Lookup tables have been constructed that address group delay and RF carrier phase advance for transionospheric signal propagation referenced to a ground-based transmitter. The tables are based on parameterizations of ray-tracing results using the Reilly ray trace/ICED, Bent, Gallagher (RIBG) model with the magnetic field turned off. The quantities of interest are incremental group and phase path lengths referenced to straight-line distance between transmitter and receiver and are designated by Δlg and Δlp (as defined, both quantities are positive). Equations are presented that relate these quantities to group delay and ionospheric Doppler shifts. Tables currently in use are three-dimensional, expressed in terms of frequency (f), elevation angle (θ), and straight-line total electron content (TECSL). At high frequencies, Δlg and Δlp can be accurately expressed by a straight-line propagation formula that is proportional to TECSL. With decreasing frequency, these quantities become greater than their straight-line counterparts due to additional refraction effects, including ray bending. Extensive ray-tracing runs have been performed under a variety of ionospheric conditions to produce scatterplots of Δlg and Δlp versus TECSL for given pairs of [f, θ] values. The current ranges in these variables are from 20 to 100 MHz and 5° to 90°, respectively. The tables contain coefficients from fourth-degree polynomial fits to the distributions of points within the scatterplots. Where scatter is problematic, the introduction of a fourth parameter, slab thickness, is shown to significantly reduce this scatter. Magnetic field effects are discussed and are shown to have a small effect on table coefficients. Discussion is also included on the use of the tables with a model ionosphere.
 The subject of this paper is parameterizations of transionospheric ray-tracing results giving group delay and RF carrier phase advance that, unlike simple formulas for straight-line propagation [see, e.g., Klobuchar, 1985], take into account refraction effects, which we categorize as ray bending. Ray bending leads to significant increases in these quantities compared with the straight-line propagation approximation at frequencies such as 30 MHz, at low elevation angles, and in the presence of significant ionization (e.g., under daytime conditions). The transmitter-receiver configuration of interest is ground-to-space. The range of elevation angles being considered in the reported work is from 5° to 90°, with emphasis on the lower values, due to greater deviations from the straight-line approximation and due also to making the parameterizations available for analyzing signals from satellite receivers whose coverage extends to the horizon. The frequency range of interest is from 20 to 100 MHz. We have not presently extended the range to lower frequencies due to the increasing likelihood of reflection that is beyond the scope of the work being reported (although results will be shown that exhibit cutoffs due to reflection).
 The quantities of interest from ray-tracing runs are group path lengths, phase path lengths, and straight-line distances from receiver to transmitter. The group and phase path lengths are recast as differences compared with straight-line distance and are designated by Δlg and Δlp. The effects of interest can be extracted from these variables, and by working with differences, the parameterizations need not deal with absolute distances to the receiver. The parameterizations are in the form of lookup tables of Δlg and Δlp that are presently three-dimensional (3-D) in the independent variables frequency (f), elevation angle (θ), and straight-line total electron content TECSL (in units of 1016 electrons m−2). The motivation for using TECSL comes from a straight-line propagation expression in terms of this variable (see section 2). A single ionospheric parameter (in this case, TECSL) is not sufficient for accurately specifying the table quantities in the presence of significant ray bending, for which details in the distribution of ionization along the ray path become important. Work is under way to extend the tables to four dimensions using slab thickness (zslab) as the fourth variable. Section 4 addresses this variable and illustrates the improvement in the specification of Δlg and Δlp by its addition.
 The ray trace/ICED, Bent, Gallagher (RIBG) model [Reilly, 1993, 1991], with improvements by Reilly up to 1996, has been used to obtain the ray-tracing results for constructing the tables. It is based on the ionospheric conductivity and electron density (ICED) profile model [Tascione et al., 1988] up to 1000 km, the Bent model [Bent et al., 1976] above this height, and the Gallagher plasmaspheric model [Gallagher et al., 1988] at still greater heights. Upgrades of RIBG later than 1996 are described and referenced by Reilly and Singh . The option is available to perform ray tracing either in the absence of a magnetic field or in its presence (in either the ordinary [O] or extraordinary [X] mode). The tables being reported do not include magnetic field effects. Results will follow, however, that illustrate the differences one obtains in Δlg and Δlp by including field effects.
 A second ray-tracing model at our disposal is that of Jones and Stephenson . This particular model inputs model ionospheres in tabulated form and is thus not restricted to one particular ionospheric model. Examples of ionospheric models that can be interfaced to such a ray-tracing model are the international reference ionosphere (IRI) model [see, e.g., Bilitza, 1997] and the parameterized ionospheric model (PIM) [Daniell et al., 1995] in addition to that used by RIBG. Experience has shown that more numerical noise is present in the Jones-Stephenson model compared with RIBG, and this is the reason for using the latter model. Three-dimensional interpolations are used in the former model to specify the plasma frequency along the ray path, whereas localized analytical techniques are employed in the latter model to ensure smoothness in this variable. Were numerical noise comparable between Jones-Stephenson and RIBG, we would not expect the derived tables to be substantially different between these models using different ionospheric models. This leads to a key point with regard to the tables: They are expected to be effectively independent of the choice of ionospheric model used to generate them. This is, in fact, precisely true with increasing frequency and elevation angle where a straight-line propagation formula can be used to specify Δlg and Δlp and is a function of only TECSLwith regard to ionospheric effects (discussed in more detail in section 2). The only basic requirement of the ionospheric model of choice is that it approximately describes the range of behavior (here excluding turbulence leading to scintillation) associated with solar activity, location (e.g., midlatitudes), and local time. In doing so, a wide range of Δlg and Δlp behavior can be sampled as is necessary for construction of the tables.
 The presentation of results is as follows. The concept behind the chosen parameterization is discussed in section 2 including a formula for straight-line transionospheric propagation. Section 3 contains the bulk of our results with their characteristic scatter that arises from the current limitation to a 3-D parameterization. Our choice of a fourth independent variable, namely, slab thickness, is discussed in section 4. Magnetic field effects on Δlg and Δlp are addressed in section 5 followed by a summary in section 6.
 The quantities being parameterized, Δlg and Δlp, are related to their respective path lengths by
where lg, lp, and lfs are group, phase, and free space (straight-line) path lengths. The basis for the chosen parameterization is their familiar behavior at high frequencies, given by
where TECSL is in units of electrons m−2 and f is in Hz [e.g., Klobuchar, 1985]. Equation (2) implies straight-line propagation and underestimates Δlg and Δlp in the presence of ray bending that becomes increasingly important with decreasing frequency, decreasing elevation angle, and increased ionization along the ray path. This is illustrated in Figure 1, which shows Δlg and Δlp as functions of frequency from 30 to 100 MHz based on ray-tracing calculations (solid curves) and equation (2) (dashed curves). Each pair of panels from left to right shows Δlg and Δlp for the same transmitter-receiver locations. In each panel, θ (referenced to the straight-line between transmitter and receiver) and TECSL are identified above the curves. The latter value is obtained by a straight-line integration through the applied model ionosphere for that given pair of transmitter-receiver locations. This value is then used in equation (2) to obtain the dashed curves. First addressing Δlg, its value at 30 MHz in the upper left panel based on ray tracing is approximately double the straight-line value, but the differences then become insignificant at the higher frequencies within the displayed range. The middle left panel shows the effect of approximately halving the ionospheric content but keeping θ fixed at 10°. As expected, smaller differences occur between the two curves, with the ray-tracing result at 30 MHz now being ∼30% greater than the straight-line value. The bottom panel shows the effect of increasing the elevation angle but with approximately the same TECSL considered in the middle panel. Still smaller differences between the curves are seen, with the ray-tracing result at 30 MHz now being ∼12% greater than the straight-line value. The differences between this and the middle panel at low frequencies illustrate the need to add θ as a third variable to our parameterization in addition to f and TECSL. Similar behavior in Δlp is seen in the three panels on the right but with smaller differences between the curves compared to Δlg, as may be expected from stationarity of the phase path length integral about the computed ray path, discussed below.
2.2. Time Delay and Doppler Shifts
 The quantities of interest derivable from the parameterization of Δlg and Δlp are ionospheric time delay Δτi and ionospheric Doppler shift Δf. Δlg is Δτi multiplied by the speed of light. The most accurate representation of Δlg comes from ray tracing or its parameterization. The straight-line representation comes from equation (2) and is less accurate. The calculation of Δf is more complicated since consideration must be given to at least two paths from the transmitter to a moving receiver. The Doppler shift is simply related to the rate of change of phase path length and its ionospheric contribution:
The time derivative can be obtained numerically from two or more positions of the receiver-transmitter pair at different times. Each position corresponds to different θ and TECSL values that will normally lead to nonzero values of the derivative. For straight-line propagation, equation (3) becomes
using equation (2) for Δlp. Note that the straight-line representations leave out any explicit dependence on elevation angle.
 If one is interested in the free space components of propagation time and Doppler shifts (that add to the above ionospheric components), they are ℓfs/c and a term like equation (3) with Δℓp replaced by ℓfs. By removing ℓfs from the total path lengths in equations (1a) and (1b), we have effectively removed the need to specify the altitude of the receiver for the ionospheric components. We note, however, that the receiver altitude for the results in this paper was chosen to be 1600 km. Δlg and Δlp have been calculated at other altitudes up to 36,000 km and confirm our expectations that minor changes arise by changing the receiver altitude (excluding changes that place the receiver in regions of significant ionization such as near the peak of the F2 layer). For example, changing from 1600 to 3200 km leads to changes in Δlg of the order of 5% with smaller changes in Δlp. Such changes in these variables and, in turn, in their associated ionospheric time delay and ionospheric Doppler shifts are considerably smaller than uncertainties arising during applications that use the lookup tables. In this context, the greatest uncertainty is expected to be that associated with the model ionosphere that specifies TECSL as one of the inputs for performing table interpolations.
3. Approach to Constructing Lookup Tables
 The sections to follow address inputs to the ray-tracing model, how the outputs are organized for obtaining the needed curves to fill the tables, uncertainties in the curves arising from limiting the independent table variables for the ionosphere, and the tables themselves. The discussion in this section is limited to representing Δlg and Δlp in the three independent variables f, θ, and TECSL. The addition of a second ionospheric variable is more fully discussed in section 4. Furthermore, magnetic field effects are not included in the present discussion but are the subject of section 5.
3.1. Making a Single Ray-Tracing Run
 The inputs needed for a run are frequency, transmitter and receiver locations, and parameters for specifying the ionosphere. Receiver location is derived from the transmitter location along with the straight-line elevation and azimuth angles specified at the transmitter. The coordinates for the receiver are obtained by first selecting the receiver altitude and then determining latitude and longitude at this altitude along the straight-line path defined by the given elevation and azimuth angles. The needed inputs for specifying the model ionosphere are sunspot number, date, and universal time.
 The ray-tracing calculation begins by launching the ray at the specified elevation and azimuth angles. A homing algorithm is used to launch subsequent rays until the ray homes on the receiver to within a specified miss distance. The outputs of interest are free-space distance, group path length, and phase path length, from which Δlg and Δlp are obtained. The straight-line TEC value between transmitter and receiver is obtained from a separate calculation.
3.2. Organizing Outputs From Multiple Runs
 Production runs are made for selected pairs of frequency and elevation angle. A uniform grid of azimuth angles from 0° to 360° is employed with the number of grid points being a function of elevation angle (more points with decreasing angle). Rays are launched from several transmitter locations spanning the latitude range 30°N to 55°N. At each location, several ionospheres are sampled through changes in sunspot number (up to values of 150), date (spanning all seasons), and UT (0 to 24 hours). The results are then displayed versus TECSL and fitted with a fourth-degree polynomial for generating the lookup tables. Figure 2 shows examples of the results in the form of scatterplots of Δlg that provide an indication of the number of production runs made at a given frequency and elevation angle. The panels on the left show scatterplots at 30 MHz for elevation angles of 5°, 10°, and 15°. The dashed curves show the straight-line approximation given by equation (2), while the solid curves show the fourth-degree fits. The panels on the right show corresponding results at 100 MHz. At this higher frequency, the ray-tracing results exhibit approximately straight-line behavior with insignificant scatter. Figure 3 shows Δlp from the same set of runs used to generate the results in Figure 2. Here the behavior is more similar to straight-line propagation with less scatter. We have previously alluded to the stationarity of the phase path integral with respect to perturbations of the ray path through fixed end points, which is a statement of Fermat's principle [Born and Wolf, 1970]. Hence bending the ray path will not result in appreciable phase path length scatter. This suggests only a minor dependence on the shape of the electron density profile and more of a dependence on an integrated quantity, like TEC, or the integral of higher-order N/f2 terms in the expansion of the square root in the index of refraction. However, in the group path length results, the effect of ray bending seems to be relatively more important in the departure and scatter effects, since group path length is not constrained by Fermat's principle.
 The TECSL cutoffs seen in Figures 2 and 3 vary with frequency and elevation angle. At 100 MHz, this cutoff is seen to decrease with increasing elevation angle. It is important to note that this behavior is not exhibiting reflection points but rather upper limits to TECSL sampled by the ionospheric model. This effect can also be seen at 30 MHz for the elevation angle of 15°. This is not the case, however, at 5° and 10° where bending becomes so severe at large TECSL values that rays are not able to reach the receiver beyond the TECSL ranges containing scatter points. Work is under way to specify cutoffs in the tables but has not progressed far enough over the full 3-D space within the tables being reported at this time. The effort is complicated by the fact that unrealistically large ionospheres are required at the higher frequencies and elevation angles in order to achieve reflection. Extending the curves in the tables to unphysical cutoffs is not a problem, however, since the curves in the vicinity of such cutoffs will not be sampled in applications of the tables using models that give realistic ionospheres.
3.3. Examining Scatter
 The scatter in Figures 2 and 3 is due to details in the electron density along the propagation path that are not taken into account using a single ionospheric parameter (TECSL). The scatter increases with decreasing frequency, decreasing elevation angle, and increasing ionospheric content due to increased ray bending that becomes increasingly sensitive to altitude behavior of the electron density as well as to horizontal gradients in NmF2 (electron density at F2 peak) and tilts in HmF2 (height of F2 peak) as functions of range within the plane of propagation. It is instructive to examine the electron density within the propagation plane for extreme scatter points at the same TECSL. A pair of such points has been selected from the upper right panel of Figure 2 at TECSL = 100. Figure 4 shows their ray paths (red curves), straight-line paths (blue curves), ionospheres, and HmF2 (nearly horizontal curves between 300 and 400 km) versus range in a flat-Earth projection. The upper and lower panels correspond to the larger (Δlg = 82.2) and smaller (Δlg = 64.6) scatter point values, respectively. Greater ray bending is seen in the upper panel and can be attributed to the ray encountering a larger NmF2 value. An examination of several images like these shows that changes in ray bending become increasing sensitive to modest changes in NmF2 once its magnitude reaches values in the vicinity of 106 cm−3 or more.
 Differences in electron density along the ray paths can be better seen in the upper panel of Figure 5 that includes overplots of the two ray paths. Cumulative TEC profiles corresponding to the density profiles are given in the lower panel. Rays 1 and 2 refer to the rays in the upper and lower panels of Figure 4, respectively. The ionospheric F2 layer for ray 1 is somewhat more compressed and, because of greater bending, keeps the ray in the vicinity of NmF2 over a longer range. The larger value of TEC for ray 1 is due to the combination of this longer range and the larger electron densities near NmF2 compared with ray 2. For the pair of rays under discussion, there is an increase of ∼16% in ray path TEC from ray 2 to ray 1. The corresponding increase in Δlg is ∼27%, thus showing an amplification in scatter compared with ray path TEC scatter. The above mentioned layer compression is conveniently represented by slab thickness zslab that is given by the ratio of vertical TEC to NmF2 (see, e.g., Klobuchar , Fox et al. , and Davies and Liu  for descriptions of its behavior under a variety of conditions). The slab thicknesses corresponding to rays 1 and 2 where they reach NmF2 are 264 and 366 km, respectively. A further examination of scatter in terms of this parameter is given in section 4 where slab thickness is introduced as a fourth independent variable for the tables.
3.4. Lookup Tables
 Scatterplot fits like those in Figures 2 and 3 have been used to construct tables of Δlg and Δlp over f and θ ranges of 20 to 100 MHz and 5° to 90°, respectively. The tables are currently 3-D (in f, θ, and TECSL) with work in progress to extend them to 4-D (see section 4). The tables specify the coefficients from the fitted curves rather than the curves themselves. Sample curves are shown in Figure 6 at 30 and 100 MHz over the TECSL range from 0 to 150. Each panel shows multiple curves to illustrate elevation angle effects versus TECSL for the labeled frequency. Presently, determination of Δlg and Δlp is allowed beyond TECSL = 150 using the table coefficients. As noted in section 3.2, work is under way to provide effective cutoffs for those combinations of f and θ where reflection can occur within the bounds of physically realistic ionospheres. Within these bounds, however, reflection does not occur over much of the 3-D space being addressed.
 We noted above that transmitter locations for the production runs used to construct the tables span the latitude range 30°–55°N. This does not place a restriction on where the tables can be used. At any location, a specification of TECSL and zslab (using both for applications involving the 4-D tables) for table interpolations provides the needed ionospheric time delay and/or ionospheric Doppler shifts. The uncertainty in these derived quantities can be expected to vary, however, from region to region. For example, more uncertainty is likely in equatorial compared with middle-latitude regions. This is a reflection of greater anticipated uncertainty in the model used to specify TECSL and zslab for interpolation purposes rather than uncertainties of the tables themselves. Nevertheless, there would be greater scatter within our scatterplots had we included equatorial ionospheres in our production runs due to more severe horizontal gradients and HmF2 behavior unique to the Appleton anomaly [Appleton, 1954] (larger values at times than occur at middle latitudes). The uncertainty associated with increased scatter is still expected to be much less than from the specification of TECSL and zslab used during table lookups.
4. Introducing Slab Thickness as a Fourth Independent Variable
 We began a discussion in section 3.3 of the cause of the scatter appearing in Figures 2 and 3. There it was noted that the degree of bending at low frequencies (e.g., 30 MHz) and low elevations angles (e.g., 5°) becomes sensitive to NmF2 in the vicinity of 106 cm−3 or greater and, in turn, that differences in ray bending (constrained by fixed TECSL) are the source of the scatter (more specifically, changes in ray path TEC associated with variable bending). NmF2 would then appear to be a candidate parameter for further characterizing the ionosphere within the lookup tables and thereby reducing the scatter. NmF2, however, is strongly correlated with TECSL and is thus not a satisfactory fourth independent table variable. The correlation becomes effectively removed by using slab thickness zslab in place of NmF2. As introduced in section 3.3, zslab is defined as the ratio of vertical TEC to NmF2. Since these latter variables track one another, at least in an approximate sense, one observes a much weaker correlation between zslab and TECSL than between NmF2 and TECSL. Since zslab is to be treated as a variable independent of TECSL, the following comments address its behavior with the constraint that TECSL is fixed. This constraint, in turn, imposes a similar constraint on vertical TEC. Thus one observes a decrease (increase) in zslab as NmF2 increases (decreases), thereby conveying information about NmF2. In general, a decreasing (increasing) zslab leads to greater (less) ray bending (keeping in mind that TECSL is being held fixed), reflecting a greater (smaller) concentration of ionization in the vicinity of NmF2.
 To examine the effectiveness of zslab in reducing scatter, plots like those in Figure 2 have been broken down into component plots for selected ranges in this variable (designated by Δzslabi where i refers to the ith slab thickness interval). To do so, however, requires a decision on where zslab is to be specified within the plane of propagation. A natural choice is to select this point in the vicinity of strongest bending. The obvious candidate point is where the straight-line path intersects NmF2. Reference is made to this path since it is a known quantity in applications of the tables where transmitter and receiver locations are specified prior to any table lookups. Results to follow will use this convention. In order to do so, NmF2 and vertical TEC have been obtained where the straight-line path intersects NmF2 for each of the ray-tracing runs producing a point within a given scatterplot. Consideration has also been given to zslab obtained 250 and 500 km up-range and downrange from the intersection point. Since, because of the possible presence of horizontal ionospheric gradients, different sets of values are obtained by moving the reference point, a redistribution of scatter points into the component plots will occur and may or may not reduce overall scatter. An examination of component plots among the five reference points for various combinations of f and θ do not show significant improvement by specifying zslab to either side of the NmF2 intersection point.
 The first illustration of how scatter appears within component plots is given in Figure 7 for Δlg at an elevation angle of 10° and at frequencies of 20, 25, and 30 MHz. The top panel within each set shows the full set of scatter points. Component plots are shown below the full scatterplots for the following four Δzslabi intervals: <280 km, 280–300 km, 300–325 km, and >325 km. These were selected from scatterplots of zslab to provide, on the average, a reasonably well distributed population of points within each Δzslabi. The larger Δlg points within the full scatterplots are seen to be associated with smaller Δzslabi or larger NmF2 values as expected from the discussion in 3.3. Overall, Δzslabi is seen to be an effective second ionospheric parameter for reducing scatter associated with only a 3-D representation of Δlg within its lookup table. Stray points such as the larger values in the middle column of the figure (25 MHz and Δzslabi between 280 and 300 km) arise from horizontal gradients in NmF2. In-plane images like those in Figure 4 lead to this conclusion. In general, it can be assumed that any remaining significant scatter after introducing slab thickness arises from such gradients. Future work will address this issue if the uncertainty in Δlg associated with the effect is deemed problematic in applications of the tables to the restricted portion of the 3-D space exhibiting gradient effects.
Figure 7 clearly shows the effect on Δlg of changing frequency. With increasing frequency, the pattern of points becomes more compressed, moves closer to the dashed line for straight-line propagation, and extends to larger TECSL values before reflection prevents further extension.
Figure 8 repeats the results from Figure 7 at 25 MHz and compares them at this same frequency with results at elevations angles of 5° and 15°. A similar reduction in scatter, here with changing elevation angle, is seen by introducing Δzslabi. The effect on the full pattern of points from the upper left to upper right panel is similar to what is seen in Figure 7 although somewhat weaker for the chosen angles.
 It should be noted, in spite of sampling a wide variety of ionospheres within the applied model, that different overall patterns of scatter can arise from the use of other ionospheric models. We address results from only the applied model here since the ray-tracing algorithm has been tailored to this model. Nevertheless, we do not expect significant deviations from the curves within the tables using other models provided they, as well as the applied model, offer physically reasonable representations of HmF2 and NmF2 under a variety of ionospheric conditions. Scatter within plots such as those in Figures 7 and 8 will exhibit differences from model to model, however, if horizontal gradients in NmF2, as discussed above, differ between models.
 As a summary comment to this section, there is no need to introduce slab thickness as a fourth variable to the tables over much of the present 3-D space. Work is currently in progress to place bounds on where it is needed and then produce curves from component plots where appropriate.
5. Magnetic Field Effects
 The RIBG model has been run in both the O and X modes to examine differences in Δlg and Δlp compared to their zero-field value. The index of refraction is smaller (larger) for the X (O) mode compared with the zero-field index, leading to larger (smaller) values in these variables. With the magnetic field turned on, the index becomes a function of · (projection of the field vector along wave vector ) and, in turn, is a function of transmitter location and the launch vector of the ray (from the perspective of how the magnetic field is sampled). It is instructive to examine field effects by observing Δlg and Δlp as a function of azimuth angle for a spherically symmetric ionosphere. Figure 9 illustrates the effects at 30 MHz and 20° elevation angle for an electron density profile under daytime conditions (the profile corresponds to a TECSL value of 138 at the selected elevation angle). The transmitter location is 25°N latitude and 25°E longitude, although similar results are obtained at other longitudes. The upper panel shows Δlg for zero-field (middle curve), for the X mode (upper curve), and for the O mode. The X mode curve possesses the largest values, as expected from its index of refraction being smaller than that for either zero-field or the O mode. The largest departure from the zero-field results occurs when looking south and is ∼7%. The difference is ∼14% compared with the O mode. The differences in absolute magnitude between the O mode and zero-field results closely follow those for the X mode but are slightly smaller. The given variations for both modes can be compared to field angle in the lower panel (the angle corresponding to · where the ray reaches 420 km).
 A comparison of curves for Δlp can be made in the middle panel. A similar behavior is seen except for smaller differences between the various curves that is consistent with earlier results shown for this variable that exhibit a weaker variation with TECSL and less scatter compared with Δlg. It should be kept in mind that the results in Figure 9 for both Δlg and Δlp are specific to a latitude of 25°N and a TECSL value of 138. They will change as these variables change and in particular show weaker (stronger) field effects for smaller (larger) TECSL values.
 Having illustrated field effects in a controlled situation (simple ionosphere and known location), we now show how they alter the distribution of points in a scatterplot. The reference plot (for zero-field Δlg) is that from Figure 7 at 25 MHz and 10° elevation angle showing the full distribution of points prior to distributing them among the various component plots. This is redisplayed in Figure 10 (top panel) along with X mode (middle panel) and O mode (bottom panel) distributions. The solid curve in each panel is the same and is the fit to the zero-field results. The same curve is used for ease in examining how the X and O mode patterns change compared to the zero-field pattern. The dominant effects are shifts in the X and O mode patterns in the directions expected from their respective indices of refraction. A secondary effect for the X mode is a slight increase in spread at the larger TECSL values that can be attributed to sampling a variety of field-line angles. The effect is less obvious for the O mode where the larger index of refraction has an overall tendency of reducing scatter (less scatter than for zero-field Δlg were the field-line angle fixed).
 On the basis of similarities in the patterns of points shown in Figure 10, one expects that component plots for the two modes to behave similarly to those already shown in Figure 7 at 25 MHz and 10° elevation angle. X and O mode component plots corresponding to their total scatterplots in Figure 10 can be seen in Figure 11. For comparison, the component plots for zero-field from Figure 7 are repeated in the left panels. Reduction in scatter by introducing slab thickness is seen to be about the same as for the zero-field results, and, as expected, slighter greater scatter is present for the X mode compared with the O mode (most clearly seen for the slab thickness interval from 280 to 300 km).
 Since differences in Δlg and Δlp arising from field effects are less than the uncertainties reflected in the various scatterplots previously shown, tables to date have not been constructed for the separate modes. If we were to do so, the same curve fitting procedure would be used as has already been done for the zero-field results. The azimuth angle effect shown in Figure 9 would be averaged out since scatterplots such as those in Figure 10 span the full range in this variable. If azimuth angle were introduced as an additional independent variable, as few as three values (e.g., 0°, 90°, and 180°) should be sufficient for which associated table curves are obtained by scaling azimuth-averaged curves using results like those in Figure 9 as a guide. When using the azimuth-dependent tables, an input azimuth value would then be associated with the most appropriate table value where input values beyond 180° would be mapped to the appropriate table value (e.g., an input value in the vicinity of 270° would be mapped to 90°).
6. Discussion and Summary
 In the previous sections we have discussed 3-D lookup tables of Δlg and Δlp currently in use for specifying transionospheric group delay and ionospheric Doppler shifts in the absence of magnetic field effects. The independent variables are f, θ, and TECSL between the receiver and transmitter. The current ranges in f and θ are 20 to 100 MHz and 5° to 90°, respectively. The tables contain coefficients to fourth-degree fits of the table variables at given values of f and θ as a function of TECSL obtained from extensive runs of the RIBG ray-tracing model. The distribution of points lies close to the curve for straight-line propagation at low TECSL values and deviates to higher values with increasing TECSL due to ray bending. Furthermore, the spread in points becomes greater with increasing TECSL due to limiting ionospheric characterization to TECSL. There is a limit to achievable TECSL due to reflection that is sensitive to f and θ as can be seen in selected scatterplots previously presented. Reflection cutoffs are not presently in the tables, but work is under way to include them.
 The scatter exhibited in TECSL plots can be reduced by a more detailed characterization of the ionosphere. The introduction of slab thickness zslab as a fourth independent variable has been shown to be effective in reducing the scatter. This variable provides information on electron density profile shape that becomes increasingly important in the presence of significant ray bending. In the vicinity of a given TECSL, a decrease in zslab corresponds to an increase in NmF2 and in turn, to increased ray bending, leading to larger values of Δlg and Δlp. The decrease in scatter by introducing zslab was illustrated in Figures 7 and 8. Work is in progress to expand the dimensionality of the current tables to 4-D with this variable. There are no plans to add variances due, in part, to the limited sampling of points in various regions of the parameter space and due also to the fact that such variances are expected to be significantly smaller than those associated with characterizing the ionosphere in applications of the tables. Further discussion on the use of the tables is given below.
 Effects on Δlg and Δlp due to a horizontal gradient in NmF2 were briefly addressed in sections 3 and 4. In the presence of a gradient, there tends to be more bending due to encountering a larger NmF2 than that associated with TECSL (i.e., than NmF2 encountered along the straight-line path). The larger NmF2 can be to either side of the straight-line path, depending on the sign of the gradient. As noted in the discussion of component plots in section 4, outliers in selected plots can be attributed to the presence of gradients. In general, gradient effects are weak in the runs used to produce the tables, since middle latitudes were selected for locating the transmitter and a climatological ionospheric model was used. Gradients become more important at low latitudes due to the Appleton anomaly (for azimuth angles near 0° or 180°) and in consideration of the effect of irregularities. For use of tables at low latitudes or in regions of irregularities (or other departures from a climatological model), one may wish to increase table variances to reflect additional scatter from irregularity gradient effects. Although the RIBG model includes gradient effects, like the day-night terminators, the midlatitude trough, the aurora, the Appleton anomaly, and others, it is difficult to get these gradients correct. A climatological model tends to diminish the gradients and ignore many others that occur. Realistic gradient effects have to be considered separately in concert with real-time data that correctly locate and characterize them. One may increase variances to reflect the uncertainties that are inevitably present due to irregularities. This is a limitation for determining ionospheric time delays and especially for determining Doppler shifts, which involve time derivatives of irregularity gradient departures from a climatological model.
 Magnetic field effects were addressed in section 5. Δlg and Δlp increase (decrease) for the X (O) mode compared with zero-field values and depend on azimuth angle, latitude of the transmitter, and the strength of the ionosphere (magnitude of TECSL). Scatterplots presented in section 5 that sample the full range of azimuth angles and a range of middle latitudes show field effects at the larger TECSL values (where the effects are largest) to be of the order of 10%. This is comparable to the uncertainty exhibited with component scatterplots, and thus there has been no attempt to represent the separate modes in our lookup tables.
 The key to effective use of the lookup tables in specifying group delay and/or ionospheric Doppler shifts is the ionospheric model used in a given application to specify TECSL and zslab (the former alone when using 3-D tables and both when using 4-D tables). The application begins by specifying transmitter and receiver locations followed by an integration through the model ionosphere along the straight-line path (or paths for successive receiver locations) to obtain TECSL. When using 4-D tables, a second integration is performed to obtain vertical TEC where the straight-line path intersects HmF2. The ratio of this TEC to its corresponding NmF2 yields zslab. Values of Δlg and/or Δlp follow by table lookup using the frequency of interest, the elevation angle corresponding to the straight-line path, and either TECSL or the combination of TECSL and zslab. Section 2 provided equations relating the Δ to group delay and Doppler shifts. The processing time to specify the Δ including determination of TECSL and zslab is much shorter than their specifications using a ray-tracing model. The largest errors are expected to arise from differences between the above TECSL and zslab values and their true values at the time and location of interest. A reduction in errors compared with a climatological specification is expected if ionospheric measurements in the vicinity of the transmitter are available for ingestion into the applied ionospheric model. The combination of lookup tables and future data assimilation models will lead to improved rapid determination of group delay and ionospheric Doppler shifts compared with present determinations, especially those that employ straight-line propagation formulas.
 Planned improvements to the current tables, as previously discussed, are to extend them to 4-D and provide reflection cutoffs. Attention may also be given to incorporating horizontal gradient effects (with the focus on NmF2 gradients) if applications arise where gradients at large TECSL values are expected to be a common occurrence.