Application of GPS radio occultation method for observation of the internal waves in the atmosphere

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111,, doi: /2005jd005823, 2006 Application of GPS radio occultation method for observation of the internal waves in the atmosphere Y. A. Liou, 1,2 A. G. Pavelyev, 3 J. Wickert, 4 S. F. Liu, 5 A. A. Pavelyev, 3 T. Schmidt, 4 and K. Igarashi 6 Received 1 February 2005; revised 10 October 2005; accepted 20 December 2005; published 24 March [1] In this study, we show that the amplitude radio occultation (RO) method, which employs high-precision global positioning system (GPS) signals, allows one to determine the vertical gradients of refractivity and monitor wave structures in the atmosphere on a global scale at altitudes ranging from 10 to 40 km. We show that the sensitivity of the RO amplitude data to the wave structures in the atmosphere with vertical periods from 0.8 to 4 km exceeds one of the RO phase data by a factor of order 10. As an example of this approach, analytical results of the Challenging Minisatellite Payload s (CHAMP) RO events are presented. Wave clusters were found in the amplitude variations of the RO signals with interior vertical periods from 0.8 to 4 km in the tropopause and lower stratosphere within the heights of km (low latitudes) to km (moderate latitudes). We demonstrate that the amplitude variations can be considered as a radioholographic image of the wave structures in the atmosphere. For internal gravity waves (GW), we show that the GW portrait, which consists of the altitude dependence of the GW phase, amplitude and vertical spatial frequency, can be retrieved from the amplitude variations of the RO signal. The GW dispersion and polarization relationships allow one to estimate the vertical profile of the horizontal wind perturbations, its gradient and the GW intrinsic phase speed. In general, when the origin and type of internal waves are not known, the height dependence of the vertical gradient of refractivity can be applied for monitoring the seasonal and geographical distributions of wave activities at different levels in the atmosphere. Citation: Liou, Y. A., A. G. Pavelyev, J. Wickert, S. F. Liu, A. A. Pavelyev, T. Schmidt, and K. Igarashi (2006), Application of GPS radio occultation method for observation of the internal waves in the atmosphere, J. Geophys. Res., 111,, doi: /2005jd Introduction [2] The internal gravity waves (GW) in the atmosphere have been observed and modeled for many years. Smallscale GW with wavelengths of a few kilometers and periods near 10 min are generated in the planetary boundary layer [Nagpal, 1979]. The medium-scale GW with horizontal wavelengths of km, vertical wavelengths of 1 10 km, and periods of 10 min to 1 hour are mainly produced by meteorological phenomena. This is due to the weather activities (fronts, cyclones, cumulonimbus convection, thunderstorms) and wind flows over 1 Center for Space and Remote Sensing Research, National Central University, Chung-Li, Taiwan. 2 Also at National Space Organization, Shin-Chu, Taiwan. 3 Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Moscow, Russia. 4 GeoForschungsZentrum Potsdam, Telegrafenberg, Potsdam, Germany. 5 Department of Industrial Design, National Cheng Kung University, Tainan, Taiwan. 6 National Institute of Information and Communications Technology, Tokyo, Japan. Copyright 2006 by the American Geophysical Union /06/2005JD irregular topography (mountain waves), which plays a decisive role in transporting energy and momentum. The atmospheric circulation and temperature regime is subsequently affected by GW [Fritts and Alexander, 2003], as they bring forth turbulence and mixing. The GW s breaking and dissipation plays an important role in the dynamics and energetics of the mesosphere [Ebel, 1984]. Dynamical simulation studies of the stratosphere and mesosphere require a broad spectrum of knowledge, which includes the GW s effects in the empirical models or analytic formulations of a realistic description of the climate changes and variations in the atmospheric, stratospheric and mesospheric global-scale flows [Pruse et al., 1996]. [3] Radiosonde and rocketsonde GW s measurements, balloon soundings, radar observations and lidar studies have been limited to ground-based sites [Eckermann et al., 1995; Sica and Russell, 1999] that are mainly situated over specific areas of the Northern Hemisphere. Recently a handful of high-resolution stratospheric satellite instruments have been tasked to detect GW [Eckermann and Preusse, 1999]. However, the observation time is quite limited. This leads to the problem of insufficient data in establishing wave climatology on a global scale, despite the good results 1of14

2 Figure 1. Key physical and geometric parameters of the RO measurements. In the case of spherical symmetry only one tangent point T exists, where the RO signal trajectory is perpendicular to the refractivity gradient. The length of coherent interaction of RO signal with medium L c connects the bending angle x(p) with the vertical gradient of refractivity dn(h)/dh at the tangent point T. from many of the ground-based and space-borne instruments [Steiner and Kirchengast, 2000]. [4] The current more advanced satellite technology incorporates high-precision GPS radio signals at two frequencies F1 = and MHz for internal waves investigation. The small satellite (e.g., MicroLab-1, launched into a near polar circular orbit of about 750 km altitude) carries a laptop sized GPS receiver to provide remote sensing of the atmosphere and ionosphere by GPS limb sounding method [Ware et al., 1996; Feng and Herman, 1999]. The advantages of space-borne observations of the Earth s atmosphere via GPS radio signals propagating through the mesosphere and stratosphere lies in the fact that the RO technique can recover atmospheric profiles above oceans as well as above land with high vertical resolution (<0.4 km) and accuracy (<1 K in temperature within the upper troposphere and lower stratosphere) [Anthes et al., 2000; Kuo et al., 2000]. The vertical wavelength of internal waves usually is far below the horizontal wavelength. Therefore the RO method is well suited for observation of the internal waves, because its vertical resolution greatly exceeds the horizontal spatial resolution. More recently, new satellites missions such as CHAMP and SAC-C have been launched for global RO control on the meteorological processes in the atmosphere and plasmas phenomena in the ionosphere [Reigber et al., 2002; Hajj et al., 2004]. Radio holographic amplitude technology has been proposed and validated to find the vertical gradients of the refractivity and temperature in the atmosphere with high accuracy and spatial resolution, which is important for GW investigation [Pavelyev et al., 2002a, 2002b; Liou et al., 2002]. [5] The RO technology of atmospheric studies consists of analyzing the phase and amplitude of the radio waves after propagating through the atmosphere. Analysis of the temperature variations found from the RO phase data produces an opportunity to measure the GW s statistical characteristics in the atmosphere [Steiner and Kirchengast, 2000; Tsuda et al., 2000; Tsuda and Hocke, 2002]. The amplitude of the RO signals presents new potential and capability for the research and simultaneous observation of the atmospheric and ionospheric waves [Sokolovskiy, 2000; Igarashi et al., 2000, 2001; Pavelyev et al., 2002a, 2002b, 2003a, 2003b; Liou et al., 2002, 2003, 2005a, 2005b]. [6] The aim of this paper is to describe possibilities in the application of the amplitude GPS RO method for studies of the internal waves in the atmosphere with a global coverage. In section 2, we find the intrinsic resolution of the GPS RO method to wave structures in the atmosphere. In section 3, we reveal the connection between the phase and amplitude variations of the RO signal, and demonstrate the high sensitivity of the amplitude variations of the RO signal to the wave structures in contrast to the phase RO data. In section 4, a method of retrieving the vertical gradient of refractivity from amplitude variations is described. In section 5, an example is given regarding the direct observation of the quasi-regular internal waves in the atmosphere, along with the application of the Hilbert transform, for revealing the wind perturbations and reconstruction of the GW portrait. In section 6, we introduce a special parameter amplitude of the vertical gradient of refractivity as a function of height to provide a preliminary analysis of the seasonal and geographical distributions of the internal wave activity at different levels in the atmosphere. 2. Intrinsic Resolution of GPS RO Method [7] The scheme of the RO experiment is shown in Figure 1. Point O is the center of the spherical symmetry of the Earth s neutral atmosphere. Radio waves emitted by the GPS satellite (point G) arrive at a receiver onboard the LEO satellite (point L) along the ray GTL, where T is the tangent point in the atmosphere. [8] At point T, the ray s distance from the Earth s surface h is minimal and the gradient of refractivity N(h) is perpendicular to the ray trajectory GTL (Figure 1). The projection of point T on the Earth s surface determines the coordinates of the RO region: latitude j and longitude l l. Record of the RO signal E(t) along the LEO trajectory is the radio hologram s envelopes at two frequencies F1 and F2, which contain the amplitudes A 1 (t) and A 2 (t), along with the phases of the radio field as functions of time. The sampling rate of the measurements (50 Hz) has been chosen to achieve the high vertical resolution. The time interval for RO measurements t depends on the orientation between the vertical direction at point T and the occultation beam path. The time t is minimal 30 s, when the orbital planes of the LEO and GPS satellites are parallel. Thus RO experiments record in a practical and simultaneous manner the impact of the internal waves on the RO signal, because the internal waves frequencies are usually well below 1/t 0.03 s 1. [9] Below we will consider the intrinsic sensitivity of the RO method. The intrinsic sensitivity characterizes the extreme value of the spatial resolution, which may be achieved by optimized methods of the RO data analysis (e.g., radio-holographic methods introduced earlier by Hocke et al. [1999], Igarashi et al. [2000], Pavelyev et al. [2002a, 2002b, 2004], Gorbunov [2002], and Jensen et al. [2003]). The intrinsic sensitivity of the RO signal to wave structures depends on the length of the coherent interaction L c between the RO signal and the atmosphere [Igarashi et al., 2001]. The value L c depends on the vertical profile of the refraction index n(r). To find this dependence we note that at point T the direction angle a of the ray trajectory 2of14

3 relative to the local horizon is zero, where radio waves propagate nearly along the layer [Igarashi et al., 2001]. As the ray inside the essential zone is nearly perpendicular to the vertical gradient of the refractivity dn(r)/dr it follows that L c = jx(p)n(r)/[dn(r)/dr]j, where x(p) is the refraction angle, p is the ray impact parameter, N(r) is the refractivity of the atmosphere at point T (Figure 1), N(r)= n(r) 1. By using analytic expressions for x(p) and N(r) designed for the exponential model of the atmosphere [Pavelyev et al., 1996; Pavelyev, 1998], one can obtain a rigorous expression for L c : L c =(2prH) 1/2 /M d, where H is the scale height of the standard atmosphere, r is the distance from the tangent point T to the center of spherical symmetry O (Figure 1), M d = [dm(r)/dr]/n(r), M(r) = n(r)r/a is the modified refraction index of the atmosphere, a is the Earth s radius. For the case of atmospheric waves, the physical optics approximation [Pavelyev et al., 1996; Pavelyev, 1998] gives the next relationship for L c : L c =(rl v ) 1/2 /M d.ifr = 6400 km, one obtains for H = 8 km, L v = 1 km, L c = 580 km, and L c = 80 km, respectively. Therefore, for internal waves with vertical period L v in the interval 1 16 km, the length L c changes in the range km. Propagation of radio waves along the essential area introduces an averaging effect within the vertical size of the Fresnel zone, owning a radius of Dh = (lxl c /4) 1/2 = [lx/(4m d )] 1/2 (rl v ) 1/4, where X is the refraction attenuation. It should be noted that the refraction attenuation X characterizes the decreasing/ increasing of the intensity of radio waves, because of the influence of the refraction effect in the atmosphere [e.g., Pavelyev and Kucherjavenkov, 1978; Pavelyev et al., 2004]. For L v = 1 km, l = 20 cm, X = M d = 1, Dh = 64 m. This Dh value characterizes the intrinsic vertical resolution of the GPS RO method. The resolution of Dh depends on the vertical period of the internal wave L v, and changes from 64 to 128 m when L v varies in the range 1 16 km. For the standard atmosphere with effective height H = 8 km the vertical resolution is equal to Dh = 170 m. One can estimate the lower diffractive limit of the length of the essential area L c and the value a at its boundary. The angle a is connected with L c : a = 0.5 L c /r. For coherent interaction of the wave with the layered structure at T, the difference a x(a) =a(1 dx/da) = am d must not exceed the angular size of the Fresnel zone at point T (Figure 1). From this condition one can obtain inequality a [lx/(2l c )] 1/2 /M d. At the boundary of the essential area the equation a = 0.5 L c /r = [lx/(2l c )] 1/2 /M d is valid, and one can find L c and a: a = 2 1/3 (lx/r) 1/3 M d 2/3, L c = 2 2/3 r 2/3 l 1/3 X 1/3 M d 2/3 and then Dh = L c [a x(a)]/2, Dh =2 2/3 r 1/3 l 2/3 X 2/3 M d 1/3. In the case X =1,M d 1, one obtains Dh =2 2/3 r 1/3 l 2/3. For X =1,l = 20 cm, M d 1, r = 6400 km, L c 32 km, Dh = 40 m. Thus the GPS RO method is sensitive to the wave structures with horizontal and vertical wavelength greater than 32 km and 40 m, respectively. It follows from this analysis that the lower diffraction limit of the intrinsic horizontal resolution of the RO method depends on the wavelength of radio waves l, the radius of the layered structure s curvature in the atmosphere r, and the vertical gradient of the modified refraction index M d,. When M d 0, the length of the coherent interaction can be high, which corresponds to the occurrences of wave-guide propagations, strong diffraction effects and low horizontal resolution. In usual conditions, the physical optics formulas for the intrinsic horizontal resolution L c =(rl v ) 1/2 /M d and vertical resolution Dh =[lx/(4m d )] 1/2 (rl v ) 1/4 are valid, which indicate dependence of L c and Dh on the vertical wavelength L v of internal wave as L v 1/2 and L v 1/4, respectively. One can obtain expression for the critical value of the vertical wavelength L vc when the coherent length L c found from the physical optics and diffraction limit are coinciding: L vc =2 4/3 r 1/3 l 2/3 X 2/3 M d 2/3 160 m for X = M d =1,l = 20 cm, r = 6400 km. It is important that this analysis renders the local character of the RO method clearly, and shows the boundaries for the physically attainable resolutions in the vertical and horizontal directions, which can be achieved by optimized methods of GPS RO data analysis. 3. Connection Between the Amplitude and Phase Variations in GPS RO Signal [10] As shown earlier, qualitatively [Igarashi et al., 2001; Pavelyev et al., 2003a, 2003b, 2004] the amplitude and phase of RO signals have different intrinsic sensitivity for observation of the internal waves in the atmosphere. The quantitative estimation can be obtained using the equations for the phase path excess and refraction attenuation of radio waves [Pavelyev et al., 2004] FðpÞ ¼ Lp ð ÞþkðpÞ R 0 Lp ð Þ ¼ d 1 þ d 2 þ pxðpþ xðpþ ¼ dkðpþ=dp where R 0 is the distance GDL, and L(p) is the distance GABL, which consists of two short lengths d 1 (GA), d 2 (BL), and arc AB (Figure 1). [11] The relationships (1) (3) connect the phase path excess F(p) with the refraction angle x(p), impact parameter p, and the main refractivity part of the phase path excess k(p). They are also valid for each physical ray in a multipath situation. The refraction attenuation X(p), which characterizes the attenuation of the intensity of radio waves due to the refraction effect in the atmosphere, can be described by equation [Pavelyev and Kucherjavenkov, 1978; Pavelyev et al., 2004] XðpÞ ¼ pp 1 R 0 jd 1 d 2 d 2 kðpþ=dp 2 þ d 1 þ d 2 j 1 s where p s is the impact parameter corresponding to the propagation in free space along the straight line GL (Figure 1). The refraction attenuation X(p) (4) depends mainly on the second derivative of the main refractivity part of the phase path excess k(p). Consequently, the amplitude variations can be more sensitive to wave structures in the atmosphere than the phase path excess. The sensitivity of the refraction attenuation X(p) to atmospheric wave structures can be characterized by factor F in comparison with the sensitivity of the phase path excess F. The factor F is equal to F =(2p/L v ) 2 D F d 1 d 2 /(D X R 0 ), where D F, D X are the noise levels in the phase path excess and ð1þ ð2þ ð3þ ð4þ 3of14

4 refraction attenuation measurements. Factor F is inversely proportional to the square of the vertical period L v, indicating high sensitivity of the amplitude data to wave structures with small vertical periods. For estimation of the magnitude F, we use the current data on the phase and amplitude noise relevant to the CHAMP RO experiment [Wickert et al., 2001]: D F ±1 mm at a distance R 0 30,000 km and D X ± %. As a result, we obtain value F in the interval for L = 1 km, d 1 = 3000 km, d 2 R 0. Therefore the sensitivity of the amplitude data to wave structures with short vertical periods 1 3 km is far beyond the corresponding value of the phase data. It should be noted that the realization of high sensitivity of the amplitude method requires the high sampling frequency of GPS RO signals (about of 50 Hz for CHAMP and GPS/MET RO missions [Wickert et al., 2001]). The sampling frequency of the planned FORMOSAT-3 s RO mission is about 100 Hz. This value is more appropriate for application of the amplitude radio-holographic method when analyzing the internal wave activity. 4. Connection Between Amplitude Variations in GPS RO Signal and Vertical Gradients of Refractivity in the Atmosphere [12] The amplitude variations can be used for obtaining the height distribution of the vertical gradient of the refractivity. The solution of the inverse RO problem for the amplitude channel of the RO signal has been described earlier [Kalashnikov et al., 1986]. For the case of circular orbits of the LEO and GPS satellites, this solution can be obtained via equations dp=dt ¼ XðÞdp t s =dt; 1 Z1 p p 0 ¼ Xtp ½ ð s ÞŠdp s p s0 p s0 where p s0, p 0 are the impact parameters corresponding to the rays GDL and GTL at the time point t = t 0. Equations (5), (6) can be used to find the vertical distribution of the vertical gradient of refractivity dn(h)/dh [Liou et al., 2002]: dnðþ=dh h ¼ n 2 ðþj h ðpþ= fp½1 þ JðpÞŠg; nh ðþ¼1þnh ðþ; 1 Z1 JðpÞ ¼ 1=p d 2 xðþ=dx x 2 x 2 p 2 p p 1=2dx dxðpþ=dp ¼ d1 1 þ d2 1 ðxðpþ 1Þ=XðpÞ ð5þ ð6þ (7) ð8þ ð9þ [13] According to (7) (9), the amplitude information can be used to retrieve the vertical gradient of the temperature profile: ½dTðÞ=dh h Š=TðhÞ ¼ ½Nh ðþš 1 dnðhþ=dh T x =Th ðþ; T x 34:16 K=km; ð10þ ð11þ where T(h) is the temperature of the dry atmosphere [K]. Equation (10) connects the vertical gradients of the refractivity and dry temperature T(h). At the height above 10 km, equation (10) may be used to find the vertical gradient of the temperature profile if the refractivity gradient is known. Equations (5) (10) can be applied for estimating the vertical gradients of refractivity and temperature, and parameters of the wave structures in the atmosphere [Liou et al., 2005a, 2005b]. [14] Examples of the amplitude variations in the upper troposphere and lower stratosphere are shown in Figure 2 for six CHAMP RO experiments provided on 23 January The choice of these events is due to their different locations. Two events fall into tropical regions (event 0140, 02 h 35 m 34 s LT, 21.9 N W (curve 1), and event 001, 02 h 09 m 51 s LT, 15.9 N W (curve 2) in Figure 2). The first event 0140 corresponds to the north Pacific tropical area, the second event 0001 relates to the Sahara desert. The third and fourth events (event 0082, 06 h 46 m 16 s LT, 75.2 N 21.1 W (curve 3), and event 0196, 05 h 17 m 59 s LT, 76.6 S W (curve 4) in Figure 2) correspond to the North and South Polar regions. The fifth and sixth events (event 0173, 14 h 58 m 06 s LT, 45.5 N 40.6 W (curve 5), and event 0158, 15 h 25 m 34 s LT, 35.0 N 11.0 W (curve 6) in Figure 2) refer to the moderate latitudes in the Northern Hemisphere. Wave structures having various intensities are seen for each event (Figure 2). The most significant are the amplitude variations in the tropical regions (curves 1 and 2 in Figure 2). In the Polar regions the intensity of amplitude variations are smaller (curves 3 and 4 in Figure 2). Amplitude variations in the moderate latitude are intermediate to the tropical and Polar areas (curves 5 and 6 in Figure 2). The amplitude variations can directly provide information on the form and spatial frequencies associated with the wave structures in the atmosphere. It is important, however, to establish the origin of the amplitude variations shown in Figure 2. The amplitude variations shown in Figure 2 can correspond to the wave propagating in the neutral gas or plasma, depending upon where the tangent point T is located: in the atmosphere or ionosphere. The radio-holographic method introduced earlier [e.g., Pavelyev et al., 2004] allows one to determine the displacement d of the tangent point T from its normal location corresponding to the spherical symmetric atmosphere. The displacement d can be caused by the horizontal gradient of the refractivity in the atmosphere or ionosphere [Wickert et al., 2004]. The evaluation results of the distance d are shown in Figure 3 for CHAMP RO events 0140, 001, 0082, 0196, 0173, 0158 (curves 1 6, respectively). As follows from Figure 3, the displacement of point T from its normal position is small ±50 km in the km altitude interval. At the altitudes above 25 km, there may be essential displacements of 4of14

5 Figure 2. Height dependence of the CHAMP RO amplitudes at frequency F1 = for CHAMP RO events 0140, 02 h 35 m 34 s LT, 21.9 N W (curve 1); 0001, 02 h 09 m 51 s LT, 15.9 N W (curve 2); 0082, 06 h 46 m 16 s LT, 75.2 N 21.1 W (curve 3); 0196, 05 h 17 m 59 s LT, 76.6 N W (curve 4); 0173, 14 h 58 m 06 s LT, 45.5 N 40.6 W (curve 5); and 0158, 15 h 25 m 34 s LT, 35.0 N 11.0 W (curve 6) (23 January 2003). point T along the radio occultation ray trajectory that may correspond to the influence of wave structures in the ionosphere. [15] Analysis of the radio holographic amplitude image of the atmosphere can be produced by using the inversion theory developed earlier [Kalashnikov et al., 1986]. According to this theory, the amplitude channel of the radio hologram can be used separately from the phase channel to obtain information on the vertical distribution of the refractivity, temperature, and its vertical gradient [Liou et al., 2002] in the upper troposphere and stratosphere. Additional information may be retrieved (in the case of GW) on the quantitative characteristics of the wave processes: vertical distribution of the wind speed perturbation and its gradient. 5. Application of the Hilbert Transform for Analysis of the Internal Waves in the Atmosphere [16] The foregoing indicates the importance of the GPS RO amplitude method in investigating the wave structures in the atmosphere with global coverage. Below we will consider in detail the method for retrieving the GW parameters from measurements of the amplitude variations of the GPS RO signal by using the CHAMP RO data. For example, we select amplitude variations observed in the RO event 0001 shown in Figure 1 (curve 2). The location of the tangent point T in this event does not have significant variations in the horizontal direction (as seen in Figure 2, curve 2). As follows, the amplitude variations observed in RO event 0001 are connected with wave structures in the neutral gas. [17] The amplitude variations have been recalculated to the perturbations in the vertical refractivity gradient of the dn(h)/dh. Variations of the vertical gradient of refractivity retrieved from the RO amplitude data are shown in Figures 4 and 5 for the CHAMP RO event 0001, 23 January The wave structure is clearly witnessed in the perturbations of the vertical refractivity gradient in the 8 40 km interval (curve 2 in Figures 4 and 5). The vertical period of the wave structure grew from km in the 8 25 km interval to 4 km in the km interval. A sharp change of the amplitude and phase of internal wave is seen at a height of about 40 km. [18] The perturbations in the vertical refractivity gradient can be recalculated to variations of the temperature and its gradient. The variations in temperature and its vertical gradient retrieved from the amplitude data are indicated for the CHAMP RO event 0001 in Figure 6, (curves 1, 2). From the analysis of Figure 6, the vertical temperature gradient increases with the height from ±2 K/km (8 20 km interval) to ±8 K/km (30 45 km interval). The temperature fluctuates from ±0.2 ±1.0 K (8 20 km interval) to ±4 K (30 40 km interval). The vertical period of the vertical temperature gradient increases from 0.8 km (8 20 km interval) to 4 km (35 40 km interval). [19] The most possible source of the observed waves in the altitude distributions of the amplitude with vertical period km is the GW activity [Steiner and Kirchengast, 2000; Tsuda and Hocke, 2002]. If the observed wave structures are caused by GW activity, the temperature variations can be related with the horizontal wind perturbations by means of the GW polarization equations. The changes in the vertical period of temperature variations Figure 3. Displacements of the tangent point T along the ray path GTL caused by wave structures in the atmosphere. The positive values correspond to displacement in direction to the point G, the negative values are related to displacement in direction to point T (Figure 1). Curves 1 6 correspond to RO events presented by curves 1 6 in Figure 2. 5of14

6 where w b 2 = g/t b G, G b /@h /km, g is the gravity acceleration, and T b is the background temperature. One can obtain a connection between the vertical gradients dv(h)/dh and dt(h)/dh from (13) dvðþ=dh h ¼ dre½ig= ðt b w b Þth ðþš=dh Re½ig= ðt b w b ÞdtðÞ=dhŠ h ð14þ Equation (14) is valid under the assumption that T b (h) and w b (h) are slowly changing at the vertical scales l h. The functions T b and w b in equation (14) may be estimated using the average values of atmospheric parameters expected in the RO region. To find the function dv(h)/dh from the second equation (14), one can implement the Hilbert transform [Rabiner and Gold, 1978]. Application of the Hilbert transform gives an analytic presentation of the real signal dt(h)/dh: dtðhþ=dh ¼ Refa t ðhþexp½if t ðþ h Šg ð15þ where a t (h) and F t (h) (real functions) are the amplitude and phase of the temperature vertical gradient. The function dv(h)/dh can further be restored from (15) by employing the Hilbert transform to the experimental data. [21] The results of restoration of the vertical profiles of the horizontal wind perturbations and its gradient are shown in Figure 7. The amplitude of the horizontal wind Figure 4. Comparison of the amplitude data (curve 1) and perturbations in the vertical gradient of the refractivity (curve 2) in the 5 30 km height interval. The increasing vertical period is observed in the km interval. can be connected with variations in the intrinsic phase speed of GW using the GW dispersion relationships. [20] The dispersion and polarization relationships will be used for analysis, which are valid for mediumfrequency cases, when the intrinsic frequency of the GW is greater than the inertial frequency f, but is well below the buoyancy frequency w b. The GW dispersion relation has the form [Fritts and Alexander, 2003; Eckermann et al., 1995]: l v ¼ 2pv i =w b ; v i ¼jc U cos jj ð12þ where l v is the vertical wavelength of the GW, U is the background wind speed, c denotes the ground-based GW horizontal phase speed, and j is the azimuth angle between the background wind and the GW propagation vectors. Equation (12) connects the vertical wavelength l v with the intrinsic phase speed of the GW v i, which can be measured by an observer moving with the background wind velocity [Eckermann et al., 1995]. A GW polarization relation was published previously [Lindzen, 1981]. This relation links the complex amplitude of the temperature variation, t(h) with the horizontal wind perturbations v(h) in GW: v ¼ Re½ig= ðt b w b Þth ðþš ð13þ Figure 5. Amplitude variations (curve 1) and changes of the retrieved vertical gradient of refractivity (curve 2). A sharp abrupt change of the phase and amplitude of internal wave is seen at a level of about 40 km (curve 2). 6of14

7 Figure 6. Vertical temperature gradient and temperature perturbations found from the vertical gradient of refractivity variations. perturbations changes from ±0.5 m/s to ±5.0 m/s, when the height increases from 10 km to 40 km (curve 1 in Figure 7). The corresponding values of the vertical gradient of the horizontal wind perturbations changes from ±2 ms 1 km 1 to ±8 ms 1 km 1, when the height increases from 8 10 km to 40 km. [22] In Figure 8, the portrait of the internal wave is shown for CHAMP RO event The portrait includes the vertical profile of the phase (curve 1) and amplitude (curve 2) of the internal wave. It is important to note that Figure 8. Vertical profiles of the phase (curve 1) and amplitude (curve 2) of internal wave. the portrait can be obtained from the amplitude variations of the RO signal without any assumption on the nature or origin of the internal wave. The phase of the internal wave (curve 1) indicates a decrease in the vertical spatial frequency of the internal wave in the km interval. The sharp changes in the phase conform to the height, where the amplitude of the internal wave is below the noise level and the coherence property of internal waves has disappeared. These regions can correspond to the boundaries of wave breaking altitudes, where the energy of the internal wave is transmitted to the turbulent structures within the stratosphere (e.g., the height interval between 4145 km, Figure 9). If the observed amplitude variations are caused by propagating GW, one can estimate the intrinsic phase speed of GW v i after differentiating the Figure 7. Horizontal wind perturbations found from amplitude variations of RO signal. Curve 1 describes the horizontal wind variations caused by propagation of GW, and curve 2 corresponds to the vertical gradient of the horizontal wind variations. Figure 9. of GW. Altitude dependence of the intrinsic phase speed 7of14

8 GW phase F(h) via the relationship (12). The results of the intrinsic phase speed evaluation are shown in Figure 9. As seen in Figure 9, the value v i (h) changes in the 2 16 m/s interval. The sharp variation of the intrinsic phase speed near the altitude of about 40 km can be related to the boundary of a GW breaking area. 6. Geographical and Altitude Distributions of Strong Atmospheric Internal Wave Events [23] It is important to point out that the amplitude variations of the GPS RO signal are sensitive to different types of internal waves (GW, Kelvin waves and other types having small vertical periods). Below our analysis is valid for all types of the internal waves. We will characterize the internal waves activity at different heights in the atmosphere by the amplitude of the Hilbert Transform q(h) [N-units/km], which is associated with the perturbation portion of the refractivity s vertical gradient. This parameter is useful for observing the distribution of the internal wave activity on a global scale. [24] Amplitude variations of the CHAMP RO signals provide an important picture of the wave activity in the km interval with global coverage. This is illustrated in Figure 10 and Figure 11 for two periods: 21 January to 3 February (Figures 10 and 11, left) and July 2003 (Figures 10 and 11, right). The geographical distribution of the wave structures with amplitudes greater than 0.6 N-units/km is demonstrated in Figure 10 for the km height interval. The general structure of the wave activity reveals interesting features, which is dependent on the season. Essentially, the wave activity is uniformly distributed at a height level of 12 km in the equatorial areas. In the North and South Polar regions, there is a strong seasonal dependence with the maximum activity during the local summer. At the heights of 14 km and 16 km, the wave activity is concentrated in the moderate latitudes in both the Northern and Southern hemispheres. At the 18 km level, most of the internal wave s energy is concentrated near the equator. The seasonal dependence is evident for some regions: Siberia has a low wave activity in the winter and a high wave activity in the summer at the 14 km height. [25] The wave activity above the North Atlantic region is weak during the summer period at the 14 km level. This may be connected to the weak meteorological activity that is dominant in the troposphere within this region. In contrast, the strong activity is seen in the northeastern part of the Asia continent and above the North America within the same level. [26] The geographical distribution of the wave activity in the stratosphere between 20 and 26 km is given in Figure 11 for the same time periods as in Figure 10. The wave activity in the km height interval is concentrated mainly in the equatorial region. The intensity of the internal waves in the equatorial areas rapidly diminishes, when the height increases. This is evident from the comparison of the absolute intensity shown for levels 20 and 22 km (Figure 11, top) and for the one shown below in Figure 11. This is connected to the decreasing air density with the height. In comparison with Figure 10, the seasonal dependence is not so evident in Figure 11. The longitudinal dependence of the internal wave intensity is essentially absent along the strong wave phenomena that are above the North Africa and Persian Gulf (20 22 km height interval). It is important that in some locations in the Northern and Southern Polar regions, there are sporadic peaks of the wave activity. It follows from Figures 10 and 11 that the amplitude method presents important information of the wave processes in the lower stratosphere within the height interval km. These images contain global and local information on the geographical distribution of the wave activity in the atmosphere, and in addition, demonstrate important features in the seasonal and regional activity. [27] The general features in the latitudinal distribution of the wave activity as a function of the season can be seen in Figures 12 and 13. The histograms of the latitude distribution of the strong CHAMP RO amplitude events with q > 0.6 N-units/km for the km and with q > 0.24 N-units/km for the km height intervals are given in Figures 12 and 13. The number of strong RO events normalized to the overall number of RO events in the corresponding latitude interval 15 in length is displayed on the vertical axes in Figures 12 and 13. The north latitude is plotted along the horizontal axes in Figures 12 and 13. Figure 12 corresponds to the Northern Hemisphere summer within the time period from 28 June 2003 to 15 July Figure 13 corresponds to the Northern Hemisphere winter within the time period from 20 January 2003 to 3 February Following the analysis of the data shown in Figures 12 and 13, there is an asymmetry in distribution of the wave activity at the 12 km level in the atmosphere. The maximal wave activity occurs in the summer polar region (Arctic in Figure 12 and Antarctic in Figure 13). The minimum wave activity is observed in the polar region s winter period (Antarctic in Figure 12 and Arctic in Figure 13). There is a slow decrease of the wave activity from the summer polar region to the winter polar region at the 12 km level. At the 14 and 16 km levels there is a strong wave activity in the moderate latitudes both in the Northern and Southern hemispheres. In the km height interval, the maximum wave activity is observed in the equatorial region. The occurrence of strong events decreases with height in the km height interval. Thus the latitudinal histograms serve as important tools to better understand the general features of the internal wave activity in the atmosphere, as compared with the detailed maps of the wave activity shown in Figures 10 and 11. More detailed analysis of the global pictures of the internal wave activity, and the studying of the possible connections with the meteorological phenomena in the atmosphere is the task for future work. 7. Conclusions [28] The applications of the aforementioned amplitude radio holographic method further extend the capabilities of the GPS RO technology. From our analysis, the amplitude variations of GPS occultation signals are very sensitive sensors to the 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, 8of14

9 Figure 10. Seasonal and geographical distribution of internal wave activity with amplitudes greater than 0.6 N-units/km in the lower stratosphere (12 18 km) for (left) period 21 January to 3 February and (right) 15 July to 30 July The height is shown in inserts located on the right sides of the plots. 9of14

10 Figure 11. Seasonal and geographical distribution of the internal wave activity with amplitudes greater than 0.24 N-units/km in the lower stratosphere (20 26 km) for (left) period 21 January to 3 February and (right) 15 July to 30 July The height is indicated in inserts on the right side of the plots. 10 of 14

11 Figure 12. Histograms of latitudinal distribution of internal wave activity in the km altitude interval for the time period 28 June to 15 July Relative number of strong CHAMP RO amplitude events (expressed in percents) with q > 0.6 N-units/km for the km and with q > 0.24 N-units/km for the km height interval is plotted versus the north latitude (expressed in degrees). 11 of 14

12 Figure 13. Histograms of latitudinal distribution of internal wave activity in the km altitude interval for the time period 20 January to 3 February Relative number of strong CHAMP RO amplitude events (expressed in percents) with q > 0.6 N-units/km for the km and with q > 0.24 N-units/km for the km height interval is plotted versus the north latitude (expressed in degrees). 12 of 14

13 indicating high sensitivity of the amplitude data to the wave structures with small vertical periods in the km interval. For CHAMP RO data, the sensitivity of the amplitude to the internal waves with vertical periods of 1 km, are greater than the sensitivity of the phase by a factor of order 10 40, depending on the signal to noise ratio. The introduced analytic method demonstrates its capability to retrieve the radio image of the internal wave-wave portrait using the amplitude data of the RO signals. The wave portrait can be restored using the Hilbert transform in the form of the analytic signal and contains the amplitude and phase of the internal wave as functions of the height. The analytic form of the internal wave presentation is convenient for the analysis of the experimental data, and can be implemented in the case of GW for determination of the GW intrinsic phase speed, and the horizontal wind speed perturbations associated with the GW influence. [29] The amplitude GPS occultation method presents a possibility to obtain the geographical distribution and seasonal dependence of the atmospheric wave activity with global coverage. The amplitude GPS occultation method reveals an asymmetry in distribution of the wave activity at the 12 km level in the atmosphere. The maximal wave activity occurs in the summer polar region. At the km levels the wave activity is centered in the moderate latitudes both in the Northern and Southern hemispheres. At km levels, most of the internal wave s energy is concentrated in the equatorial areas. The local seasonal dependencies are clear for some regions, e.g., Siberia at the height of 14 km in the winter, owns a low wave activity and a high wave activity in the summer. Therefore the amplitude radio holographic method has great promise to be effective in investigating the climatology of the wave activity in very large height intervals in the upper troposphere and stratosphere. [30] Acknowledgments. We are grateful to the GeoForschungZentrum Potsdam for delivering the CHAMP RO data. We are grateful to the National Science Council of Taiwan, R.O.C., for financial support under the grants NSC M and NSC M Work has been partly supported by the Russian Fund of Basic Research, grant In addition, assistance was provided by the Russian Academy of Sciences, program OFN-16 and OFN-17. References Anthes, R. A., C. Rocken, and Y.-H. Kuo (2000), Applications of COSMIC to meteorology and climate, Terr. Atmos. Oceanic Sci., 11, Ebel, A. (1984), Contribution of GWs to the momentum, heat and turbulent energy budget of the upper mesosphere and lower thermosphere, J. Atmos. Terr. Phys., 46(9), Eckermann, S. D., and P. Preusse (1999), Global measurements of stratospheric mountain waves from space, Science, 286, Eckermann, S. D., I. Hirota, and W. A. Hocking (1995), GW and equatorial wave morphology of the stratosphere derived from long-term rocket soundings, Q. J. R. Meteorol. Soc., 121, Feng, D. D., and B. M. Herman (1999), Remotely sensing the Earth s atmosphere using the Global Positioning System (GPS) The GPS/ MET data analysis, J. Atmos. Oceanic Technol., 16, Fritts, D. C., and M. J. Alexander (2003), Gravity wave dynamics and effects in the middle atmosphere, Rev. Geophys., 41(1), 1003, doi: /2001rg Gorbunov, M. E. (2002), Canonical transform method for processing radio occultation data in the lower troposphere, Radio Sci., 37(5), 1076, doi: /2000rs Hajj, G. A., C. O. Ao, B. A. Iijima, D. Kuang, E. R. Kursinsky, A. J. Manucci, T. K. Meehan, L. J. Romans, M. de la Torre Juárez, and T. P. Yuanck (2004), CHAMP and SAC-C atmospheric occultation results and intercomparisons, J. Geophys. Res., 109, D06109, doi: /2003jd Hocke, K., A. Pavelyev, O. Yakovlev, L. Barthes, and N. Jakowski (1999), Radio occultation data analysis by radio holographic method, J. Atmos. Sol. Terr. Phys., 61, Igarashi, K., A. Pavelyev, K. Hocke, D. Pavelyev, I. A. Kucherjavenkov, S. Matugov, A. Zakharov, and O. Yakovlev (2000), Radio holographic principle for observing natural processes in the atmosphere and retrieving meteorological parameters from RO data, Earth Planets Space, 52, Igarashi, K., A. Pavelyev, K. Hocke, D. Pavelyev, and J. Wickert (2001), Observation of wave structures in the upper atmosphere by means of radio holographic analysis of the RO data, Adv. Space Res., 27(6 7), Jensen, A. S., M. S. Lohmann, H.-H. Benzon, and A. S. Nielsen (2003), Full spectrum inversion of radio occultation signals, Radio Sci., 38(3), 1040, doi: /2002rs Kalashnikov, I., S. Matugov, A. Pavelyev, and O. Yakovlev (1986), Analysis of the features of RO method for the Earth s atmosphere study (in Russian), in Electromagnetic Waves in the Atmosphere and Space, pp , Nauka, Moscow. Kuo, Y.-H., S. Sokolovskiy, R. A. Anthes, and F. Vandenberghe (2000), Assimilation of GPS radio occultation data for numerical weather prediction, Terr. Atmos. Oceanic Sci., 11, Lindzen, R. S. (1981), Turbulence and stress owing to GW and tidal breakdown, J. Geophys. Res., 86(C9), Liou,Y.-A.,A.G.Pavelyev,C.-Y.Huang,K.Igarashi,andK.Hocke (2002), Simultaneous observation of the vertical gradients of refractivity in the atmosphere and electron density in the lower ionosphere by RO amplitude method, Geophys. Res. Lett., 29(19), 1937, doi: / 2002GL Liou, Y.-A., A. G. Pavelyev, C.-Y. Huang, K. Igarashi, K. Hocke, and S. K. Yan (2003), Analytic method for observation of the GW using RO data, Geophys. Res. Lett., 30(20), 2021, doi: /2003gl Liou, Y. A., A. G. Pavelyev, J. Wickert, T. Schmidt, and A. A. Pavelyev (2005a), Analysis of atmospheric and ionospheric structures using the GPS/MET and CHAMP radio occultation database: A methodological review, GPS Solution, 9, , doi: /s y. Liou, Y. A., A. G. Pavelyev, and J. Wickert (2005b), Observation of gravity waves from GPS/MET radio occultation data, J. Atmos. Sol. Terr. Phys., 67, Nagpal, O. P. (1979), The sources of atmospheric GWs, Contemp. Phys., 20, Pavelyev, A. (1998), On the possibility of radio holographic investigation on communication link satellite-to-satellite, J. Commun. Technol. Electron., 43(8), Pavelyev, A. G., and A. I. Kucherjavenkov (1978), Refraction attenuation in the planetary atmospheres, Radio Eng. Electron. Phys., 23(7), Pavelyev, A., A. V. Volkov, A. I. Zakharov, S. A. Krytikh, and A. I. Kucherjavenkov (1996), Bistatic radar as a tool for Earth investigation using small satellites, Acta Astronaut., 39, Pavelyev, A., K. Igarashi, C. Reigber, K. Hocke, J. Wickert, G. Beyerle, S. Matyugov, A. Kucherjavenkov, D. Pavelyev, and O. Yakovlev (2002a), First application of the radioholographic method to wave observations in the upper atmosphere, Radio Sci., 37(3), 1043, doi: / 2000RS Pavelyev, A. G., Y. A. Liou, C. Y. Huang, C. Reigber, J. Wickert, K. Igarashi, and K. Hocke (2002b), Radio holographic method for the study of the ionosphere, atmosphere and terrestrial surface from space using GPS occultation signals, GPS Solutions, 6, Pavelyev, A., J. Wickert, Y.-A. Liou, K. Igarashi, K. Hocke, and C.-Y. Huang (2003a), Vertical gradients of refractivity in the mesosphere and atmosphere retrieved from GPS/MET and CHAMP radio occultation data, in First CHAMP Mission, Results for Gravity, Magnetic and Atmospheric Studies, edited by C. Reigber, H. Luhr, and P. Schwintzer, pp , Springer, New York. Pavelyev, A. G., T. Tsuda, K. Igarashi, Y. A. Liou, and K. Hocke (2003b), Wave structures in the electron density profile in the ionospheric D and E-layers observed by radio holography analysis of the GPS/MET radio occultation data, J. Atmos. Sol. Terr. Phys., 65(1), Pavelyev, A. G., Y. A. Liou, and J. Wickert (2004), Diffractive vector and scalar integrals for bistatic radio holographic remote sensing, Radio Sci., 39, RS4011, doi: /2003rs Pruse, J. M., P. K. Smolarkiewicz, and R. R. Garcia (1996), Propagation and breaking at high altitudes of GWs excited by tropospheric forcing, J. Atmos. Sci., 53, Rabiner, L., and B. Gold (1978), Theory and Application of Digital Signal Processing, Prentice-Hall, Upper Saddle River, N. J. Reigber, C., H. Lühr, and P. Schwintzer (2002), CHAMP mission status, Adv. Space Res., 30(2), of 14

14 Sica, R. J., and A. T. Russell (1999), Measurements of the effects of GWs in the middle atmosphere using parametric model of density fluctuations. Part I: Vertical wavenumber and temporal spectra, J. Atmos. Sci., 56, Sokolovskiy, S. V. (2000), Inversion of RO amplitude data, Radio Sci., 35(1), Steiner, A. K., and G. Kirchengast (2000), GW spectra from GPS/MET occultation observations, J. Atmos. Oceanic Technol., 17, Tsuda, T., and K. Hocke (2002), Vertical wave number spectrum of temperature fluctuations in the stratosphere using GPS occultation data, J. Meteorol. Soc. Jpn., 80(4B), Tsuda, T., M. Nishida, C. Rocken, and R. H. Ware (2000), A global morphology of GW activity in the stratosphere revealed by the GPS occultation data (GPS/MET), J. Geophys. Res., 105, Ware, R., et al. (1996), GPS soundings of the atmosphere from low earth orbit: Preliminary results, Bull. Am. Meteorol. Soc., 77, Wickert, J., et al. (2001), Atmosphere sounding by GPS radio occultation: First results from CHAMP, Geophys. Res. Lett., 28, Wickert, J., A. G. Pavelyev, Y. A. Liou, T. Schmidt, C. Reigber, K. Igarashi, A. A. Pavelyev, and S. S. Matyugov (2004), Amplitude scintillations in GPS signals as a possible indicator of ionospheric structures, Geophys. Res. Lett., 31, L24801, doi: /2004gl K. Igarashi, National Institute of Information and Communications Technology, 4-2-1, Nukui-Kita Machi, Koganei-shi, Tokyo , Japan. (igarashi@nict.go.jp) Y. A. Liou, Center for Space and Remote Sensing Research, National Central University, Chung-Li 320, Taiwan. (yueian@csrsr.ncu.edu.tw) S. F. Liu, Department of Industrial Design, National Cheng Kung University, Tainan 701, Taiwan. (liusf@mail.ncku.edu.tw) A. A. Pavelyev and A. G. Pavelyev, Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Fryazino, Vvedenskogo sq. 1, Moscow, Russia. (pvlv@ms.ire.rssi.ru) T. Schmidt and J. Wickert, GeoForschungsZentrum Potsdam, Telegrafenberg, D Potsdam, Germany. (wickert@gfz-potsdam.de) 14 of 14