for accurate GPS velocity tracking.

Introduction of tropospheric and ionospheric correction into pseudovelocity is performed by the formula:

\[ \dot { S } _ { i,j,k,3 } = \dot { S } _ { i,j,k,2 } + \dot { \Delta } _ { trop } ^j + \dot { \Delta } _ { ion } ^j. \]

Corrected Pseudovelocity: Combines tropospheric and ionospheric corrections with raw pseudovelocity for accurate GPS navigation.