Vertical profiling of atmospheric refractivity from ground-based GPS

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13 - 4 LOWRY ET AL.: REFRACTIVITY PROFILING FROM GROUND-BASED GPS Figure 2. Measured excess phase path DS GPS as a function of time on 8 December 1999. tive interference commonly results in loss of L2 phase lock for satellites at low elevation [e.g., Anderson, 1994]. Most of the information about the vertical distribution of refractivity is contained at low satellite elevation angles less than 5° [e.g., Gaikovich and Sumin, 1986], and of the 70 processed satellite rises and sets depicted in Figure 2, only 27 had usable observations at those elevations. On average, we collected 24 usable rises and sets per day over the course of the experiment. 3. Modeling of Excess Phase Path [11] GPS excess phase path was modeled using an integral expression derived from ray theory. Assuming a spherical, radially symmetric refractive medium, a ray is described by Snell’s law as [e.g., Born and Wolf, 1964]: rnðrÞ sin a ¼ a; ð1Þ in which a is the angle of ray propagation from the vertical (zenith) direction (Figure 3), n is the refractive index, and a is a constant for a given ray (termed the impact parameter, if the ray is not trapped). The phase path between points 1 and 2 on a ray is S ¼ Z 2 1 nðrÞdl; ð2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where dl = dr 2 þ r 2 dq 2 in polar coordinates. Substituting dr = dlcosa and using Snell’s law (1), equation (2) may be expressed as S ¼ ¼ Z r2 r 1 rn 2 ðrÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dr r 2 n 2 ðrÞ a 2 Z x2 x dn ½1 n dx ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Šx x 1 p dx; ð3Þ x 2 a 2
LOWRY ET AL.: REFRACTIVITY PROFILING FROM GROUND-BASED GPS 13 - 5 2 α 2 Once S and e are calculated, the spherical angle q between points 1 and 2 is given by α 1 ε q ¼ arcsin ½a=x 1 Š arcsin ½a=x 2 Š þ e: ð7Þ 1 r 1 β r 2 The elevation angle b is then b ¼ arctan r 2 cos q r 1 ; ð8Þ r 2 sin q and the path in vacuum S 0 is 0 θ S 0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r1 2 þ r2 2 2r 1 r 2 cos q: ð9Þ Figure 3. Graphical depiction of the geometric parameters of refractive bending. where x rn(r) is the refractional radius. Equation (3) permits calculation of the phase path S for a given impact parameter or for given zenith angle of the ray at a point r (unless the ray has a perigee point between 1 and 2, in which case (3) consists of separate terms describing the ray sections on either side). The excess phase path is simply DS = S S 0 (where S 0 is the path in vacuum, i.e., the distance between the antenna 1 and GPS satellite 2) and is a function of the elevation angle b of the straight line connecting 1 and 2. Excess phase path DS cannot be calculated explicitly for given b; rather, we calculate DS and b for a given impact parameter a using the bending angle e of the ray connecting satellite and receiver: e ¼ Z 2 1 dl r ; ð4Þ where r is the local radius of curvature of the ray. In polar coordinates, r ¼ ½r 2 þ ðdr=dqÞ 2 Š 3=2 r 2 þ 2ðdr=dqÞ 2 rðd 2 r=dq 2 Þ : ð5Þ Substituting (5) into (4) and using Snell’s law (1), the expression for e is e ¼ ¼ Z r2 a r 1 Z x2 a x 1 dn=dr pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dr nðrÞ r 2 n 2 ðrÞ a 2 dn=dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx: ð6Þ nðxÞ x 2 a 2 In practice, we define the center of sphericity (r =0)to be the local center of curvature of the reference ellipsoid at r 1 in the direction of r 2 . Then we calculate DS and b on a dense mesh of impact parameters a and interpolate the discrete model of DS(b) totheb of our observations using cubic splines. 4. Parameter Search for Best Fit Model [12] To parameterize the GPS measurement of excess phase path DS GPS (b), we assume an array of refractivity models N(r) =10 6 [n(r) 1], calculate for each a modeled excess phase path DS mod (b), and compare with the measured DS GPS (b) to determine which provides the best fit in a least squares sense. The model design was motivated partly by a desire to characterize anomalous refractivity gradients. Normally, the vertical refractivity gradient is negative, dN/dr < 0, so that rays are bent toward the Earth’s surface, and most often the radius of curvature of rays is greater than the Earth’s radius r E . However, when the radius of ray curvature at a tangent point is equal to the spherical radius, i.e., dn/dr = n/r ’ 160 N km 1 , the ray is critically refracted, and when dn/dz < n/r,itis referred to as superrefraction. Rays having tangent points within a superrefracting layer are trapped. [13] The refractivity model used here (Figure 4) consists of a constant normal gradient ( 10 N-units km 1 ) beginning at the receiver altitude Z 1 r 1 r E , a critically refractive gradient ( 160 N-units km 1 ) between altitudes Z A and Z B , and a constant gradient between Z B and 6 km altitude. There are only two independent variable parameters in the model, corresponding to the altitudes Z B and Z A at top and bottom of the critically refractive gradient. Refractivity at the receiver, N 1 , is calculated from pressure, temperature, and relative humidity (P, T, n) collected at the surface meteorological sensor. Refractiv-
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Vertical profiling of atmospheric refractivity from ground-based GPS

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