## Refraction

Ionospheric refraction and group path delay errors are especially important for radar ranging and surveillance problems. The last row in Table 4-8 indicates the amount of wedge refraction that is introduced as a radiowave penetrates the troposphere and then traverses the ionosphere. Figure 4-8 gives the geometry.

EFFECT |
Comment |
UNITS |
FORMULAE |

Faraday Rotation |
See Table 4-5; Section 4.3.1.4 |
Rad |
2.97 (1(y2) f"2Hi (EC) |

Group Path Delay |
See Table 4-5; Section 4.3.1.2 |
Sec |
~ 1.34 (10~7)f} (EC) |

Phase Path length |
See Section 4.3.1.2 |
Meters |
-40.5 f2 (EC) |

Group Path Length |
See Section 4.3.1.2 |
Meters |
-40.5 f 2 (EC) |

Phase Change |
See Section 4.3.1.2 |
Rad |
-8.44 (10~7) f '1 (EC) |

Doppler Shift |
Hertz |
-1.34(10~T)f'1 d/dt (EC) | |

Time Dispersion |
See Table 4-5 |
Sec/Hz |
-2.68 (10~7) f'3 (EC) |

Phase Dispersion |
Rad/Hz |
-8.44 (10~T) H (EC) | |

Wedge Refraction |
See Table 4-5; Section 4.3.1.1 |
Rai |
40.5f2 d/dx (EC) |

Note: The units of EC are electrons/m2, and Hl is that component of the geomagnetic field that lies along the ray path. The units are MKS (i.e., ampere-turns/meter). The Total Electron Content (TEC) is related to the EC by a secant factor. We have EC = TEC sec q> where <p is the ray zenith angle based upon a mean ionospheric height of ~ 400 km.

Note: The units of EC are electrons/m2, and Hl is that component of the geomagnetic field that lies along the ray path. The units are MKS (i.e., ampere-turns/meter). The Total Electron Content (TEC) is related to the EC by a secant factor. We have EC = TEC sec q> where <p is the ray zenith angle based upon a mean ionospheric height of ~ 400 km.

200 Space Weather & Telecommunications

The amount of "wedge" refraction due to the ionosphere is given by:

where \Nds is the integrated electron density along the ray path (or the slant electron content, EC), and "d/dx" implies a transverse gradient. The refraction x is in radians, and MKS units are used throughout.

Figure 4-8: Radiowave trajectory through the troposphere and ionosphere. Note that the refractive index n < 1 in the ionosphere, n = 1 in free space, and n > 1 in the troposphere. The apparent elevation angle is higher than the actual (straight line) elevation to the space vehicle After Millman [1967].

Figure 4-8: Radiowave trajectory through the troposphere and ionosphere. Note that the refractive index n < 1 in the ionosphere, n = 1 in free space, and n > 1 in the troposphere. The apparent elevation angle is higher than the actual (straight line) elevation to the space vehicle After Millman [1967].

Assuming a rather sharp 10% gradient over 10 km and a frequency of 100 MHz, we arrive at a value for x of 4 x 10 3 radian (i.e., 4 milliradians or ~ 1/20 degree). At 1 GHz, the frequency dependence drives the wedge refraction level to a negligible number ~ 40 microradians. Figure 4.9 shows the total refraction error at 100 and 200 MHz as a function of altitude.

Elevation angles of 0, 5 and 20 degrees are exhibited. It is seen that at 100 MHz, the difference between the total daytime refraction (i.e., including the ionosphere and troposphere) and the daytime tropospheric refraction is ~ 6 milliradians for an elevation angle of 0 degrees.

Figure 4-9: Total refraction error at 100 and 200 MHz as a function of altitude with elevation angle as parameter. The refractive error increases dramatically as the elevation angle moves toward the horizon, and as the frequency is reduced. (Note Mc/s = MHz.) Original illustration courtesy of George Millman [1967].

Figure 4-9: Total refraction error at 100 and 200 MHz as a function of altitude with elevation angle as parameter. The refractive error increases dramatically as the elevation angle moves toward the horizon, and as the frequency is reduced. (Note Mc/s = MHz.) Original illustration courtesy of George Millman [1967].

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