P sin1 isin1 R2

sin sin i2

Two different expressions (Eqs. 6 and 7) can be used to obtain the connections between the impact parameters ps and p and the refraction angle *(p). From Eqs. (1) and (6) or (7) one can obtain by time differentiation of the phase excess p) and the central angle 0 the relationship connecting the Doppler frequency Fd of the RO signal with the impact parameters p and ps:

1di1 dt

dt d0 dt

1 dR1 1 dR2

(d1sR1)—1 + (d2sR2) —2 dt dt d0 dp, dps dt d1s = (R2 — ps2)1/2, d2s = R2 — ps2)

where d1s, d2s are the distances GQ and QL, correspondingly. After substitution of Eq. (10) into Eq. (9) one can obtain:

Fd = —(p — ps Ml i + dt) t — ( p2 — ps2)( 21 + fi2)j

Equations (9), (10), (11), and (12) are valid for general case of non-circular orbits of the GPS and LEO satellites. Usually in RO experiments the absolute values of difference p — ps and vertical velocities dR1j2/dt are far below (by factor 10—2 —10—4) the absolute magnitudes of the impact parameter p (or ps) and vertical velocity of ray perigee dps/dt. Under condition |(p — ps)dR1j2/dt| ^ ps|dps/dt| one can obtain from Eq. (12) a simple formula for estimation of the difference p — ps on the Doppler frequency

dts d2s) dt

The values ps, dps /dt, d1s, d2s can be delivered from orbital data and the phase delays $1>2(p) are the objects of measurements and given in the phase parts of radio-holograms at frequencies f1 and f2. Below we will consider new relationships, which connect the phase acceleration a = d2&(t)/dt2, Doppler frequency Fd(t), and refraction attenuation of radio waves X(t). Under conditions s

and by use of equation dp/dt - dps /dt & [X(t) - 1]dps/dt (Liou et al. 2006), one can obtain fromEq. (13):

dt dt2

Equations (15) and (16) relate the refraction attenuation and the excess phase path acceleration via a classical dynamics equation-type, first published in 2006 (Liou and Pavelyev 2006). The coefficient m, having the dimension s2/m, is a slowly changing function of vertical velocity dps/dt of the ray perigee T and distances di, d2. Equation (15) indicates equivalence between variations of the excess phase path acceleration a, derivative of the Doppler frequency Fd(t), and refraction attenuation X(t). Usually during RO experiments parameters m and dps/dt are known from orbital data because the location of the spherical symmetry center O and its projection on the line of sight—point Q are supposed to be known, and the distances GT d1 and TL d2 can be estimated as d1,2 — (2 — p2)1/2. Therefore, Eq. (15) gives the possibility to recalculate the phase acceleration a and/or Doppler frequency Fd to the refraction attenuation X p. This is useful for excluding systematic errors from phase and/or amplitude data. This is also useful for the estimation of absorption in the atmosphere. The refraction attenuation Xa is determined from amplitude data as a ratio of intensity of radio waves propagating through the atmosphere Ia (t) to its intensity in free space Is:

The experimental value Xa (dimensionless) is the product of the refractive and absorption contributions. However, the phase acceleration depends on the refraction effect only. This gives a possibility to determine the absorption in the atmosphere Y(t) as a ratio:

This possibility must be investigated in detail because in future satellite RO missions measurements of absorption effects due to water vapor and minor atmospheric gas constituents are planning and difficulties will consist of removing the refraction attenuation effect from amplitude data. Equations (18) indicate the feasible way to solve this problem. Also Eqs. (18) may be useful for estimation the conditions for communication in the Ku/K bands between two LEO satellites in a radio occultation geometry (Martini et al. 2006).

Phase variations of the RO signal as function of time $ (t ) at each GPS frequency f1 and f2 contain slowly and quickly changing parts $s(t) and $f (t):

After substitution of Eq. (19) in Eq. (15) one can obtain:

Equation (20) is valid under condition: |d2$s(t)/dt2| < |d2$f (t)/dt2|, which is fulfilled if the influence of irregularities in the atmosphere and ionosphere is far below the effect of the standard atmosphere (or ionosphere) near the ray perigee. The contribution of layered structures to phase excess variations in some cases may be considered as a quasi-periodical process, and the second derivative d2$(t)/dt2 may be presented in the form:

where vp is the vertical velocity of the RO ray near the perigee T. Parameter «2 depends on the vertical period Xv of the layered structure. After substitution of Eq. (21) in Eq. (15) one can obtain:

kv2=4n 2A2;

where kv is the vertical wave number of the vertical quasi-periodical structure in the atmosphere (ionosphere), d1, d2 are the distances along the ray GTL from the points L and G up to the ray perigee T, respectively, R0 is the distance GL. Equation (22) connects the high-frequency part of phase path excess variations $ f (t) with refraction attenuation changes X(t) - 1 of the RO signal. Note that the coefficient c0 in Eq. (22) does not depend on the vertical velocity of the ray perigee. Equations (22) allow one to recalculate refraction attenuation variations of GPS radio-holograms to phase path excess variations and vice versa. The form of phase path excess variations $ f (t) is similar to the form of intensity variations X(t) — 1. It follows from Eqs. (22) that variations of the intensity of the RO signal are proportional to phase path excess oscillations and inversely proportional to the second power of the vertical spatial period of the layered structure. One can estimate from Eqs. (22) the value of parameter c0 by comparison of amplitude and phase variations of the RO signal.

As follows from our analysis the practical algorithm for revealing the contribution of the lower ionosphere in the phase data can be described by:

where (t)) denotes the trend in the phase data. An alternative approach to find (t)) consists in averaging of the phase path excess over a sliding interval; the size of the sliding averaging interval must be long enough to account for the long-scale influence of the atmosphere or upper ionosphere.

Usually during RO experiments the parameter m (Eqs. 15 and 16) is a slowly changing function of time. Parameter dps/dt depends on the velocity components v, w of the GPS and LEO satellites, respectively. Components v, w are perpendicular to the straight line GL in the plane GOL. The components v, w are positive when oriented in direction to the point O and are negative in the opposite case. Components v, w are connected with the parameter dps/dt by:

d ps d1s

dt R0

Equations (15), (16), and (24) can be used to find the distance LT d2s from simultaneous observations of phase and intensity variations of radio waves. To achieve this, one can find m from Eq. (15) from the ratio of refraction attenuation changes to phase acceleration variations and then the distance d2S can be evaluated from the relationship d2s = 2mw

Equations (15) and (25) may be applied for the location of the tangent point T (or locally spherical symmetric layers) in the propagation medium.

3 Connection Between the Phase Acceleration and Intensity Variations: Experimental Validation

The phase acceleration a calculated as the second temporal derivative of the phase path excess and intensity variations X(t) - 1 at the frequency f1 are shown in Fig. 2, (curves 1 and 2, respectively) for CHAMP RO events 0136 (January 14, 2003) and 0023 (September 21, 2003). As seen in Fig. 2, there is a good correspondence between variations of the phase acceleration and the refraction attenuation of the RO signal. The coefficient m is different in RO events 0136 and 0023. The average ratio of the refraction attenuation and the phase acceleration m is about 1.0 s2/m in the 5-40 km height interval for RO event 0136 (Fig. 2, left panel) but for event 0023 (Fig. 2, right panel) is about 1.5 times greater. As follows from orbital data, the change in parameter m during these RO events is about 10%. As follows from

10 12 14 16 18 2(1 22 14 26 28 30 32 34 36 38 4(1 S 10 If 20 25 30 35 40

height |km| height [Ion]

Fig. 2 Phase acceleration a and refraction attenuation variations X — 1 at the first GPS frequency f1 (curves 1 and 2, respectively) for two CHAMP RO events: No. 0136 on January 14, 2003 (left panel) and No. 0023 on September 21, 2003 (rightpanel)

10 12 14 16 18 2(1 22 14 26 28 30 32 34 36 38 4(1 S 10 If 20 25 30 35 40

height |km| height [Ion]

Fig. 2 Phase acceleration a and refraction attenuation variations X — 1 at the first GPS frequency f1 (curves 1 and 2, respectively) for two CHAMP RO events: No. 0136 on January 14, 2003 (left panel) and No. 0023 on September 21, 2003 (rightpanel)

analysis of data shown in Fig. 2, Eqs. (15) and (16) are valid and they may allow one to locate the layered structures in the atmosphere and ionosphere, which are responsible for the variations of the intensity and phase acceleration of radio waves in the satellite-to-satellite links.

An example of the determination of the displacement D = d2s — d2 is shown in Fig. 3 for the FORMOSAT-3/COSMIC RO event 0006, April 23, 2006, 15 h 54 m 28 s LT, with geographical coordinates 9.5°S, 288.9°W. Curves 1 and 2 in Fig. 3, left panel, demonstrate good correspondence between the refraction attenuation Xp estimated from the phase acceleration a and parameter m by use of formula 1 — Xp = ma, and Xa evaluated from amplitude data, respectively, at the first GPS frequency f1.

The results of evaluation of the displacement D by use of Eqs. (15) and (25) are shown in Fig. 3, right panel. According to Fig. 3 (right panel) the displacement D is bound between ±50 km in the 10-25 km altitude interval and between ±100 km

tidsMlt |km| tidsMlt I kill |

Fig. 3 Left panel: comparison of the refraction attenuations Xa and Xp calculated from the phase (curve 1) and amplitude (curve 2) data. Right panel: displacement D of the tangent point T calculated by use of Eqs. (15) and (25)

tidsMlt |km| tidsMlt I kill |

Fig. 3 Left panel: comparison of the refraction attenuations Xa and Xp calculated from the phase (curve 1) and amplitude (curve 2) data. Right panel: displacement D of the tangent point T calculated by use of Eqs. (15) and (25)

in the 25-40 km height interval. It follows that the phase acceleration has the same importance for RO experiments as the well-known Doppler frequency.

4 Wave-Breaking Effect and Determination of Internal Wave Parameters

The amplitude channel of the radio-hologram can be used to obtain information on the vertical distribution of refractivity, temperature, and their vertical gradient (Pavelyev et al. 2002, 2003; Liou et al. 2002, 2003, 2006). Amplitude variations of the RO signal depend mainly on the high-frequency part of the derivative of the refraction angle on the impact parameter df (p)/dp. For obtaining the corresponding variations in the vertical gradient of refractivity dN(h)/dh the low frequency part in the function df (p)/dp corresponding to the low frequency noise in the amplitude data has been excluded by numerical filtration. The remaining high-frequency part of dfh(p)/dp was transformed by use of the Abel transformation technique (Liou et al. 2002), to find perturbations in the vertical gradient of refractivity dNh(h)/dh. This procedure does not need an optimization technique because (1) absence of the low frequency noise in the function dfh(p)/dp and (2) high sensitivity of the amplitude to high-frequency variations in the vertical gradient of refractivity (variations of the intensity of the RO signal are inversely proportional to the second power of the vertical spatial period of the layered structure).

Variations in the vertical gradient of refractivity retrieved from the RO amplitude data are shown in Fig. 4 for CHAMP RO events No. 0140 and No. 0001,

height [kin) height [km]

Fig. 4 Vertical gradient of refractivity perturbations at the first GPS frequency f in the 1030 km (left) and 30-65 km height interval (right). Abrupt changes in the amplitude and phase of refractivity perturbations are seen between the altitudes 38 km and 40 km (curves 1 and 2, right panel) and 45-50 km (curve 1, right panel). Curves 1 and 2 correspond to CHAMP RO events No. 0140 (02 h 35 m 34 s LT, 21.9°N, 172.5°W) and No. 0001 (02 h 09 m 51 s LT, 15.9°N, 330.0°W), January 23, 2003, respectively

height [kin) height [km]

Fig. 4 Vertical gradient of refractivity perturbations at the first GPS frequency f in the 1030 km (left) and 30-65 km height interval (right). Abrupt changes in the amplitude and phase of refractivity perturbations are seen between the altitudes 38 km and 40 km (curves 1 and 2, right panel) and 45-50 km (curve 1, right panel). Curves 1 and 2 correspond to CHAMP RO events No. 0140 (02 h 35 m 34 s LT, 21.9°N, 172.5°W) and No. 0001 (02 h 09 m 51 s LT, 15.9°N, 330.0°W), January 23, 2003, respectively

January 23, 2003 (curves 1 and 2, respectively). The wave structure is clearly seen in the perturbations of the vertical refractivity gradient in the 10-45 km interval (curves 1 and 2, left and right panels in Fig. 4). The vertical period of the wave structure grew from 0.8-1.0 km in the 10-25 km interval and from 2-4 km in the 30-40 km interval. Abrupt changes in the amplitude and phase of the refractivity perturbations are seen between the altitudes 38 km and 40 km (curves 1 and 2, right panel) and 45-50 km (curve 1, right panel). These changes may be connected with wave-breaking altitudes where the wave energy dissipates to eddies and turbulence. Previously the wave breaking effect has been observed from space during CRISTA experiments (Eckermann and Preusse 1999) at altitudes between 30 km and 40 km. There are indications for GW breaking at altitudes of about 50-60 km (Ern et al. 2006). Note that phenomena of wave breaking shown in Fig. 4 are the first direct observations by use of the GPS RO method.

5 Integral Behavior of Wave Activity in the Years 2001-2003

The wave activity in the atmosphere over a global scale can be characterized by the RO index j: probability of strong wave amplitudes exceeding the fixed level of the vertical gradient of refractivity q. This index has a global importance for the description of wave activity in the atmosphere as seen in Figs. 5 and 6 based on the analysis of CHAMP RO data for the time period 2001-2003. The magnitude j has been defined in this analysis as a ratio of numbers of intense wave amplitude greater than q > 0.6 N-units/km for the 12-16 km and with q > 0.24 N-units/km for the 18-26 km to total number of measurements over the Earth globe. The value j is marked in percents at vertical axis in Figs. 5 and 6. The smooth curves are obtained as approximation of the experimental data (broken lines) by least squares method. The data, shown in Figs. 5 and 6, are relevant to three year time interval September 2001-September 2003, with three data gaps: (1) from October 15, 2001, up to February 28, 2002; (2) from May 16, 2002, up to October 31, 2002; (3) from January 1 up to January 12, 2003. The seasonal and annual changes of wave activity are seen at all altitudes 12-26 km. For example, changes with period of about 12 months are seen at altitudes 12 km, 16 km, and 20-26 km (Figs. 5 and 6) for period January-September 2003. The phases of one-year oscillations at altitudes 12 km and 16 km are opposite to the phases of wave activity variations at the heights 20-26 km. The wave activity behavior at the altitudes 14 km and 18 km is different from that at the heights 12 km, 16 km, and 20-26 km. The wave activity at the heights 14 km and 18 km is increasing when the time goes to the end of the considered period—September 2003. This may be connected with tropopause effects. Changes in tropopause height, significant variations in refractivity gradients near the tropopause, and other phenomena can produce strong scintillations in amplitude variations of the RO signal. Analysis of these effects is a task of future investigation. At altitudes 12 km, 16 km, and 20-26 km the wave activity is gradually diminishing by 10-40% when time changes from September 2001 to September 2003. This diminishing may be connected with

Fig. 5 Changes in internal wave activity (averaged over the Earth globe) during 2001 and 2003 at different altitudes in the atmosphere: 12 km and 14 km (top left and right panels, respectively), 16 km and 18 km (bottom left and right panels, respectively)

changes in the intensity of meteorological processes, types of atmospheric circulations, and, very likely, with reduction in solar activity in the considered period of time. In that respect internal wave activity may be considered among the other important parameters described in various publications (e.g., Suh and Lim 2006, and references therein), which characterize effects of solar activity on the Earth's atmosphere.

6 Conclusion

Comparative theoretical and experimental analysis of phase and amplitude variations of GPS radio-holograms discovered new relationships, which relate the refraction attenuation and the excess phase path acceleration via a classical dynamics equation-type. The advantages of the introduced relationship consist in (1) a

Fig. 6 Changes in internal wave activity (averaged over the Earth globe) during 2001 and 2003 at different altitudes in the atmosphere: 20 km and 22 km (top left and right panels, respectively), 24 km and 26 km (bottom left and right panels, respectively)

possibility to separate the layered structure and turbulence contributions to the RO signal; (2) a possibility to estimate the absorption in the atmosphere by dividing refraction attenuations found from amplitude and phase data; (3) a possibility to locate the tangent point in the atmosphere with accuracy in the distance from the standard position of about ±100 km. The suggested method has a general importance (for example, it may be applied for analysis of amplitude and phase variations in the trans-ionospheric satellite-to-Earth links). We showed also that amplitude variations of GPS occultation signals are very sensitive sensors to internal waves in the atmosphere. The sensitivity of the amplitude method is inversely proportional to the square of the vertical period of the internal wave, indicating high sensitivity of amplitude data to wave structures with small vertical periods in the 0.8-4 km interval. By use of the amplitude data analysis the internal wave breaking phenomenon has been observed at the first time in RO practice at altitudes between 38 km and 45 km.

The amplitude GPS occultation method presents a possibility to obtain the geographical distribution and seasonal dependence of atmospheric wave activity with global coverage. At the altitudes 12 km, 16 km, and 20-26 km the wave activity is gradually diminishing by 10-40% from September 2001 to September 2003. This diminishing may be connected with changes in the intensity of meteorological processes, types of atmospheric circulations, and, very likely, with reduction in solar activity.

Acknowledgements We are grateful to the GeoForschungZentrum Potsdam for delivering the CHAMP RO data and to UCAR/CDAAC (Boulder, CO, USA) for the provision of FORMOSAT-3/ COSMIC data. We are grateful to the National Science Council of Taiwan, R.O.C., for financial support under the grants NSC 92-2811-M008-001, NSC 91-2111-M008-029, and the Office of Naval Research (ONR) of USA under grant N00014-00-0528. Work has been partly supported by the Russian Fund of Basic Research, grant No. 06-02-17071. In addition, assistance was provided by the Russian Academy of Sciences, program OFN-16 and OFN-17.

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