Analytical formulas for refraction of radiowaves from exoatmospheric sources

Authors


Corresponding author: I. Holleman, Radboud University Nijmegen, Institute for Molecules and Materials, PO Box 9010, 6500 GL Nijmegen, Netherlands. (i.holleman@science.ru.nl)

Abstract

[1] We present new analytical formulas for the atmospheric refraction of radiowaves from exoatmospheric sources, such as the Sun. The refraction formulas are derived from the so-called Effective Earth's Radius Model (k-Model) or 4/3-Model which is widely used for atmospheric radars. The new formulas for the refraction angle as a function of elevation are compared to numerical results from the atmospheric refraction routines of the Starlink positional astronomy library and to refraction observations of operational weather radars in the Netherlands and Finland. It is concluded that the refraction formulas from the k-Model are in good agreement (within 0.02°) with the reference calculations and radar observations. As the k-Model is used in numerous radar applications, it is expected that these easy-to-use refraction formulas, which are consistent with this physical model, can be of wide use.

1 Introduction

[2] Approximate or semiempirical analytical expressions for the refraction of radiowaves from exoatmospheric sources—such as the Sun—are useful for many radar applications. For instance, radio signals from the Sun are used for monitoring of the antenna alignment and the receiving chain of weather radars [Huuskonen and Holleman, 2007; Holleman et al., 2010a, 2010b]. As weather radars scan close to the horizon, the intercepted radio signals originate from a rising or setting Sun and thus atmospheric refraction is an issue.

[3] The energy paths of radiowaves in the atmosphere can be approximated by rays, and hence, their propagation can be obtained from the ray-tracing method [Bean and Dutton, 1966; Doviak and Zrnić, 1993]. The exact differential equation that specifies the ray path in a spherical stratified medium, like the Earth atmosphere, was first given by Hartree et al. [1946]. The ray propagation can be accurately calculated from this differential equation using, e.g., the Starlink positional astronomy library [Disney and Wallace, 1982; Starlink, 2011]. For many applications, however, modest precision is required, and easy-to-use analytical expressions are preferred. In Huuskonen and Holleman [2007], we presented empirical formulas for the refraction of radiowaves by fitting “Sonntag-like” equations [Sonntag, 1989; Bennett, 1982] to the Starlink calculations. Although these formulas are sufficiently accurate, we find it unsatisfactory that they do not originate from a physical model.

[4] The Effective Earth's Radius Model (k-Model) or 4/3-Model is commonly used to describe the propagation of radiowaves in a spherically stratified atmosphere [Bean and Dutton, 1966; Doviak and Zrnić, 1993]. The main assumption of this model is that the refractivity decreases linearly with increasing height. Bean and Dutton [1966] noted that “its success is due to the 4/3-Model being in essential agreement with the average refractivity structure near the Earth surface which largely controls the refraction of radio rays at small elevations.” Later, Robertshaw [1986] used the k-Model to approximate determinations of grazing angle, ground range, and slant range for higher altitude paths.

[5] In this paper, we present analytical formulas in closed form for the refraction angle of radiowaves from exoatmospheric sources as a function of elevation. These analytical formulas are derived from the k-Model and thus are fully compliant with this physical model. As this model is used in numerous radiowave applications, it is expected that these simple refraction formulas can be of wide use.

2 Derivation of Refraction Formulas

[6] In this section, the analytical formulas for the refraction angle as a function of elevation are derived. First, the k-Model is introduced, and the propagation of radiowaves according to this model is presented. Then the geometry of exoatmospheric refraction is discussed, and the link with the k-Model is detailed. Finally, the analytical formulas for the refraction angle of exoatmospheric sources as a function of their apparent (as seen by observer) or true (as seen without refraction) elevations are derived.

2.1 Effective Earth's Radius Model

[7] Using well-known trigonometric equations, it is straightforward to derive that the height h above the surface of a ray, leaving an antenna at elevation θ with respect to the Earth surface, can be written as follows:

display math(1)

where symbols r and a refer to the range along the propagation path and the Earth radius (6371 km), respectively. Here it has been assumed that the radiowaves propagate along straight lines, and thus, the effect of atmospheric refraction is not accounted for.

[8] The k-Model (4/3-Model) is explained in many textbooks on atmospheric radars, e.g., in Bean and Dutton [1966] and Doviak and Zrnić [1993]. The crucial assumption of the model is that the derivative of the refractive index n with respect to the height is constant. It can then be shown that the effect of the atmospheric refraction on the propagation can be incorporated by the following modification of the Earth radius [Doviak and Zrnić, 1993]:

display math(2)

where the parameter k is the so-called “Effective Earth's Radius Factor.” From this formula, the vertical gradient of the refractive index is calculated to be as follows:

display math(3)

The curvature of the propagation path, i.e., the inverse of the radius of curvature rc, is given by the following [Doviak and Zrnić, 1993]:

display math(4)

with s being the on-ground distance of the ray. In the right-hand side of the upper equation, the derivative of the height with respect to on-ground distance math formula is approximated by “ tanθ” for sa. The lower equation is obtained by substitution of the refractive index gradient according to equation ((3)). For a constant refractive index gradient, the equations above imply that radiowaves propagate along rays with constant curvature.

[9] According to the k-Model, the original range-height equation (equation (1)) with the effect of atmospheric refraction included becomes the following:

display math(5)

where the elevation θanow refers to the apparent elevation at the surface as the refraction will slightly change the elevation along the atmospheric propagation path. Via inversion of the range-height equation (equation (5) ) the refraction-corrected range as a function of apparent elevation and height is obtained as follows:

display math(6)

where the second-order term in h/ahas been neglected as it is much smaller than the first-order term.

2.2 Geometry of Exoatmospheric Refraction

[10] Figure 1 schematically shows the propagation path of radiowaves from exoatmospheric sources received by an antenna at the Earth surface. Note that the thickness of the Earth atmosphere is severely exaggerated in order to better visualize the refraction. In an atmosphere with a constant refractive index gradient, the “equivalent height” heis defined as the height where n has dropped to unity. The equivalent height thus is the following:

display math(7)

where n0represents the refractive index at the Earth surface and equation ((3)) is used to evaluate the vertical derivative of the refractive index. The equivalent range re is defined as the length of the propagation path required to reach the equivalent height and is obtained by substituting the equivalent height into equation ((6)).

Figure 1.

Schematic drawing of the refraction geometry of radiowaves from exoatmospheric sources and approximations made in the k-Model. The radiowaves enter the atmosphere in the upper right of the figure and are received by an antenna at the top of the sphere. Note that the drawing of the Earth and its atmosphere is not to scale.

[11] The elevation of the ray from the radiowave source at the top of the atmosphere is denoted as the “true elevation.” The atmospheric refraction angle τ is defined as the difference between the apparent elevation θaand the true elevation θtas follows:

display math(8)
display math(9)

where all angles are given in radians and τa/trepresents the refraction angle as a function of the apparent/true elevation. In the k-Model, the refraction angle is the ratio of the equivalent range re and the radius of curvature of the propagation path rc. This is also indicated schematically in Figure 1.

2.3 Formulas for Exoatmospheric Refraction

[12] The refraction angle τaof radiowaves from exoatmospheric sources as a function of the apparent elevation (referred to as “apparent refraction”) can be written as follows:

display math(10)

where the apparent refraction angle is given in radians. The equivalent range re is obtained from equation ((6)) with the height given by equation ((7)), and the curvature 1/rc is substituted using equation ((4)).

[13] A similar equation can be obtained for the refraction angle of exoatmospheric sources as a function of the true elevation (referred to as “true refraction”). Using equations ((4)) and ((8)) and the well-known trigonometric addition theorem, the sine of the apparent elevation can be approximated by the following:

display math(11)

for low elevations (cos2θ≈1) and “short” ranges (rrc). It will become clear below that the impact of this “low-elevation” approximation on the obtained refraction formulas is negligible.

[14] By insertion of the above result in the height equation (equation ((5))), subsequently solving for r, and setting the height to he, one obtains the following:

display math(12)

where the length of the atmospheric propagation path is now given as a function of the true elevation. Using this range equation, the true refraction of the radiowaves becomes the following:

display math(13)

with the true refraction angle τtagain given in radians.

2.4 Value of Effective Earth's Radius Factor

[15] Observations of the vertical gradient of the refractive index suggest that it is approximately equal to −1/4a for radiowaves at the Earth surface, and according to equation ((2)), this corresponds to an Effective Earth's Radius Factor of k=4/3. Hence, the k-Model is often referred to as the “ 4/3-Model” [Bean and Dutton, 1966; Doviak and Zrnić, 1993]. Refraction from exoatmospheric sources is not only occurring in the surface layer but also at higher altitudes in the atmosphere. At higher altitudes, the vertical gradient of the refractive index is smaller (closer to zero), and thus, it is not a priori clear what the best overall value for k is but it will be closer to one. In the spirit of the 4/3-Model, we therefore decided to test also k values of 5/4 and 6/5 in the remainder of this study. It should be noted that the refractive index gradient in the lowest kilometer may have a substantial day-to-day variability, i.e., 25% or more [Bech et al., 2007].

[16] For the surface refractivity n0, the reference value of N0=(n0−1)×106=313 is used throughout this paper [Doviak and Zrnić, 1993].

3 Analysis of Refraction Formulas

[17] Figure 2 shows the calculated refraction angle as a function of the apparent and the true elevation from the k-Model (equations ((10)) and ((13)). For elevations above roughly 6°, the curves are overlapping, but the refraction at zero apparent elevation (0.83°) is clearly different from that at zero true elevation (0.64°). According to equations ((10)) and ((13)), the true and apparent refraction at zero elevation are the following:

display math(14)
display math(15)

and thus, the ratio is only dependent on k. For higher elevations (sin2θ≫10·n0−1, i.e., θ≫3°), the true and apparent refraction both converge to the following:

display math(16)

which explains the overlapping of the curves in Figure 2 for higher elevations. In this case, the refraction is independent of the vertical derivative of the refractive index (related to k). This high-elevation approximation, which agrees with Bean and Dutton [1966], is plotted in Figure 2 for comparison. It is noted that the high-elevation limit of the true elevation formula behaves well although an approximation valid for low elevations only was made during its derivation.

Figure 2.

Radiowave refraction angle as a function of the true and apparent elevation calculated using the formulas from the 4/3-Model. In addition, the high-elevation approximation of the refraction equations and the apparent refraction as constructed from the true elevation are shown.

[18] Via a plotting exercise, the consistency of the formulas for the true and apparent refraction can be investigated. For this, the true refraction (equation ((13))) is plotted with a modified x-axis. From equation ((8)), it can be understood that a curve (x,y) with plotting values (θt+τt(θt), τt(θt)) should equal the refraction angle τa as a function of the apparent elevation θa. The dashed curve in Figure 2 is constructed using this procedure. It is clear that the refraction as function of the apparent elevation (equation ((10))) overlaps perfectly with the constructed curve, and thus, the mutual consistency of the derived equations for the true and apparent refraction is demonstrated.

4 Comparison With Calculations

[19] The Starlink Project was a British astronomical computing project (hardware and software) which supplied general purpose data reduction software [Disney and Wallace, 1982]. The project was stopped in 2005, but the software continues to be developed at the Joint Astronomy Center (Hawaii) and is open source [Starlink, 2011]. The true and apparent refraction of radiowaves can be calculated using the atmospheric refraction routines of the Starlink positional astronomy library (SLA). For this, an ambient temperature of 288.15 K, a temperature lapse rate of −6.5 K km−1, and a surface pressure of 1013.25 hPa are chosen according to the Standard Atmosphere [NOAA, 1976]. The wavelength is set at 5.0 cm (note that the k-Model and thus our refraction equations are valid for all radiowave frequencies), and the relative humidity is set at 20, 60, or 100%. For the surface temperature and pressure prescribed by the Standard Atmosphere, the reference surface refractivity of N0=313 corresponds to a relative humidity of just below 60%, which is also the optimum value according to Huuskonen and Holleman [2007].

[20] Figure 3 shows a comparison of the refraction formulas according to the k-Model with the numerical results from the atmospheric refraction routines of the Starlink library. Analytical results for the 4/3-Model, the 5/4-Model, and the 6/5-Model are shown. For higher elevations (not shown), the apparent and true curves from the different k-Models overlap very well with those from Starlink with a 60% relative humidity. For elevations below 10° and decreasing toward zero, the apparent and true curves from the 4/3-Model migrate toward those from Starlink with a relative humidity of 100%, while those from the 5/4-Model continue to follow the curves from Starlink with a 60% relative humidity rather accurately. The 6/5-Model underestimates both the apparent and true refraction for 60% relative humidity. The propagation path through the atmosphere at these lower elevations is much longer, and therefore, the refraction is much more sensitive to the actual shape of the refractivity profile. The k-Model assumes a linear decrease of the refractivity, but an exponential decay as incorporated in the Starlink routines is much closer to the actual profile. At the surface, math formulaapproximates best the refractivity profile, but at higher altitudes, the gradient has decreased, and math formula appears to better approximate the “effective” gradient for the whole atmosphere. The difference at an apparent elevation of 1° between the refractivity from the k-Models and that from Starlink with 60% relative humidity is 0.067°, 0.017°, and −0.021° for the 4/3, 5/4, and 6/5-Models, respectively. From the figure, one can conclude that for the 5/4-Model, the deviations are comparable to a 10% variation of the relative humidity. The natural variability of the refractivity profile is often larger [Bech et al., 2007], and thus, the refraction formulas from the 5/4-Model are potentially useful.

Figure 3.

Comparison of the refraction calculated from the k-Model with different k factors and that from the Starlink positional astronomy library (SLA) for different relative humidities. The left panel shows the results for the apparent elevation, and the right panel shows those for the true elevation.

5 Comparison With Observations

[21] Atmospheric refraction of radiowaves from the Sun can be observed with the weather radars operated by the National Meteorological Services, in our case the Royal Netherlands Meteorological Institute (KNMI) and the Finnish Meteorological Institute (FMI). Huuskonen and Holleman [2007] introduced the detection of solar signals by weather radars and the use for determining the antenna pointing. The method was extended to monitoring of the receiving chain and the differential reflectivity [Holleman et al., 2010a, 2010b]. Weather radars collect reflectivity data as a function of antenna azimuth and elevation, and from these so-called polar volume data, the rainfall products are derived. For the solar analyses, incidental Sun signals are detected automatically in the polar volume data, and the power, elevation, azimuth, and date/time stamp (for calculation of the azimuth and elevation of the Sun) are stored. The solar signatures collected during a certain period are analyzed using a linear model in order to extract information on, e.g., the antenna pointing. Details of the detection and analysis procedures are given in the papers referenced above.

[22] The analysis of the solar data can be limited to a single elevation of the radar antenna. Using the same linear model as mentioned above, the elevation bias of the radar antenna with respect to the Sun can be extracted. By repeating this “single-elevation” analysis for all elevations, a radar observation of the atmospheric refraction angle as a function of the apparent elevation is obtained [Huuskonen and Holleman, 2007, Figure 5]. Figure 4 shows the results of single-elevation analyses for weather radars in De Bilt (Netherlands) and in Vimpeli (Finland). For both radars, data over a 3 month period were used: January–March 2010 for De Bilt and July–September 2010 for Vimpeli. In the figure, the elevation bias is plotted as a function of the elevation reading of the radar antenna (apparent elevation). In addition, the atmospheric refraction angles according to the different k-Models are plotted. The reading of the radar antennas may have a (mechanical) bias, and therefore, the radar data have been shifted such that the bias against the high-elevation refraction formula (equation ((16))) is zero for elevations above 10°. It is evident that the radar refraction data match very well to the 5/4-Model and reasonably well to the 4/3 and 6/5-Models. The root-mean-squares errors of the radar refraction data against the 5/4-Model are only 0.016° for De Bilt and 0.024° for Vimpeli, and they are clearly larger for the 4/3 and 6/5-Models: 0.024 and 0.026° for De Bilt and 0.038 and 0.032° for Vimpeli, respectively. It is concluded that also in comparison with the radar refraction observations, the 5/4-Model performs better than the “standard” 4/3-Model.

Figure 4.

Comparison of the radiowave refraction as a function of the apparent elevation as calculated from the k-Models with that extracted from observations of solar signals by C-band weather radars from the Netherlands (De Bilt, 52.10°N/5.18°E) and Finland (Vimpeli, 63.10°N/23.82°E). For Vimpeli, the observation at 45° is not shown, but it was used in the bias correction. NL, Netherlands; FIN, Finland.

6 Summary and Conclusions

[23] In this paper, we presented new analytical formulas describing the refraction angles of radiowaves from exoatmospheric sources. The formulas are derived from the Effective Earth's Radius Model (k-Model) which is commonly used in radar applications to describe the propagation of radiowaves through the atmosphere. The crucial assumption of this model is a linear decrease of the refractivity with increasing height implying that radiowaves propagate along rays with constant curvature. Using the k-Model, analytical expressions for the atmospheric refraction as a function of the apparent and true elevation are derived.

[24] The formulas for the true and apparent refraction have been checked for their mutual consistency. In the high-elevation limit, the refraction formulas become identical (as expected), and their functional form is in agreement with that found in textbooks. At lower elevations, the refraction formulas have been compared with the numerical results from the routines of the Starlink library, which is a well-established piece of open source software. Good agreement between the formulas based on the k-Model and the reference results from the Starlink library is found for k=5/4 and a fair agreement for k=4/3.

[25] Observations of the radiowave signals from the Sun by operational weather radars contain information about the atmospheric refraction. Refraction data from radars operated by the Netherlands and Finnish meteorological services have been compared with the apparent refraction according to the k-Model. The root-mean-squares error of the k-Model formulas against the refraction data from the weather radars is 0.016° for De Bilt and 0.024° for Vimpeli. Also, in the comparison with the radar refraction observations, the 5/4-Model performs better than the “standard” 4/3-Model.

[26] It is concluded that the new formulas for the refraction of exoatmospheric radiowaves derived from the k-Model, especially with k=5/4, are in good agreement with reference calculations for the Starlink library and radar refraction observations. These formulas are derived from a physical model that is widely used to describe the propagation of radiowaves in the atmosphere, and they are sufficiently accurate for many applications. As these easy-to-use formulas provide the true and apparent refraction in a closed analytical form, it is expected that they will prove useful in many cases.

Acknowledgments

[27] Hans Beekhuis (KNMI) is gratefully acknowledged for his assistance in the daily running of the software for extraction of solar signatures from the weather radar data and the provision of archived data.

Ancillary