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Compact airborne Raman lidar for profiling aerosol, water vapor and clouds

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Abstract

A compact airborne Raman lidar system, which can perform water vapor and aerosol measurements both during nighttime and daytime is described. The system design, setup and the data processing methods are described in the paper. The Raman lidar was tested on University of Wyoming King Air research aircraft (UWKA) during the Wyoming King Air PBL Exploratory Experiment (KAPEE) in 2010. An observation showing clouds, aerosols and a dry line is presented to illustrate the lidar detection capabilities. Comparisons of the water vapor and aerosol measurements using the Raman lidar and other in situ airborne instruments show good agreement.

© 2014 Optical Society of America

1. Introduction

Water vapor and aerosols are two important atmospheric parameters. The determination of aerosol concentration profiles has important applications in areas such as heterogeneous atmospheric chemistry, weather and climate research, and pollution monitoring [1]. Water vapor plays a major role in the radiation budget and the water cycle with important implications for weather and climate [2]. Raman lidar is a stand-alone remote sensor, which has proven to be an effective tool to characterize aerosol and water vapor distribution [36]. The technique operates by transmitting a pulse of laser light at a fixed wavelength and simultaneously recording signals at wavelengths corresponding to the inelastic Raman shifts of nitrogen and water vapor. The elastic-backscattered signals corresponding to Rayleigh and aerosol scattering also are detected.

The aerosol and water vapor distribution are highly heterogeneous in the atmospheric boundary layer due to their variable sources and sinks, and the complexity of atmospheric motion and mixing. Satellite remote sensing partly meet the needs of their observations, but has coarse spatial resolution and lack of flexibility. Measurements with better accuracy and resolution can be provided by airborne lidars, which can sample relatively large interesting areas in a short time period and follow the atmospheric dynamics developments. Kiemle et al. have used an airborne differential absorption lidar (DIAL) to study water vapor transportation [7, 8]. However, a DIAL system with the optical parametric oscillator (OPO) laser is much more expensive, less robust, and more difficult to operate than a Raman lidar. Airborne Raman lidar also has been deployed successfully for water vapor and aerosol measurements [9, 10]. To advance our understanding of the interactions of atmosphere and land-surface and to accurately quantify aerosol effects, we developed a compact airborne Raman lidar to work together with other in situ measurements for the boundary layer aerosol, water vapor, and cloud measurements from the University Wyoming King Air (UWKA) research aircraft.

Airborne research at the University of Wyoming began nearly 50 years ago. Since the mid-1960s, researchers at the university have utilized three different aircraft to acquire measurements throughout the lower troposphere [11]. The UWKA, the most current aircraft, is a specially instrumented Raytheon Beechcraft King Air 200T (twin turbo-prop) designed and used for atmospheric research.

The compact Raman lidar was deployed on UWKA and tested during the Wyoming King Air PBL Exploratory Experiment (KAPEE) in 2010. In addition to the Raman lidar, there are also other in situ instruments [12] for aerosol and water vapor measurements during the experiment. These in situ measurements can be used to calibrate or to evaluate lidar measurements.

2. System setup

The design goal of the compact Raman lidar is to provide synergetic measurements with other in situ instruments from the UWKA to better study planetary boundary layer (PBL) aerosol and water vapor. The compact Raman lidar needs to work during daytime. Therefore, a powerful laser is needed to provide strong water vapor Raman scattering signals under the strong daylight solar background. However, the laser power and the telescope aperture are both quite limited for an airborne system, especially when there are many other instruments onboard a small aircraft, such as UWKA. The lidar also needs to provide measurements as close to aircraft as possible to better combine with in situ measurements. This requests a shorter blind area and results in a wider field of view (FOV) of the telescope; however, to suppress the solar background, the FOV should be as narrow as possible. Furthermore, in order to provide higher horizontally resolved measurements from aircraft, the averaging time for each lidar profile should be as shorter as possible. This means that the long-time averaging method, which is usually used to beat down noise in ground-based lidar systems, is not the best option for airborne lidar systems to improve signal qualities. All of these conflicts made our design quite challenging. Fortunately, for PBL studies, the extended detection range is not as vital as for a ground-based system, since the aircraft can fly at different altitudes. Thus, the system design focused near range measurements by taking advantage of strong r−2 dependent signals.

A flashlamp-pumped Nd:YAG laser (Bigsky CFR400 GRM), which provides ~50 mJ output of 355nm at 30 Hz, is used as the transmitter. The 355nm wavelength is chosen to meet several criteria; weaker solar background, stronger Raman scattering and higher limits for eye-safety at 355nm than at 532 nm. The emitting divergence of the laser beam is reduced from original 1.8 mrad to 0.36 mrad with a 5X beam expander. A half-wavelength-plate (HWP) is installed close to the laser-emitting window to adjust the polarization direction of the outgoing laser and to implement the calibration method for depolarization measurements [13]. A coaxial arrangement was chosen to provide near field measurements and to make required window port smaller (see Fig. 1).

 figure: Fig. 1

Fig. 1 Photograph of the Raman lidar inner structure.

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The receiver is a Cassegrain telescope with a 12 inch aperture. The primary mirror is mounted on a rigid optical bench together with the laser and a light shading tube of anodized aluminum. The tube is installed in the telescope to block most scattered daylight that directly falls onto the field-stop pinhole. The secondary mirror of the telescope is mounted on the top of the tube (Fig. 1). The lidar has four detector channels: two channels for parallel and perpendicular elastic scattering and two for water vapor and nitrogen Raman scattering. All of the channels use narrow band interference filters with FWHM (full width at half maximum) of 0.3 nm (Barr Associates) and photomultiplier tube (PMT) as detectors (Hamamatsu H5873). A 200M, 12-bit A/D data acquisition system, instead of a photon counting system, was selected to record single shot raw data, which provides a 3 meter horizontal resolution while the aircraft cruising speed is around 90 m/s and up to 0.75 meter vertical resolution. Different horizontal and vertical averaging can be selected during post data analyses for different measurement targets. A simplified instrument schematic diagram is shown in Fig. 2.

 figure: Fig. 2

Fig. 2 Schematic diagram of the compact airborne Raman lidar.

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The Raman lidar inner structure is shown in Fig. 1. The transmitter, the telescope and the receiver optics are all mounted on the same optical bench, which provides the needed stability to maintain the optical alignment in a vibration environment. The blue line in Fig. 1 is a sketch of the outgoing laser beam. The lidar is downward looking and is mounted above the window in the cabin of UWKA. A list of the main system parameters is given in Table 1.

Tables Icon

Table 1. System parameters of the compact airborne Raman lidar

3. Data processing

There are four optical channels in the airborne Raman lidar system, two elastic scattering signals in parallel and perpendicular polarization and two Raman scattering signals of nitrogen and water vapor. The background subtracted and overlapped corrected backscattered signals at 355 nm are expressed as,

P=ηPLβτ2r2,P=ηPLβτ2r2,
where P|| and P are the total backscatter (molecular plus aerosol) power of parallel and perpendicular channels, respectively, η is the system constant, PL is the laser power, β|| and β are parallel and perpendicular components of the volume backscatter coefficients. The factor τ2 accounts for the atmospheric two-way transmittance, and r is the range from the lidar.

The ratio of the total perpendicular-polarized backscatter coefficient to the total parallel-polarized backscatter coefficient is called the linear volume depolarization ratio δv,

δV=ββ=CAPP,
where CA is the calibration constant to account for different gain effects of the two channels (including the optical transmittances and the electronic amplification in each channel). The detail of the method used for calibration is described in [13].

The aerosol backscatter coefficient can be determined based on a combination of elastic and Raman scattering retrieval methods [1]. The lidar equations for the elastic and Raman backscattered signals are,

PE=KE(βa+βm)τE2r2,PN=KNβNτN2r2,
where PE and PN are the overlap corrected signals from the elastic scattering channel (λ0 = 355 nm) and nitrogen Raman scattering channel (λN = 386 nm), KE and KN are system constants of the two channels, τE2 and τN2 are two-way atmospheric transmittances for λ0andλN, βaandβmare backscatter coefficients due to aerosol and air molecule at 355 nm respectively, βN is the Raman scattering coefficient due to nitrogen at 386 nm. The procedure used for the signal overlap correction is described in [14], and the minimum range for complete overlap is around 90 m. Aerosol backscatter coefficient can be derived from Eq. (3),
βa=RPβNRKRTβm,
where the range-dependent signal ratio isRP=PE/PN, the range-dependent transmittance ratio isRT=τE2/τN2, and the ratio of system constants isRK=KE/KN. Since the transmitted wavelength and the nitrogen Raman wavelength are nearly same, the difference between the two transmittances can be ignored (RT1) or corrected iteratively. The values ofβm and βN can be calculated with air density [15]. The air density is altitude dependent and can be estimated withn(z)=n(z0)exp[(zz0)/H], where n(z0)is the air density at flight level and H is the scale height. Both of them can be derived from the King Air in situ measurements (pressure, temperature and humidity). The constant ratio RKcan be calibrated with clear air data (aerosol-free assumption) or other in situ instrument measurements.

The water vapor mixing ratio can be determined by the ratio of signals from two Raman scattering channels [16],

WVM=CτNPVτVPN,
where PV is the overlap corrected signals from the water vapor Raman scattering channel (λV = 407 nm), τN and τV are one way backward atmospheric transmittances forλN andλV. Since the two wavelengths are quite close, the transmittance difference can also be ignored for short-range measurements (τN/τV1). However, the transmittance difference due to air molecule can be reliably estimated with air density profiles, and the contribution due to aerosol also can be roughly estimated from aerosol backscattering coefficient profiles in general. C is the system calibration factor, which can be calibrated with airborne in situ measurements.

The maneuverability of the aircraft platform facilitates easy water vapor calibration using onboard sensors. Within a well-mixed PBL, the near field lidar measurements can be used to compare with in situ measurements directly for calibration and correction. Vertical profiles from stacked short flight legs at different levels and slow spiral descent or ascent flights can be used to calibrate the measurements. With such spatial measurements, a calibration error less than 3% is archived, as illustrated in Section 4.

The random error in determining βa and WVM can be calculated by applying the standard error propagation formulas [17] to Eqs. (4) and (5),

σβa2βa2=σRP2RP2+σRK2RK2+σRT2RT2,σWVM2WVM2=σC2C2+σPV2PV2+σPN2PN2,
where the calculation errors ofβmand βN are ignored for simplification, which is reasonable due to the atmosphere stability and the small uncertainties in pressure and temperature measurements. The error budget is usually dominated by the random errors in the lidar signals [18].

The raw data of lidar signals are averaged or smoothed to reduce random errors. Ten shots are usually averaged to provide 30 meters horizontal resolution. A parabolic low-pass FFT filter is employed to smooth the vertical structure data with a filter window function expressed as [19]:

W(f)={1,ifffC11(ffC1)2(fC2fC1)2,iffC1ffC20ifffC1
where fC1 is the pass frequency and fC2 is the stop frequency. The vertical window length of the filter is 10 points (7.5 meter) within 200 m, 20 points (15 meter) at 200 - 500 m range and 40 points (30 meter) at 500 - 800 m range. Examples of the retrieved aerosol backscatter and water vapor are described in Section 4.

4. Results and comparisons

The compact Raman lidar was installed on the UWKA during KAPEE from May 26 to June 26, 2010. Data from total 41 research flight hours were collected under different meteorological conditions over southern Wyoming and northern Colorado. It demonstrated that the lidar system is easy to operate with good alignment stability during the whole project. Examples from KAPEE are presented here to illustrate the capability of the system.

Figure 3 shows a ten-minute segment of Raman lidar observation from 17:40 to 17:50 UTC on June 21. In Fig. 3, the top three cross-sections are Raman lidar measured (a) aerosol backscattering coefficients, (b) depolarization ratios and (c) water vapor mixing ratios. Figure 3(d) shows the vertical wind velocity (black line) and temperature (red line) at the flight level from in situ probes.

 figure: Fig. 3

Fig. 3 A ten-minute segment of Raman lidar observations on June 21, 2010: (a) aerosol backscattering coefficients (β, km−1) plotted as10log10(β); (b) depolarization ratios; (c) water vapor mixing ratios; (d) vertical wind velocity (black) and temperature at the flight level. The two-abscissa axes are UTC time (upper) and distance (lower). The extending black region in a, b, and c indicate the surface below and the top line is just below the aircraft flight level. The wind was blowing from left to right at ~7 m/s while aircraft flying from south to north almost along the wind.

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Figure 3 clearly shows the large spatial variations of aerosol and water vapor in the PBL. Around 17:40:30, there are clouds detected at 2.6 km altitude. Lower than molecule depolarization near cloud top indicating water clouds consistent with estimated cloud temperature (warmer than 15°C). The water vapor mixing ratio indicates that very moist environment for aerosols in the PBL, which is consistent with lower depolarization within the high aerosol region. Furthermore, the depolarization are lower at the top of the aerosol plumes, which indicates these aerosols are closer to spherical shape due to hygroscopic growth.

The aircraft flew across a dry line towards the dry side at 17:47:30, and the dry line has a slightly tilted interface with water vapor gradient over 10g/kg/km. Meanwhile, there is a sharp transition of aerosols as well. The flight level vertical wind velocity shows an updraft right at the dry line. The dry line is an important trigger mechanism for convective initialization. During KAPEE, many cases were observed, and one such case was studied in detail by Bergmaier et al. [20].

Two single profiles of the lidar measured mixing ratio and backscattering coefficient profiles at 17:41:30 UTC are shown in Fig. 4. The profiles were measured at 10:41 AM local time with strong solar background. Even in such a condition, the compact Raman lidar system can provide of water vapor measurements with less than 10% relative error up to 500 meters below the flight level, while the system can provide backscattering coefficient measurements with less than 10% the relative error at 800 meters range. These measurement capabilities provide an important tool to study the PBL structure and processes [20].

 figure: Fig. 4

Fig. 4 Lidar measured profiles of (a) water vapor mixing ratios and (b) aerosol backscattering coefficients measured at 17:41:30 June 21, 2010. Error bars show the standard deviation of measurement uncertainty. Note that the y-axis is range from the aircraft.

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Figure 5 shows comparisons of water vapor mixing ratios and aerosol backscattering coefficients measured by lidar and in situ instruments for the whole flight on June 21. In Fig. 5(a), lidar measured water vapor mixing ratios (black line) are averaged values from 30 m to 37.5 m below the aircraft while the red line shows water vapor mixing ratios measured at the flight level by the onboard LI-6262 (LICOR) [21], which has a 1% accuracy for water vapor measurements. Figure 5(b) shows aerosol backscattering coefficients from lidar (averaged from 30 m to 37.5 m below the aircraft) and from PCASP-100 probe (PMS Inc.) [22]. PCASP provides dry aerosol size distributions, and it measures particles ranging from 0.1 μm to 3.0 μm with 20% diameter accuracy and 16% concentration accuracy. Mie scattering algorithm was used to calculate the backscattering from measured size distrubutions by correcting aerosol refractive index and size according to ambinet relative humidity [23]. Figure 5(c) and (d) show that lidar and flight-level measurements agree well except for some differences due to vertical inhomogenity of aerosol and water vapor. The adjusted R2 for the linear fitting in Fig. 5(c) is 0.98 and in Fig. 5(d) is 0.97.

 figure: Fig. 5

Fig. 5 Comparisons of (a) water vapor mixing ratios and (b) aerosol backscattering coefficients measured by lidar (black) and in situ instruments (red) for the whole flight on June 22, 2010. (c) and (d) are scatter plots for the two comparisons measured 30 to 37.5 m below the aircraft.

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5. Summary

In this paper, a compact airborne Raman lidar with four receiving channels are described. The lidar has been successfully built to work together with other instruments on UWKA. A laser operating at 355 nm with 50 mJ pulse energy and a 12 inch telescope provided needed sensitivities for boundary layer water vapor and aerosol measurements both at daytime and nighttime. The Raman lidar has been deployed during KAPEE. Throughout the project, it has exhibited the needed reliability for turn-key operation from aircraft. The unique design of sharing the same optical bench for the receiver and the transmitter hold the alignment for the entire campaign. To provide effective synergy with other in situ measurements, the Raman lidar can provide near range measurements as close as 30 m.

Measurement examples during KAPEE are provided to illustrate its measurements capabilities for PBL clouds, aerosols and water vapor. The mixing ratio and backscattering coefficient profiles show that the system can provide measurements with the relative errors less than 10% within 500 meters at 30 m horizontal resolution, even under strong daylight condition. Comparisons with in situ measurements confirm that the compact lidar measurements are accurate.

The high spatially resolved aerosol and water vapor structure in the PBL provided by the compact Raman lidar are demonstrated to be very useful for atmospheric dynamics studies, such as dry line triggered convection. Furthermore, the synergy of Raman lidar measurements with in situ sampling provides a unique set of measurements to better characterize boundary layer aerosols and water vapor and to advance our understanding on related processes controlling their spatial variations. But, the compact Raman lidar has limited measurements range during daytime due to low laser pulse energy, which may require different flight legs to provide a complete vertical structure for a deep PBL. With the successful development of the compact Raman lidar system, a more powerful Raman lidar with additional temperature measurements capabilities is under development new with further funding from NSF.

Acknowledgments

This work is supported by National Science Foundation (NSF) under Awards AGS-0645644 and AGS-1337599.

References and links

1. A. Ansmann, M. Riebesell, and C. Weitkamp, “Measurement of atmospheric aerosol extinction profiles with a Raman lidar,” Opt. Lett. 15(13), 746–748 (1990). [CrossRef]   [PubMed]  

2. A. G. U. Special Report, “Water Vapor in the Climate System, December” (American Geophysical Union, 1995). http://www.eso.org/gen-fac/pubs/astclim/espas/pwv/mockler.html.

3. C. R. Philbrick, “Raman lidar measurements of atmospheric properties,” Proc. SPIE 2222, 922–931 (1994). [CrossRef]  

4. J. E. Goldsmith, F. H. Blair, S. E. Bisson, and D. D. Turner, “Turn-Key Raman Lidar for Profiling Atmospheric Water Vapor, Clouds, and Aerosols,” Appl. Opt. 37(21), 4979–4990 (1998). [CrossRef]   [PubMed]  

5. R. K. Newsom, D. D. Turner, B. Mielke, M. Clayton, R. Ferrare, and C. Sivaraman, “Simultaneous analog and photon counting detection for Raman lidar,” Appl. Opt. 48(20), 3903–3914 (2009). [CrossRef]   [PubMed]  

6. D. N. Whiteman, K. D. Evans, B. Demoz, D. O. C. Starr, E. W. Eloranta, D. Tobin, W. Feltz, G. J. Jedlovec, S. I. Gutman, G. K. Schwemmer, M. Cadirola, S. H. Melfi, and F. J. Schmidlin, “Raman lidar measurements of water vapor and cirrus clouds during the passage of Hurricane Bonnie,” J. Geophys. Res. 106(D6), 5211–5225 (2001). [CrossRef]  

7. C. Kiemle, A. Schäfler, M. Wirth, A. Fix, and S. Rahm, “Detection and Analysis of Water Vapor Transport by Airborne Lidars,” IEEE J-STARS 6(3), 1189–1193 (2013).

8. T. F. Refaat and H. E. Elsayed, “Advanced Atmospheric Water Vapor DIAL Detection System,” Technical Report, (NASA Langley Technical Report Server, 2000). http://dl.acm.org/citation.cfm?id=871423.

9. W. S. Heaps and J. Burris, “Airborne Raman lidar,” Appl. Opt. 35(36), 7128–7135 (1996). [CrossRef]   [PubMed]  

10. D. N. Whiteman, K. Rush, S. Rabenhorst, W. Welch, M. Cadirola, G. Mcintire, F. Russo, M. Adam, D. Venable, R. Connell, I. Veselovskii, R. Forno, B. Mielke, B. Stein, T. Leblanc, S. Mcdermid, and H. Voemel, “Airborne and Ground-Based Measurements Using a High-Performance Raman Lidar,” J. Atmos. Ocean. Technol. 27(11), 1781–1801 (2010). [CrossRef]  

11. A. Rodi, “King of the air: The evolution and capabilities of Wyoming’s observation aircraft,” Meteorol. Technol. Int. 44-47 (2011).

12. Z. Wang, J. French, G. Vali, P. J. Wechsler, S. Haimov, A. Rodi, M. Deng, D. Leon, J. R. Snider, L. Peng, and A. Pazmany, “Single aircraft integration of remote sensing and in situ sampling for the study of cloud microphysics and dynamics,” Bull. Am. Meteorol. Soc. 93(5), 653–668 (2012). [CrossRef]  

13. B. Liu and Z. Wang, “Improved calibration method for depolarization lidar measurement,” Opt. Express 21(12), 14583–14590 (2013). [CrossRef]   [PubMed]  

14. B. Liu, Z. Zhong, and J. Zhou, “Development of a Mie scattering lidar system for measuring whole tropospheric aerosols,” J. Opt. A, Pure Appl. Opt. 9(10), 828–832 (2007). [CrossRef]  

15. E. D. Hinkley, Laser monitoring of the atmosphere (Berlin and New York, 1976), Chap. 4.

16. D. N. Whiteman, S. H. Melfi, and R. A. Ferrare, “Raman lidar system for the measurement of water vapor and aerosols in the Earth’s atmosphere,” Appl. Opt. 31(16), 3068–3082 (1992). [CrossRef]   [PubMed]  

17. P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, (New York, 1992).

18. D. N. Whiteman, “Examination of the traditional Raman lidar technique. II. Evaluating the ratios for water vapor and aerosols,” Appl. Opt. 42(15), 2593–2608 (2003). [CrossRef]   [PubMed]  

19. OriginLab software user reference, “Window for low-pass parabolic filter,” http://wiki.originlab.com/~originla/wiki2/index.php?title=X-Function:Fft_filters.

20. P. T. Bergmaier, B. Geerts, Z. Wang, B. Liu, and P. Campbell, “A Dryline in Southeast Wyoming. Part II: Airborne In Situ and Raman Lidar Observations,” Mon. Weather Rev.in press.

21. LI–COR, Inc., “LI-6262 CO2/H2O Analyzer Instruction Manual,” http://www.mmm.ucar.edu/abl/forest/docs/licor_6262.pdf.

22. P. M. S. Inc, “Passive Cavity Aerosol Spectrometer Probe,” https://www.eol.ucar.edu/raf/Bulletins/B24/pcasp100.html.

23. J. R. Snider, S. Guibert, J. L. Brenguier, and J. P. Putaud, “Aerosol activation in marine stratocumulus clouds: 2. Köhler and parcel theory closure studies,” J. Geophys. Res. 108(D15), 8629 (2003). [CrossRef]  

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Figures (5)

Fig. 1
Fig. 1 Photograph of the Raman lidar inner structure.
Fig. 2
Fig. 2 Schematic diagram of the compact airborne Raman lidar.
Fig. 3
Fig. 3 A ten-minute segment of Raman lidar observations on June 21, 2010: (a) aerosol backscattering coefficients (β, km−1) plotted as10log10(β); (b) depolarization ratios; (c) water vapor mixing ratios; (d) vertical wind velocity (black) and temperature at the flight level. The two-abscissa axes are UTC time (upper) and distance (lower). The extending black region in a, b, and c indicate the surface below and the top line is just below the aircraft flight level. The wind was blowing from left to right at ~7 m/s while aircraft flying from south to north almost along the wind.
Fig. 4
Fig. 4 Lidar measured profiles of (a) water vapor mixing ratios and (b) aerosol backscattering coefficients measured at 17:41:30 June 21, 2010. Error bars show the standard deviation of measurement uncertainty. Note that the y-axis is range from the aircraft.
Fig. 5
Fig. 5 Comparisons of (a) water vapor mixing ratios and (b) aerosol backscattering coefficients measured by lidar (black) and in situ instruments (red) for the whole flight on June 22, 2010. (c) and (d) are scatter plots for the two comparisons measured 30 to 37.5 m below the aircraft.

Tables (1)

Tables Icon

Table 1 System parameters of the compact airborne Raman lidar

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

P = η P L β τ 2 r 2 , P = η P L β τ 2 r 2 ,
δ V = β β = C A P P ,
P E = K E ( β a + β m ) τ E 2 r 2 , P N = K N β N τ N 2 r 2 ,
β a = R P β N R K R T β m ,
WVM=C τ N P V τ V P N ,
σ β a 2 β a 2 = σ R P 2 R P 2 + σ R K 2 R K 2 + σ R T 2 R T 2 , σ WVM 2 WV M 2 = σ C 2 C 2 + σ P V 2 P V 2 + σ P N 2 P N 2 ,
W(f)={ 1,iff f C1 1 (f f C1 ) 2 ( f C2 f C1 ) 2 ,if f C1 f f C2 0iff f C1
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