The relation between the refractivity of air at radio frequencies and atmospheric variables is given in Eq. (1) from Ruieger (2002) where N is total atmospheric refrac-tivity, Nd and Nw, respectively the first term and the second term (which is within the brackets) in Eq. (1), are the contributions of the dry gas and water vapor components of the atmosphere, Pd is dry partial pressure in hPa, T is temperature in K, e is partial pressure of water vapor in hPa, and the constants are those of the "best average" case. The constant for the dry component of refractivity assumes a CO2 mixing ratio of 375 ppm.
N = Nd + Nw = 77.6890y + (71.2952- + 375463—j (1)
The impact of dry turbulence has received much attention in the literature due in particular to its impact on astronomy. The wet refractivity term has received less attention and is only relevant at microwave frequencies slow enough to respond to the motion of the permanent dipole of the water molecule. Since scintillations at ATOMMS frequencies are dominated by wet rather than dry turbulence over much of the troposphere, we have developed a parameterization of wet turbulence described in Otarola (2008).
At radio frequencies, in the warmer regions of the troposphere, the fluctuations in the refractive index tend to depend more on humidity fluctuations and less on temperature fluctuations (e.g., Coulman and Vernin 1991). The structure constant of the air index of refraction fluctuations, C2n, is the variance of the difference in index of refraction variations at points separated by one meter. C can be written as shown in Eq. (2). This is possible under the assumption that the power spectral density of the dry, wet, and cross terms share similar spectral characteristics with the only exception being the power level. Equation (2) is similar to that found in the works of Wesely (1976) and Coulman (1985) except that it is written in terms of the structure constants associated with the dry and wet components of air refractivity, C1nA and C^w, respectively, and the covariance of the variations in the dry and wet components of the air refractive index, Cnd,nw.
Therefore, the strength of the turbulence, C^L, where L is the path length through the atmosphere depends on fluctuations in both the dry and wet components of the air index of refraction, C^ and C1nw. While the literature includes some empirical formulations for the determination of the C1nA structure constant as a function of altitude (e.g., Fenn et al. 1985), little information is available about how to estimate the contribution to the turbulence strength from fluctuations in humidity. Combining our analysis of aircraft observations of temperature, pressure, and humidity at various altitudes levels in the atmosphere with a theoretical framework, we have derived a relationship between the wet component of the index of refraction structure constant and the mean wet refractivity that is given in Eq. (3) (Otarola 2008).
The procedure we have developed to model and account for the effects that turbulent variations in real refractivity will have on ATOMMS retrieval system performance is outlined here. Under the condition of weak turbulence (weak scintillations), the appropriate equation for computing the expected amplitude variations for a signal propagating through a turbulent medium is given by Eq. (4) (Frehlich and Ochs 1990),
where A is the amplitude that is varying due to scintillations, A0 is the mean amplitude of the signal, k is the signal wavenumber, C2 is the refractive index structure parameter, L is the total distance from transmitter to receiver, and the integration proceeds along the propagation direction, specified here as the x direction. In order to utilize this equation, one must specify values for C'2 along the propagation path. As mentioned above, for microwave signals, the atmospheric refractivity depends on both the dry air density (dry part) and water vapor density (wet part). Turbulent variations in both components will result in amplitude scintillations on our signals. Figure 4a shows a representative profile of the dry part and two realizations of the wet parameterization for Mid-Latitude Summer (MLS) and Arctic Winter (ArW) conditions, respectively. Figure 4b shows simulated profiles of the turbulent amplitude fluctuations for 20 GHz signals using the C'2 profiles in Fig. 4a and
Eq. (4). As Fig. 4b shows, we estimate, based on Eq. (4), that turbulent scintillations measured during the high altitude aircraft-to-aircraft occultations will be a factor of 2-3 smaller than those of the LEO-LEO occultations due to the shorter aircraft-to-aircraft path length. The amplitude scintillation fluctuations will be reduced significantly with amplitude ratioing as indicated in Fig. 2.
Because ATOMMS utilizes calibration tones to mitigate unwanted instrumental and atmospheric effects, a key quantity is the difference between optical depths measured at two (or more) frequencies, defined as At = t(fi) - t(f2), rather than the optical depth at a single frequency, t(fl). The optical depth difference is proportional to the logarithm of the ratio of the two signal amplitudes. Therefore, the appropriate quantity for retrieval error estimation is the residual uncertainty in the amplitude ratios, rather than the amplitude uncertainty at an individual frequency.
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