Radio occultation bending angle and impact parameter errors caused by horizontal refractive index gradients in the troposphere' A simulation study

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. Dll, PAGES 11,875-11,889, JUNE 16, 2001 Radio occultation bending angle and impact parameter errors caused by horizontal refractive index gradients in the troposphere' A simulation study S. B. Healy Met Of[ice, Bracknell, England. Abstract. Radio occultation (RO) bending angle and impact parameter values are derived from a Doppler shift measurement, assuming spherical symmetry. The purpose of this work is to illustrate the errors that arise when this assumption is not valid. Doppler shift values have been simulated for ray paths through a threedimensional refractive index field derived from a mesoscale model forecast, which has a horizontal grid of 12 km by 12 kin, and includes water vapor. These have then been inverted, making the spherical symmetry assumption. It is demonstrated that refractive index gradients perpendicular to the ray path can cause errors in both the bending angle and impact parameter values, but the latter is the more significant. It is shown that the impact parameter value at the tangent point can differ by around m from the derived value. This can cause an effective bending angle error exceeding - 10% near the surface. A statistical analysis of the errors caused by horizontal gradients for simulations through 54 mesoscale forecasts, using fixed spacecraft trajectories and tangent point locations, is presented. In general, the bending angle errors are found to be - 3% near the surface. A new set of analytical expressions for errors has been derived. These are based on integrating the horizontal gradients along the ray path and are found to be in good agreement with the simulation results. The implications of this work for the assimilation of RO data into numerical weather prediction models are discussed and areas of future work are outlined. 1. Introduction Radio occultation (RO) measurements of the Earth's atmosphere using the GPS satellite constellation are a relatively new source of meteorological data [Kursinski et al., 1996; Rocken et al., 1997], with potential applications in both operational numerical weather prediction (NWP)and climate research [Leroy, 1997]. The technique is based on measuring how radio waves are bent by refractive index gradients in an atmosphere. It can be shown that by making some simplifying assumptions, such a the atmosphere being locally spherically symmetric, that this information can be inverted with an Abel transform to give a vertical profile of refractive index and subsequently temperature under additional assumptions. In order to use these data effectively, it is important to have a good physical understanding of the origin, magnitude, and correlation of measurement errors. For example, statistically optimal (variational) retrieval techniques used extensively in NWP require an accurate knowledge of the observation error covariance Copyright 2001 by the American Geophysical Union. Paper number 2001JD /01/2001 JD ,875 matrix. Recently, Zou et al. [1999] and Palmer et al. [] have presented variational assimilation/retrieval approaches based on the direct use of R bending angle measurements. One of the perceived advantages of this approach over the direct assimilation of refractivity [Healy and Eyre, ] is that it is expected to be more accurate in regions with strong horizontal gradients, such as weather fronts, despite the fact that a number of authors [Kuvsinski et al., 1997; Ahmad and Tyler 1999; Zou et al., 1999] have noted that the horizontal gradients cause bending angle and impact parameter errors. Consequently, the aim of this work is to illustrate the nature of the bending angle and impact parameter errors that arise through the assumption of local spherical symmetry, when the radio wave propagates through realistic meteorological conditions. To investigate this, we have simulated or "forward modeled" occultations within the domain of an NWP forecast model at mesoscale resolution (12 krn by 12 kin). The results have then been processed with standard techniques, which implicitly assume spherical symmetry, and the errors in the bending angle have been rived. This paper is an attempt to produce a clear physical description of these errors, which will prove useful for determining their relative importance in the

2 11,876 HEALY: RADIO OCCULTATION BENDING ANGLE ERRORS overall measurement error covariance matrix. It is also relevant to the development of "fast" bending angle forward operators, as it provides some insight into reasonable approximations that can be made to speed up the calculation, without a significant loss of accuracy. In this work, we extend the analysis given by Kursinski et al. [1997] and Ahmad and Tyler [1999]. In the former, Kursinski et al. presented errors in derived refractivity caused by "along-track" horizontal gradients, by ray tracing through a 40 km resolution mesoscale model. The results indicated that the along-track gradients, parallel with the ray path, cause refractivity er- rors of order -- 1% near the surface, falling linearly with 2. Theory of RO height to - 0.2% at 10 km and then remained relatively constant up to 30 km. However, these simulations ne- The theory of RO measurements using G PS satellites glected horizontal refractive index gradients perpendic- has been described by a number of authors [Kursinski ular to the ray path (E. R. Kursinski, personal com- et al., 1997; Rocken et al., 1997] and is only outlined munication, 1999), which cause out of plane bending. here. The technique is based on measuring how radio In contrast, Abroad and Tyler [1999] included perpen- dicular gradients in their error simulations, but these were based on refractive index fields derived from the output of a relatively coarse global forecast model, with a grid resolution 2.5 ø latitude by 3.75 ø longitude, primarily used for stratospheric modelling [Swinbank and O'Neill, 1994]. They presented simulations for an occultation located at a latitude of 60øS, where the forecast ß model has a horizontal grid size of approximately 280 km by 210 km (compare with 12 km by 12 km used in this work). Consequently, Ahmad and Tyler noted that their results were not influenced by horizontal structure in the refractive index field with a spatial scale less than a few hundred kilometers. In addition, errors caused by horizontal variations in the water vapor in the lower troposphere could not be investigated, because the stratospheric model does not include humidity. In this work we have investigated measurement errors caused by realistic horizontal gradients in the troposphere, which are on a much smaller scale. It is shown that perpendicular gradients can cause a significant measurement error, when water vapor is included in the simulation. It will be demonstrated that there is no simple relationship between the impact parameter at the tangent point and the value derived from the measured Doppler shift, if the atmosphere is not spherically symmetric. In addition, we will present (Appendices A and B) new, physically based analytical expressions which relate the errors in the derived bending angle and impact parameter to the along-track and perpendicular horizontal refractive index gradients integrated along the ray path. This approach is more general than that used by Abroad and Tyler [1999], which is based on a geometrical argument, whereby departures from spherical symmetry are represented as locally spherically symmetric structures, with the center of curvature offset in three dimensions from the center of mass. Their approach is valid for large-scale gradients where, by definition, the change in the radius of curvature of the atmosphere along an individual ray path and during the occultation is slow, but this is not generally the case. In section 2 the basic physics of the RO measurement will be briefly outlined, followed by a description of how the measurements are processed, emphasizing any assumptions that are being made. The potential errors arising from horizontal refractive index gradients are described in section 3, and the derivation of refractive index fields and gradients from the mesoscale model is given in section 4. The "ray tracer" is described in section 5. The results are presented in section 6, followed by the discussion and conclusion in sections 7 and 8, respectively. waves transmitted by the GPS satellite are bent by refractive index gradients before being received at a low Earth orbiting (LEO) satellite (see Figure 1). In gen- eral, the equation of the ray path through the atmosphere is [Born and Wolf, 1986] d dr d (n ss) - Vn, (1) where s is the path length (the distance from the transmitter), r is the position vector of the ray along the path, and n is the refractive index. If the atmosphere is spherically symmetric, the calculation of the ray path can be made considerably simpler by using Bouguer's formula, nr sin b = a = const, (2) where r is the radius value, is the angle between the ray path and the local radius vector and a is a constant for a ray path, known as the impact parameter. Note also that spherical symmetry ensures that the ray path remains in the "occultation plane", which throughout this work is defined as the plane containing both satellites and the origin of the coordinate system. For a spherically symmetrical atmosphere, the bending angle rr can be written as a function of the impact parameter, in integral form, GPS V T -Vy T y LEO y V R Figure 1. The geometry of the radio occultation measurement.

3 HEALY: RADIO OCCULTATION BENDING ANGLE ERRORS 11,877 a-- -2a ov d Inn f (x2- )1/2 &' (3) where x -nr. The variation of bending angle with impact parameter can be inverted with an Abel transform [Fjeldbo et al., 1971] to recover the refractive index profile, n(x) -- exp (a 2 -- X 1/2 ' The refractive index is related to the total pressure, temperature, and water vapor pressure P, T and Pw through n_ l + lo_6 (clp c Pw) +, (5) where f is the frequency, c is the speed of light, V T and V t are the transmitter and receiver velocity vectors, respectively, k T and k t are the unit vectors of the ray at the transmitter and receiver and k is the unit vector of the straight line connecting the transmitter and receiver. Note the "true" bending angle a is simply a = cos- 1 (k T. k t). The additional Doppler shift can only be inverted to yield derived bending angle and impact parameter values (c d and ad, respectively) by making assumptions about the horizontal refractive index gradients. A common approach is to assume spherical symmetry through Bouguer's formula, thereby constraining the ray path to the occultation plane. When the ray path remains in the occultation plane, a much simpler expression for fd can be derived, f - -f(v (cos - 1)+ V sin - where Cl (= 77.6 K/hPa) and %(= 3.73 x 105 K2/hPa) c are known constants [Bean and Dutton, 1968]. This is often rewritten as n - lq- 10-6N, where N is the refrac- V (cos 7-1) + Vy a sin 7), (7) tivity. For a dry atmosphere (Pw = 0) the refractivity is directly proportional to density and the refractivity which is in a more convenient Cartesian coordinate sysprofile can be used to integrate the hydrostatic equation tem, where the "x" direction is parallel to the straight dp/dz = -pg, and a vertical temperature profile is deline between the satellites, pointing toward the receiver rived from the ideal gas law, since P = prt = NT/cl. (see Figure 1). This can be inverted with an iterative calculation to give the bending angle ad = fi + 7 and If the water vapor is not negligible, it is possible to dethe impact parameter ad, using Bouguer's formula (2), rive temperature or humidity information from the refractivity profile using a priori information, but in general the solutions have poor error characteristics as unrt sin(; bt + fi) = ri sin(; br + 7) = aa, (8) certainties in the measurement and a priori information are not accounted for [Healy and Eyre, ]. Alternatively, both temperature and humidity information can be retrieved simultaneously from bending angle or where r T and r t are the radius of the transmitter and receivers, respectively, noting that the refractive index is assumed to be unity at both satellites. The angles ½T and ;b t are illustrated in Figure 1. refractivity profiles in a statistically optimal way using It should be emphasized that formally deriving a "optimal estimation" or "variational" techniques [Zou bending angle from the Doppler shift is an ill-posed ctal., 1999; Healy and Eyre, ; Palmer ctal., ]. problem, because there are an infinite number of combi- The bending angles and impact parameters are not nations of fi and 7 that are consistent with the measured measured quantities; the receivers on the LEO satel- Doppler. The problem is made well posed by making lites measure the phase and amplitude of the signal. assumptions about the horizontal refractive index gra- The bending induced by the atmospheric refractive in- dients. For example, Lindal [1992] has described a gendex gradients introduces an additional phase delay. The eralized ray-tracing inversion technique, based on the derivative of phase delay with respect to time gives a assumption that the refractive index is constant along Doppler shift value that differs from what would be ex- geoids. However, it is more common when processing pected if the ray followed the straight-line path between RO measurements of the Earth to invert the data usthe satellites, and is related to the angle the ray makes ing Bouguer's formula, as outlined above, based on an with the straight-line path at the satellites. Note that a priori assumption of spherical symmetry. Clearly, the Doppler shift does not contain any information on if this assumption is made, we can only guarantee ray path, other than the difference of tangent vectors c (ad) = c a(aa)if the atmosphere is spherically symat the satellites [Melbourne ctal., 1994]. metric. For setting occultations, the satellites are moving away from each other and the radio signal at the re- 3. Measurement Errors Caused by ceiver is shifted to a lower frequency. (At optical fre- Horizontal Refractive Index Gradients quencies this would be described as a "red shift".) The atmospheric bending increases the size of the shift to the 3.1. Along-Track Horizontal Gradients lower frequency. Neglecting relativistic effects, the addi- Along-track horizontal gradients refers to departures tional Doppler shift caused by the atmospheric bending from spherical symmetry caused by gradients in the ocfa can be written as cultation plane. Since these do not cause the ray to leave the plane, it is convenient to consider the problem - f(v ß k - v ß k - (V - k), (6) c in plane polar coordinates (r, O) (see Figure 1).

4 11,878 HEALY: RADIO OCCULTATION BENDING ANGLE ERRORS The horizontal gradients mean that Bouguer's for Perpendicular Horizontal Gradients: mula (equation(2)) is no longer exact, and consequently Out of Plane Bending the impact parameter a is not constant over the ray Refractive index gradients perpendicular to the ocpath. In physical terms this situation is analogous to cultation plane can cause out of plane bending, leading the rate of change of angular momentum of a classical to errors in both the derived bending angle and impact particle being equal to the applied torque, with the imparameter values. The errors arise because the meapact parameter value being the ray equivalent of the sured Doppler shift contains a component that is deangular momentum [Melbournet al., 1994]. It can be pendent on the satellite velocities perpendicular to the shown [Gorbunov et al., 1996] that the rate of change occultation plane. However, this is interpreted as plaof impact parameter with the path is nar bending when deriving and a from the Doppler shift. where (On/OO)r is the partial derivative of refractive index with respect to 0 at a fixed radius. Therefore the difference in impact parameter values at the satellites is given by integrating the along-track horizontal refractive index gradient over the ray path, ar--at--aa--/(o ) r ds, (10) where a T and a n are the impact parameter values at the transmitter and receiver, respectively. It is not immediately obvious how to interpret the single impact parameter value derived from the Doppler shift assuming spherical symmetry aa if the real impact parameter value is changing along the ray path as a result of the horizontal gradients. However, in Appendix A it is shown that the derived value can be written in terms of the impact parameter values at the satellites. We find aa _ a r T cos T t + rt cos b t - kr n cos b n (an- at)' (11) where (an- a T) is given by (10), k is a scalar, which depends on the satellite velocities and occultation geometry (Appendix A, equation (A5)). The angles b t = b t + fi and b n - b n + 7 are illustrated in Figure 1. Note that there is no clear relationship between ad derived from the Doppler shift and the "true" impact parameter value at the tangent point. The latter can only be determined by integrating (9) along the ray path between the transmitter and the tangent point. The bending angle error that arises as result of inverting the Doppler shift, assuming the impact parameter has a constant value ad given by (8), is considered in Ap- pendix A. It is shown that the error is proportional to (an-a T). This leads to some interesting results. For example, if a two-dimensional (2-D) refractive index field contained gradients which had perfect reflection symmetry about the tangent point, then a n -a T, and the bending angle derived from the Doppler would be correct. However, we would not be able to determine the impact parameter at the tangent point, without knowledge of the horizontal gradients. To first order, the additional Doppler shift can be written as (Appendix B) A f a... f ( vf e t- Vf en ), (12) ½ where e t and e n are the angle the ray makes with the occultation plane at the transmitter and receiver, respectively, and, similarly, Vf and Vf are the respective satellite velocities along the z axis, perpendicular to the occultation plane. Note that the orbital geometry means that Af. --(f/c) x Vile n is usually a good approximation. Using (1), to first order the values of e t and e n are related to the refractive index gradients perpendicular to the occultation plane by en--et' '/l(on) ds, (13) where the integral is taken along the ray path. Expressions for the subsequent errors in the derived bending angle and impact parameter values are given in Appendix B. Although the errors for a given occultation are dependent on the refractive index gradients, perpendicular velocity vectors and orientation of the satellites specific to that event, it is possible to make some general points before presenting any numerical results. First, for a setting occultation the atmospheric Doppler shift is to a lower frequency, meaning fa < 0. If the additional Doppler shift caused by the perpendicular gradients is positive (Afa 0), both the derived bending angle and impact parameter values will be smaller than the true values. Conversely, if A fa < 0, both the derived bending angle and impact parameter values will be larger than the true values. 4. Derivation of the Refractive Index Profiles and Gradients From the Mesoscale Forecast Field In the simulations it is assumed that the Earth is spherical and the ionospheric contribution to the refractive index can be neglected. These assumptions are justified only by the fact we are attempting to investigate measurement errors caused by horizontal gradients in the neutral atmosphere. Residual ionospheric signal or misplacing the origin of the coordinate system will increase the measurement errors, but these have been

5 HEALY: RADIO OCCULTATION BENDING ANGLE ERRORS 11,879 discussed by other workers [Kursinski et al., 1997; Syndergaard, 1999a] and would only confuse the issues that are being addressed here. The refractive index profiles are derived from the Met Office's unified model [Cullen, 1993], at operational mesoscale resolution, which is centered on the United Kingdom. The data are specified on a 182 by 146, latitude/longitude grid, but with the North Pole rotated to 37.5øN, 177.5øE. The rotated pole ensures that the grid, which is equally spaced in latitude and longitude with a spacing of 0.11 degrees in each direction, corresponds to an almost equal spacing on the surface of the sphere with a separation of 12.3 km over the domain. The pressure, temperature, and specific humidity are specified on 38 vertical levels between the surface and approximately 5 hpa (- 35 kin), from which a continuous refractive index profile is derived. For a ray po- sition vector in (rotated) polar coordinates r, A, q, the radius, latitude, and longitude, respectively, the four model horizontal grid points that surround the ray are identified. The hydrostatic equation is solved and a continuous refractive index profile is defined at each grid point. To achieve this, it is assumed that temperature and log(specific humidity) vary linearly in geopotential bending angles. For a smooth, exponentially decaying height between the model levels. The refractive index refractivity profile, the impact parameter varies by less (n) and radial gradient (On/Or) for the radius value are than 10-3 m along the ray. The Abel transform refraccalculated at each grid point and the values are interpo- tivity values agree with original profile to within 0.03%. lated (bilinear) to the ray coordinate. Similarly, the gra- It does not appear possible to validate the bending andients with respect to latitude and longitude gles and impact parameters calculated for an arbitrary and On/O b ) are evaluated between the grid points and atmosphere which includes significant horizontal grathen linearly interpolated to the ray coordinate. dients, against any obvious analytical solution. How- The refractive index and the gradient values above ever, in simulations neglecting perpendicular gradients the highest level of the mesoscale model, within the hor- we have found the variation of impact parameter along izontal domain, are evaluated as above, but assuming the ray path is in agreement with (9), which is an exthe atmosphere is dry and isothermal with tempera- act, analytical relationship for the ray path. Note that tures given by the uppermost model level. Outside the the evaluation of the impact parameter a, using (2), domain the refractivity (10c(n- 1)) is assumed to have is not required during the ray tracing. However, we an exponential form and be spherically symmetric. It have found that calculating the variation of a along the is defined as N(r) = 330exp(-(r- r )/8 km), where path, and comparing it with that found by integrating r is the radius of the Earth. This is a very simple model, but the satellite orbits are such that most of the bending occurs within the mesoscale domain. In general, the ray paths enter into the mesoscale domain at heights of 70 km above the surface. Consequently, assumptions about the refractivity field outside the model domain have very little affect on the bending angle and impact parameter values of ray paths in lower troposphere. Note that, the refractive index values are always set to unity for all heights greater than 100 km above the surface. 5. Description of the Ray Tracer To perform the simulation or forward modeling of the measurement described in this work, it is necessary to solve the ray equation (1) with a general approach that does not assume spherical symmetry. In this work it is solved with a variable step length, fifthorder Runge-Kutta-Fehlberg method, based on the al- gorithm outlined by Press et al. [1992]. The method monitors the truncation error of the solution and ad- justs the step length to keep this below a predefined tolerance. A maximum permissible step length is set to ASmax 6 km and the minimum value is hsmi n "'" 60 m. The calculation is performed in Cartesian coordinates, although refractive index is evaluated on the rotated latitude/longitude grid. The ray path that intersects both satellites is found with an iterative "shooting" approach, whereby the direction cosines of the ray at the transmitter are adjusted until the receiver is intersected, to within a specified tolerance. In this work the tolerance is 15 cm. The algorithm is too computationally expensive to be implemented for the operational assimilation of RO data into an NWP system, but it may be useful for the development and validation of "fast" bending angle models. The accuracy of the ray tracer can be demonstrated in a spherically symmetric medium in two ways. First the impact parameter along the ray path can be monitored, as this should be constant by definition. In addition, it should be possible to reconstruct the original refractive index field from an Abel transform of the calculated (9), provides a useful consistency check for a simulated ray path. 6. Results The simulations performed in this work used a Met Office mesoscale 6 hour forecast for the United Kingdom on December 8, 1998, valid for hours. This case was chosen as it was known to contain a frontal system and it was expected to give rise to larger than normal effects of horizontal gradients. The weather chart can be found in the Daily Weather Summary, issued by the Met Office (Daily weather summary, December 1998; available from the National Meteorological Library, Bracknell, Berkshire, England). Figure 2 illustrates the mesoscale model domain (light-shaded region). A "representative" tangent point and corresponding 300 km section of an occultation plane have been superimposed. The ray propagates from the northwest toward the southeast of the United Kingdom.

6 11,880 HEALY: RADIO OCCULTATION BENDING ANGLE ERRORS (- 18.0, 16.9, 62.8) track term does not significantly modify the simulated ray path (or bending angle). Melbournet al. [1994] pointed out that physically the contribution of this term must be small because the ray is propagating almost parallel to when most of the bending occurs, and the change in bending angle at any point on the ray path is only sensitive to gradients perpendicular to the ray path. They also stated that the principal contribution of the along-track gradient to the ray path (or bending angle) is through the variation it causes in the 5OOO 4000 (-12.2, 43.6) 5000 Figure 2. The domain of the mesoscale model (shaded region). The longitude and latitude of the domain corners are given in the brackets (longitude, latitude). A representative tangent point and occultation plane are indicated with the solid square and straight line, respectively. The ray propagates to the southwest of the United Kingdom. 1 ooo Horizontal distance (kin) Since the occultation plane varies during the measurement, the plane shown in this figure is defined by average satellite positions and the tangent point is given by the location of closest approach of the straight-line path between these positions. We did not have an observed G PS/MET occultation for this time and location, but the satellite coordinates and velocities used in the simulations were derived from a real GPS/MET occultation (a Jet Propulsion Laboratory (JPL) file from 1995 ' :33gps-met25.1eve11', which gave the temperature retrieval at 69.2øN, 82.6øW [Kursinski et al., 1996, Figure 2A]), that were transformed to give the tangent points approximately at the center of the mesoscale model domain Errors Caused by Along-Track Gradients For a 2-D refractivity field, in plane polar coordinates with unit vectors and, the full expression for the change in bending angle, including along-track gradients is given by [Eyre, 1994] ( ) 1(0 ) 5r 5c -- 1 On rs0+ (14) n 0 n r r which can be derived from (1) [Born and Wolf, 1986]. In this work both the refractive index n and the radial gradient On/Or vary within the occultation plane and are evaluated as a function of 0, but we approximate the along-track problem by neglecting the (On/OO)r term in (14) when calculating the ray path in the plane. Given the aim of this section this approximation may initially seem surprising. However, in practice, the along- (b) (c) 5OOO looo 5OOO 4000, Horizontal distance (km) [00 i I Horizontal distance (kin) 150 go Figure 3. Cross sections of the temperature(k), specific humidity (g/kg) and refractivity (N units) (3a, 3b, and 3c, respectively) through the 300km section of the occultarian plane illustrated in figure 2.

7 HEALY: RADIO OCCULTATION BENDING ANGLE ERRORS 11, difference between the impact parameter values at the satellites (ar--at), given by integrating the along-track gradients along the ray path (10), divided by the distance between the satellites. More precisely, the error in the bending angle can be written as (A9) 40OO 0 i i i I i i l Bending ongle error (%) Figure 4. The percentage error in the bending angle derived from the Doppler shift as a function of tangent height as a result of along-track gradients. dial component of the gradient. From (14) it is clear that at the tangent point the along-track contribution to the total bending angle is zero by definition, because 5r/r = cot 50 = 0, where ½ is the angle between the ray tangent vector and local radius vector. More generally, in simulations we have found that it contributes of order % of the total bending angle value, which is comparable to the numerical accuracy of the model. We have also found that the variation of impact parameter a along a path which is calculated neglecting the (On/OO)r term in (14) still satisfies (9) to a good approximation. Typically, the total impact parameter difference ar-a T, calculatedirectly using a = nr sin & (2) at the satellite locations, agrees with the numerical integration of (9) along the entire path to within ---1 cm. As a result, the errors in the derived bending angle and impact parameter values can be related to (On/00)r us- ing (9) and (10), despite the fact that the along-track gradient term in (14) is not used explicitly to determine the ray path between the satellites. The along-track horizontal temperature, specific humidity, and refractivity cross sections are illustrated in Figure 3, for the "representative" tangent point and occultation plane indicated in Figure 2. Figure 4 shows the percentage error as a function of tangent point height (which is found during the calculation of the ray path and is given by atan/n- re) of the bending angle derived from the Doppler shift ad, assuming spherical symmetry when compared with the "true" value, c = cos-z(k T- k t), evaluatedirectly from the difference in the ray tangent vectors at the satellites. As noted earlier, both c and a0 are consis- tent with the measured Doppler shift, but an error, Ac = c 0 - cos-l(k T' k t), arises because c 0 is derived assuming that the impact parameter is constant along the ray path, when it actually varies according to (9). It is clear from Figure 4 that these errors Aa are very small, but they agree with the analytical expressions given in Appendix A, where it is shown that the bending angle error is related approximately to the (x + cos cos (x where, as noted earlier (section 3.1), k is a scalar, which depends on the satellite velocities and occultation geometry (A5). In this example, its numerical value is k For the results illustrated in Figure 4, a z > a T, but k, so the derived bending angles errors are negative (i.e., the derived values are smaller than the true values). The largest error, at around 3.5 kin, corresponds the biggest difference in the impact parameter values at the satellites, with a -a T = f., (On/OO)rds _ 80 m. Note that the errors illustrated in Figure 4 are of a similar magnitude to the absolute accuracy of the ray-traced bending angle values, which is of order %. However, an error estimate, Ac = c a- cos-l(k T. k t), is not sensitive to the absolute accuracy with which we can evaluate the bending though the refractivity field (although clearly a high accuracy is desirable). This is because a simulated excess Doppler shift value fa is consistent with the corresponding c by definition. The fact that c may not be perfectly accurate for a given refractivity field is not crucial because the derived bending angle c a is inverted without any prior knowledge of the refractivity field, other than the values at the satellite positions (n = 1). Therefore the difference between c and arises because of the assumption of spherical symmetry in the inversion. Figure 5 illustrates that the error in the derived impact parameter aa is far more significant. In this case we define the impact parameter error as aa-atan where atan is the value at the tangent point, evaluated during the simulation. Toward the surface we find the impact parameter errors can be over m. It is shown in OOO O0 Impact parameter error (m) Figure 5. The error in the derived impact parameter value as a result of along-track gradients.

8 _ 11,882 HEALY: RADIO OCCULTATION BENDING ANGLE ERRORS i i i,,, x "..,., 7 -,--,,,,., xl 06-5O x106 ' xl 06 -loo Path length s(m) Figure 6. The variation of the impact parameter value along a ray path with a tangent point 1200m above the surface. The asterisk marks the position of the tangent point. Appendix A that a0 can be written in terms of the impact parameter values at the satellites, but the value at the tangent point depends on the along track gradient, integrated over the path between the transmitter and the tangent point. Therefore, using (10), the impact parameter error can be written as _ ad -- atan -- r T cos T rtcos _ kr T R R fs (Oft) - ds values. In this example the horizontal gradients cause a change to the refractive index field which is almost sym-. dr, metrical either side of the tangent point. In contrast, where, as noted, the first integral is taken over the if the gradients lead to an asymmetric perturbation, whole path (s) between the satellites, but the second atari would be bounded by a T and a R. This illustrates is only evaluated along the part of the path between an important point, implicit in the expression for ad the transmitter and the tangent point (ST). It should derived in Appendix A, namely, that the Doppler meabe emphasized that the impact parameter error definisurement does not provide any information about atari, tion given above is not the same as that used by Ah- the impact parameter value at the tangent point. This mad and Tyler [1999]. They looked at the differences between the impact parameter values derived, assuming spherical symmetry, from the Doppler shift, evaluated for the incorrect and correct centers of curvature. How- ever, that approach does not account for the fact that the impact parameter value varies continuously along xl 06,,, x106 o xl E xl06 a---e3 True erived x106 - t I,,, Bending angle (rod) Figure 7. The true and derived bending angles as a function of impact parameter x106 ',,,, i Bending angle error (%) i i i I, i i - 1; Figure 8. The effective error in the derived bending angle, caused by the impact parameter errors. the ray path when the ray encounters along-track horizontal gradients (see(9)). For example, Figure 6 shows the variation of impact parameter, calculated with (2), as a function of path length for a ray path with a tangent point height 1200 m above the surface. It is clear that impact parameter changes significantly over a distance of 100 km near the tangent point. In this case the impact parameter difference at the satellites is only m, but atari is over 100 m less than either of these can only be estimated by making a priori assumptions about the nature of the horizontal refractive index gradients, and clearly the simplest is spherical symmetry. This is significant since the derived impact parameter value ad is sometimes used in "fast forward models" [e.g., Eyre, 1994] to define the tangent point height, giving a point on the ray path from which to calculate the bending angle. Generally, if aa is wrong by around 100 m, the calculated bending angle will be in error by around 1-2%, using Ac (Oc /Oa)Aa and assuming that the bending angle falls exponentially with impact parameter, with a scale height of km which is reasonable for a dry atmosphere. However, the precise value of the error depends on the bending angle profile scale height, which tends to be much smaller near the surface as a result of the water vapor gradients. This becomes apparent when both o (atan) and c a(aa) are plotted together, as illustrated in Figure 7 for bending angles from km up to 3 km above the surface. Note that consecutive derived and true impact param-. _ t,i+ 1 eter values (a /- a +1 and a[an "tan) near the surface are only separated by 10 m (Figure 7), but in both cases the profiles remain monotonic. (However, we have

9 ß., HEALY' RADIO OCCULTATION BENDING ANGLE ERRORS 11,883 found examples with other mesoscale fields where the derived impact parameter begins to increase toward the surface.) Figure 8 shows the "effective bending angle error", 100x (ozct(ad)--oz(ad))/oz(ad), as function of impact parameter, when the "true" bending angle profile has been interpolated to the derived impact parameter values c (ad). Near the surface the effective bending angle errors caused by an uncertainty in the impact parameter of > 100 m can be up to - 20% because the scale height is c /(Oc /Oa) _ 600 m. Comparable scale heights are evident in measured GPS/MET bending angle profiles that get close to the surface. Although the primary aim of this work is to investigate the bending angle and impact parameter errors, it is useful to comment on the magnitude of the errors in the refractivity profile derived from the Abel transform of c d(ad) for these simulations. For this profile, the largest refractivity error is 1.3%, where the "true" refractivity values used to define the errors are evaluated directly from mesoscale model field, for the derived tangent point height and location. Alternatively, comparing the refractivity profiles derived by an Abel transform of c d(ad) with the true bending angle profile interpolated on to the observed impact parameters c (ad) leads to refractivity differences as high as 2.4%. It should be emphasized that these are differences rather than errors, because in general the Abel transform of the true bending angle profile c (ad) will not produce the true refractivity values, because of the horizontal gradients. Nevertheless, in both cases, is clear that the O0O 4OOO 2OOO O0 Impact parameter error (m) Figure 10. As Figure 5, but refractive index gradients perpendicular to the ray path are included Errors Caused by Perpendicular Gradients For the results presented in this section, all the horizontal refractive index gradients are included when calculating the ray path between the satellites. This means that the ray is not constrained to a fixed plane. Refractive index gradients perpendicular to the occultation plane do not significantly affect how the im- pact parameter value varies along the ray path, and to a good approximation this variation is governed by just the planar gradients. Therefore the analysis given above, in terms of not knowing atan, is still valid. Howbending angle errors exceeding 10% are not translating ever, the perpendicular component of the Doppler shift into similar errors/differences in the refractivity profiles. introduces an additional error in both the derived bend- This is a result of a combination of the smoothing in- ing angle and impact parameter values, which depends troduced by the Abel transform [Melbourn et al., 1994] on the out of plane bending angles e T and e R and the and, to a lesser extent, the partial cancellation of the refractivity error as a result of the error in the radius value, as discussed by Syndergaard [1999b]. Note that in the Abel transform the contribution of a bending anperpendicular satellite velocities V T and Vfi. In this example the magnitude of perpendicular velocities IV TI and IVfil, are 1.05 and 7.1 km s -, respectively. We have found that the latter is one of the largest valgle value c is proportional to the impact parameter ues of IV l in 600 GPS/MET occultations provided by separation a + - a, which tends to be small near the the Jet Propulsion Laboratory. Therefore the errors in surface, where the largest errors occur. the derived bending angle and impact parameter values caused by perpendicular gradients presented here will.+. o i i x 106 t' ', - ' x106 & []---D True --- Derived x106 o 2OOO _E xl 06 0 O' -O.O O. 10 Bending angle error (%) Figure 9. As Figure 4, but refractive index gradients perpendicular to the ray path are included x ( Bending angle (tad) Figure 11. As Figure 7, but perpendicular gradients are included.

10 11,884 HEALY: RADIO OCCULTATION BENDING ANGLE ERRORS x106 ' ' xl 06 E o x106 (J c E xl 06 eter errors exceeded 100 m near the surface. The errors are larger when perpendicular gradients are included, as illustrated in Figures 14 and 15. The largest bending angle and impact parameter errors are 0.05% and - 25 m, respectively. These values should be compared with - 0.1% and 60 m when humidity gradients are included in the simulation, as shown in Figures 9 and 10. To summarize, the errors are significantly underestimated if water vapor is not included in the simulations x Bending angle error (%) Figure 12. As Figure 8, but perpendicular gradients are included. be toward an upper limit for the out of plane bending defined by e T and e i. Figure 9 illustrates the errors in c a. Note that these are an order of magnitude bigger than the bending angle errors shown in Figure 4. In this example, if the ray paths that go close to the surface are not corrected to account for the perpendicular gradients, the LEO will be missed by typically 30 m. The angle the rays make with the tangent plane at the GPS and LEO satellites is only of order I prad and 10 prad, respectively, but this is large enough to modify the Doppler shift by around - 0.1%. Figure 10 shows the impact parameter error aa - attn. Both oz(at n) and o a(aa) and the percentagerrors in (c d(au)- o (au))/o (ad)) are presented in Figures 11 and 12, respectively. In this case, the largest derived refractivity error is 1.4%. The impact parameter and effective bending angle errors found when perpendicular gradients included are smaller than in Figures 5 and 8. In this particular case, which is a setting occultation, the additional Doppler component (12) is positive (Afa > 0), reducing the values of both the derived bending angle and impact parameter (see section 3.2) and correcting some of the positive error shown in Figure 5. This cancellation is a purely fortuitous result, specific to this reft'active index field, but it nevertheless illustrates that the derived ira.- pact parameter value, which may be incorporated into a "fast bending angle model", is sensitive to the perpendicular gradients Contribution of Water Vapor The simulations in 6.1 and 6.2 have been repeated assuming a dry atmosphere (P = 0) in order to illustrate the relative contribution of water vapor gradients to the errors outlined above. Figure 13 illustrates the horizontal refractivity gradients when water vapor is not included. These are considerably smoother than those shown in Figure 3c. When the perpendicular gradients are neglected, the magnitude of the largest bending angle and impact parameter errors are % and 11 m, respectively (figure omitted). The latter should be compared with the results in section 6.1, where it was shown that the largest impact param Preliminary Statistical Analysis A preliminary statistical analysis (a more comprehensive analysis will be the subject of future work) of bending angle errors caused by horizontal gradients has been performed in order to assess how representative the results shown in sections 6.1 and 6.2 are. The error statistics have been derived by simulating occultations, using the same satellite position/velocity data and tangent point locations, through 54 randomly chosen 6 hour mesoscale fordcasts, dated between November 1998 and October 1999, valid at either or hours. The root mean square (RMS) of the effective bending angle errors (where true bending angle profile has been interpolated to the derived impact parameter values as shown in Figures 8 and 12) as a function of tangent point height are illustrated in Figure 16. Some caution is required when interpreting the statistical errors near the surface, because it is not always possible to successfully calculate the ray path for each set of satellite coordinates for every mesoscale forecast. This leads to sample numbers which are less than 54, and are as low as 19. Noting this limitation, in general, the bending angle errors are larger when perpendicular gradients are included, reflecting the fact that the largest root mean square (RMS) impact parameter errors are - 73 m when they are included, compared with- 35 m when omitted. For the former the largest effective bending angles errors are - 3% around I km above the surface, which are about a factor of 3 lower than those 5OO0 40O0,E 3000 looo , 240. i i 260 i i i i I i I 26[ Horizontal distance (km) Figure 13. A contour plot illustrating the along-track refractivity gradients derived from the mesoscale forecast field when water vapor is not included.

11 HEALY: RADIO OCCULTATION BENDING ANGLE ERRORS 11,885 E ß 6000 ' o. 1 o Figure 14. -o.o5 cluded in the simulation. o.oo Bending ongle error (%) I o.o5 O, 0 As Figure 9, but water vapor is not in- shown in section 6.2, because that case was expected to produce large errors. Note that the RMS refractivity errors are less than 1.4%, in reasonable agreement with Kursinski et al. [1997]. 7. Discussion The simulations have illustrated that errors in the bending angle and impact parameter values (c a and aa) derived from the measured Doppler shift fa arising as a result of horizontal gradients are significant for a high-resolution atmospheric field which was known to contain a weather front. This case was chosen so ance matrices for bending angle and refractivity will be the subject of future work.) However, it is useful to illustrate how the observation errors increase for real- istic and important meteorological conditions, such as weather fronts, as it was anticipated that the direct assimilation of bending angles would largely overcome the difficulties associated with the assumption of spherical symmetry. In this work we have found that for realistic satellite orbits along-track horizontal refractive index gradients lead to negligible errors in the derived bending angles. However, as the impact parameter varies along the ray path, it is not possible to determine the impact parameter value at the tangent point atari from the measurement. This is important if the derived value aa is to be used to determine the tangent point height, which is an approximation sometimes used in fast models for simulating the bending angle [Eyre, 1994; Palmer et al., ]. Perpendicularefractive index gradients should not be ignored when performing an error analysis. These lead to a modified Doppler shift value, which includes a component dependent on the satellite velocities perpendicular to the occultation plane. Although this is not strictly a measurement error, it results in errors in c a and aa, which are derived assuming planar bending. However, once again the errors in c a are relatively small when compared with the effective bending angle errors introduced by mapping the impact parame- ter errors into bending angle space (Ac _ (Oce/Oa)Aa). It is interesting to note that horizontal gradients primarily cause errors in the derived impact parameters, as this may partially account for well-known difficul- ties with a subset of real GPS/MET measurements. It that the bending angle errors that arise in such regions could be clearly identified and investigated. Conhas been observed that in some measurements c a(aa) is not monotonic, because impact parameter values begin sequently, the largest errors derived with this mesoscale to increase near the surface, but the derived bending forecast are about a factor of 3 bigger than those derived in a preliminary statistical analysis derived from angles appear well behaved. We have found one clear example of this whilst performing simulations through 54 other forecasts, when perpendicular gradients are in- the mesoscale fields. cluded. (A more comprehensive statistical analysis and Throughout this work for convenience we have asthe derivation of statistical measurement error covari- sumed a spherical Earth with no ionosphere in or , ' '... '... '... 8OOO : i: c full,. npg _ 6000 ß of Impact parameter error (m) Figure 15. As Figure 10, but water vapor is not included in the simulation. i'!., RMS bending angle error (%) Figure 16. The RMS bending angle errors derived from 54 mesoscale forecasts, when perpendicular gradients are included (squares) and neglected (triangles)).

12 11,886 HEALY: RADIO OCCULTATION BENDING ANGLE ERRORS der to concentrate on measurement errors caused by horizontal gradients. In practice, the measurements are processed by approximating the geoid with an ellipsoid based, local radius of curvature [Syndergaard, occultation plane, perhaps using the horizontal weighting function described by Abroad and Tyler [1998], we 1999a] and removing the ionospheric contribution to the would expect these errors in "interpretation" to be rebending by taking a linear combination of the L1 and duced further. These issues will be investigated more L2 bending angle values [Vorob'ev and Krasil'nikova, fully in future work. 1994]. Both of these processing steps are approxima- The assimilation of bending angle is a more computions which will produce some error, and it is useful tationally expensive option [Zou et al., 1999], but this to compare the magnitude of these with the horizontal is usually justified because it is assumed to be a more gradient errors. First, Kursinski et al. [1997] estimate accurate approach in regions with horizontal gradients, that approximating the geoid with an ellipsoid-based such as weather fronts. If the derived bending angle radius of curvature results in refractivity errors of or- profile did not require the assumption of spherical symder AN/N _ 0.002%. In contrast, in sections 6.1 and metry, this would undoubtedly be a more accurate op- 6.2 the refractivity errors caused by horizontal gradient errors were of order -- 1% near the surface. Inspection of GPS/MET bending angle profiles near 50 km sug- gests that the residual ionospheric noise is of order / rad. Near the surface, where the horizontal gradient errors are largest, the bending angle values are tad, indicating that the bending angle errors caused by residual ionospheric noise are of order %. The work contained in this paper has some implications when assessing the relative merits of different options for assimilating RO data into a NWP model. It is unlikely that temperature and/or humidity profiles derived directly from the refractive index profiles (as outlined in section 2) will be used since the error characteristics of this approach are complex and not amenable to optimal assimilation [Healy and Eyre, ]. This leaves the possibility of direct assimilation of refractivity or bending angle. The former is a relatively simple approach. It overcomes the water vapor ambiguity and also has the advantage of not requiring extrapolation above the NWP model atmosphere. Note also that the refractivity values are essentially a weighted sum of bending angle values, which tends to lead to some cancellation of the errors in a(ad). However, additional errors arise if the output of the Abel transform is interpreted as the vertical refractivity profile at a specific location, ignoring tangent point drift or the limb found that the fractional errors of the derived refrac- tivity values tend to be smaller than the fractional errors of the bending angles used in the Abel transform. This indicates that the smoothing introduced by the Abel transform more than compensates for any error in "interpretation". In addition, if the derived refractivity profile was assumed to be an average over the tion than assimilating refractivity. However, this work suggests that bending angle measurement errors will be largest in these regions and it is also apparent that some of the assumptions made to enable the fast bending angle calculations, primarily the use of the derived impact parameter values, are less accurate here. Zou et al. [1999] noted that the measured Doppler shifts could be assimilated instead of bending angle, thereby removing the spherical symmetry assumption completely. However, this is a computationally expensive option and is not currently feasible. Alternatively, it may be possible to correct the derived bending angle errors, based on the horizontal gradients contained in the NWP forecast, but this is questionable because NWP forecast will also contain errors. Another option is to vary the observation error covariance matrix, according to the NWP forecast. For example, this work suggests the assumed bending angle errors should be increased if the NWP forecast indicates that the measurement is in a region with significant horizontal gradients. It is clear that the continued development of fast forward models is essential if bending angles are to be assimilated operationally into NWP systems. However, it is equally important to specify any assumptions made about the measurement clearly, paying particular attention to forward model errors that arise as a result of these assumptions. For example, a simple fast forsounding nature of the measurement. In this case the ward model, neglecting the limb sounding nature of the refractivity errors, which are vertically correlated, will measurement, tangent point drift and horizontal grabe [Melbournet al., 1994] dients, would imply that the measurement contained vertical profile information at a nominal tangent point location. It is possible to estimate the forward model AN(x) "' 10-6; 1 j oo (a _3 2)l/2dad, Aa(ad) (17) errors through simulation, by comparing a bending angiven gle derived from a full ray tracing solution Doppler shift Aa(aa) -- aa(ad) - a,(aa), (18) aa(aa) with the fast forward model value given the same impact parameter, a! (a0 )(Note that a(atan) is not used where as(ad) is the theoretical (i.e., not measur- in the comparisons because this is not the measured able), spherically symmetrical bending angle profile quantity.). Given a statistically significant number of that would produce the correct refractivity profile at cases, we can derive a forward model error covariance the specified location. Using this definition, we have matrix of the form F -< (c d(ad) -- cq,(ad))(c d(ad) -- cq,(ad)) ' >. (19) If the fast model was made more sophisticated by including, for instance, along-track gradients, the forward

13 HEALY: RADIO OCCULTATION BENDING ANGLE ERRORS 11,887 model error should be reduced, since the models would include a better description of the underlying physics of the measurement. The relative trade-off of forward model accuracy against computational expense could then be determined. 8. Conclusions It has been demonstrated that realistic horizontal re- fractive index gradients, derived from a mesoscale forecast on a 12 km by 12 km horizontal grid which includes water vapor, can cause significant errors in RO bending angle profile as a function of impact parameter (,a(aa)) derived from a Doppler shift. It is found that along-track gradients do not significantly affect the derived bending angle, but they produce a variation in the impact parameter along the ray path, and it is not possible to determine the impact parameter value at the tangent point from the measurement, without making a priori assumptions about the nature of the horizontal gradients. Refractive index gradients perpendicular to the ray path introduce an additional component to the measured Doppler shift caused by out of plane bending. This results in an error in both the derived bending angle and impact parameter values as these assume planar bending, but the error in the latter is more significant. where the integral is along the path taken between the satellites. The error & - & + &7 can be derived In the simulations we have shown examples where bend- in terms of &a as follows. Since both the true and ing angle errors near the surface ( 1-2 kin) can exceed derived bending angles will be consistent with the ob , depending on the gradient of the bending angle served Doppler shift, we can write profile. However, a preliminary statistical analysis suggests that more generally this figure is around.- 3-4% RMS. The RMS of the refractiviw errors are generally less than 1.4ø76 near the surface. New analytical expressions for the errors, based on integrating the horizontal gradients along the ray path have been derived which are in good agreement with the simulation results. The results indicate that both bending angle measurement and forward model errors will increase in regions with strong horizontal gradients. The ray tracer developed for this work may be useful in assessing the error in proposed forward models of bending angle and this, along with error estimates and correction strategies based on the NWP forecast, should be considered. In addition, a more comprehensive statistical analysis of the errors associated with horizontal gradients will be the subject of future work. Appendix A' Errors Caused by Along-Track Horizontal Gradients The approach adopted in this work is more general than that presented by Abroad and Tyler [1999], in which the error caused by horizontal gradients was analyzed in terms of a shift in the origin of the coordinate system. Formally, their approach leads to bending angle and impact parameter differences rather than errors, unless the atmosphere is locally spherically symmetric in one of coordinate systems by definition. However, this is unlikely to be the case when the atmosphere con- tains significant refractivity structure over a relatively small horizontal distance, as illustrated in Figure 3c, for example. The geometry is illustrated in Figure 1. Neglecting horizontal gradients perpendicular to ray path, the excess Doppler shift can be written as fa -- f(vf(cosfi-1)+vy sinfi- and the bending angle is c V n (cos '/- 1) + Vy n sin '/), = +, (A1) (^2) where a, fi, and 7 are the "true" values. However, in practice, the bending angle value derived from the Doppler contains an error Aa -- c d-a, because the impact parameter values at the satellites differ from each other as a result of the horizontal refractive index gra- dients Aa = a n-a r=r nsin(0 n+3,)-rrsin(0 r+fi) = /(O - ), ds, (A3) Afa - L((_Vfsinfi+V cosfi)a c + (V sinv+v cos7)a7)-0, (A4) giving a relationship between Aft and Vfi sin 7 + V cos) 7 For the satellite orbits used in this work, the numerical value of k In the inversion of the Doppler shift, the derived impact parameter aa is assumed constant and equal to - sin(4 * + - sin(4 a + (A6) where r = r + fi and = + 7. Expanding (A6) for small values of Aft and A7 and equating to (A3), a-- r cos fl - r a cos a 7 - ds. Eliminating 7, using 7 - k fl, we can write a - 4 r - 4 a' where the denominator is of order the distance between the satellites. The error in the total bending angle is then

14 11,888 HEALY: RADIO OCCULTATION BENDING ANGLE ERRORS Aa = A] + A-/= (1 + k)a], (A9) and, using (A6), the derived impact parameter value can be written as a = r T sin( T q- A ) a T q- r T cos b TA/, (A10) which, after inserting the expression for A], becomes aa ~ a " q- r r T cos T, cos & " - kr n cos &a(aa- a ")' (All) Appendix B' Errors Caused by' Out of Plane Bending The out of plane bending caused by horizontal gradients perpendicular to the ray path affect the measured Doppler shift, from which the bending angle and impact parameter values are subsequently derived. In the presence of out plane bending (using the vector notation outlined in section 2.1), the Doppler shift is given by - f(v ß k - v ß k - (v - c where the scalar products can be expressed as k), (B1) V T. k T - Vf cos e T cos + V cos e r sin + V sin e V r.. k r. _ V n cos e a cos 7 - Vy (e2) n cos e ' sin 7 + Vfi sin e a A f ~ f--(vzt e T Vfie ), ½ (B6) which can usually be approximated with A fa - -(f/c)vfie n. Strictly, the additional Doppler shift is not a measurement error, but it produces an error in the derived bending angle and impact parameter values because they are evaluated assuming planar bending. In a similar approach to Appendix A, the errors in the derived bending angle and impact parameter values can be written in terms of the additional Doppler shift. (Note that from this point the approach is equally applicable for any source of Doppler shift error Afa and is not specific to out of plane bending.) Afa will cause errors A] and A7, Afa -- f-((-v T sin 1 + Vy T cos Clearly, {A9), (A10), and (All} could all be written in ½ terms of (A3), the integral of the along-track horizontal gradients along the ray path. + (V sin 7 + Vy a cos 7)A7). Note that the measured Doppler shift does not pro- As the bending angle is derived assuming a constant vide any information on the impact parameter value at impact parameter (aa), using (8) we can obtain the rethe tangent point. This can only be estimated with lationship between A] and A7, the aid of additional, a priori information, such as the assumption of spherical symmetry, for example. r T cos T (B7) zx - a cos a zx/s - A/S, ( 8) where n 8 for the GPS/MET geometry. Hence by substituting A7 = na/ into (B7), A/ can be written in terms of the additional Doppler shift, Aft - _v T sin 1 + Vy T cos 1 + n(v sin 7 + Vy a cos 7)' (B9) where A(= c/f)is the wavelength of the signal. This can be simplified further using Af ~ -(f/c)l/za R and replacing the sin of angles with the angle and the cos terms with unity, a/s ~ -v?a (-V / + V + n(v 7 + Vya)) ' (B10) The error in the total bending angle caused by perpendicular gradients is then (v ß - k = - Aa = A] + A7 = (1+ (Bll) where the z axis is perpendicular to the x- y, occultaand differentiating aa = r "sin d " where d "= r + fl, tion plane and e T and e a represent the angles between gives the change derived impact parameter the ray vector and the x-y plane at the transmitter and receiver, respectively. Using (1), to first order these are Add r T cos dt (B12) related to the refractive index gradients perpendicular to the occultation plane by Note that (B9) can be substituted for A into (Bll) and (B12), providing a clear link between the change in the Doppler shift value caused by the perpendicular gradients A fd and the errors Aa and Add. en--et~ l(0n) It is interesting to note that out of plane bending only causes a second-order change in bending angle values determined directly from the difference in unit tangent where the integral is taken along the ray path. For small values of e "and e R, the additional Doppler shift is found by subtracting {A1)from(B1), using (B2), vectors, a = cos-l(k T. k t). Denoting the change in (B3), and {B4), to give the "true" bending angle as Aat, we can write 1 Aat = -(1- (k T. kr)2)l/2 A(kT' kr)' (B13) where, to first order, the difference in the scalar product caused by the out of plane bending is given by

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