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emission. Its radiance also called brightness Bν(T) is given by the Planck’s law: B(T,ν) = Bν(T) = 2hν3 c 2 1 , (327) exp(hν/kT)−1 where h is Planck’s constant, k is Boltzmann’s constant and c is the vacuum speed of light. The factor 2 in the numerator accounts for both polarizations according to the usual convention. The related emission from a real body – often called grey body – at the same temperature is AνBν(T) where Aν is the fraction of incident energy absorbed from a certain direction. In the general theory, scattering into and from other directions can lead to both losses and gains to the intensity along a given propagation direction and can be taken into account in both the terms Sν and αν. But if we assume a local thermodynamic equilibrium, so that each point into the elementary volume is characterized by the same temperature T(s), the strict requirement of balance between the energy absorbed and emitted by any particular volume element leads to Kirchoff’s law and for the S term we can suppose: Sν(s,T) = [αν(s)(1−Aν)]Bν(T). (328) For our application we will consider a no-scatter isotropic medium. In these hypothesis the source term Sν(s,T) expresses only the locally generated contribution to the radiation, and the absorption coefficient αν(s) becomes a local scalar characteristic of the medium that describes a true loss of energy from the radiation field into the medium. Moreover, for a perfectly reflecting or transmitting body, Aν is equal to zero and incident energy can be assumed to be redirected or pass through the body without being absorbed. Under these hypotheses for an upward looking radiometer we can rewrite Eq. (328) as: Sν(s,T) αν(s)Bν(T), (329) where Bν(T) is always given by Planck’s function Eq. (327). Operating in the microwave frequency range ν < 300 GHz, according to the Rayleigh- Jeans approximation of Planck’s law, by expanding the exponential term exp(hv/kT) and truncating to the second term of Eq. (327) exp(hv/kT) = 1+(hv/kT)+(hv/kT) 2 /2!+... ∼ = 1+(hv/kT) , we obtained for Bν(T) a well-known approximated linear form (see Fig. 173): from which Eq. (329) becomes: Bν(T) ∼ = 2kv2 T(s), (330) c2 Sν(s) = αν(s)Bν(T) ∼ = αν(s) 2kv2 T(s) (331) c2 Bν(T) is the surface brightness, which is the flow of energy across a unit area, per unit frequency, from a source viewed through free space in an element of solid angle. Since the brightness and the intensity have the same units (according to Fig. 174) the two are locally equivalent in this case. It follows: Iν(s) ≡ Bν(s) ∼ = 2kv2 T(s), (332) c2 in which Iν is expressed as linear function of its relative thermodynamic temperature T(s) as usually in the microwave frequency range formulation. SubstitutingEq. (331)formofSν(s) in Eq.(326)inthe hypothesisofnoscattering,in local thermodynamic equilibrium, plane horizontallystratified, perfectly transmitting (or reflecting) atmosphere we obtain the following radiance power transfer differential equation: dIν = −αν(s)Iν(s)ds+ 2kν2 c 2 αν(s)T(s)ds. (333) 262 <strong>DTU</strong> Wind Energy-E-Report-0029(EN)
Figure 173: Rayleigh-Jeans’ approximation assumed into the Planck’law Figure 174: The surface brightness Bν(T) of a blackbody emitter as viewed through free space in the solid-angle element dΩ produces a flow of energy given by the specific intensity Iν ≡ Bν Referring to the conventions reported in Figure 175(a), the previous differential equation provides the following integral solution: s0 Iν(0) = Iν(s0)exp − αν(s 0 ′ )ds ′ + 2kν2 c2 s0 s0 αν(s)T(s)exp − αν(s 0 0 ′ )ds ′ ds. (334) If we define the integral function of the absorption αν(s) along the path as optical depth τν(s) τν(s) s 0 αν(s ′ )ds ′ , (335) we can rewrite Eq. (334) in the following more readable form for the ground level measured <strong>DTU</strong> Wind Energy-E-Report-0029(EN) 263
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Remote Sensing for Wind Energy DTU
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Author: Alfredo Peña, Charlotte B.
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4 Introduction to continuous-wave D
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8 Nacelle-based lidar systems 157 8
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12 Complex terrain and lidars 231 1
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1 Remote sensing of wind Torben Mik
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Figure 2: Calibration, laboratory w
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Figure 3: Example of scatter plots
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1.2.3 Summary of sodars Most of tod
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1.3.3 Wind lidars Measuring wind wi
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Figure 6: CW wind lidars (ZephIRs)
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Further developments Furthermore, n
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2 The atmospheric boundary layer S
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Figure 9: Large spatial scale varia
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Du3 Dt Du1 Dt Du2 Dt The three mome
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Figure 13: Consensus relations betw
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ψ z L ∼ − 5 L . For unstable c
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Figure 15: Behavior of the turbulen
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Figure 17: Newly developed models t
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The value of q0 at the surface is d
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the spray is the source of icing on
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u∗2 u∗1 u1(h) = u∗1 k ln h
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Figure 26:Land-seabreeze system,whe
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Figure28:Three dimensionalpicture o
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sufficient information. Finally, we
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Mann, J. (1998) Wind field simulati
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(2010) and comparison under differe
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To get the velocity field from the
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τ(k) [Arbitrary units] 10 3 10 2 1
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(Koopmans, 1974; Bendat and Piersol
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anemometer was installed at each en
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and fSw(f) u 2 ∗ = 1.05n 1+5.3n 5
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and n = 0.468. This spectrum implie
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to be calculated. We do that on a m
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Notation A Charnock constant neutra
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Maxey M. R. (1982) Distortion of tu
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4.2 Basic principles of lidar opera
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4.2.5 Wind profiling in conical sca
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4.3.1 Behaviour of scattering parti
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the beam radius at the output lens.
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from which the value of VLOS is der
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the atmosphere. The SNR 4 for a win
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individual line-of-sight wind speed
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A general approach to mitigating th
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from ±VH sinδ (if the tilt is tow
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as a down draught (of the same abso
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Table 7: Combined results from 28 Z
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in Eastern Jutland between January
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Figure 56: Normalized power curves
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the concept. Developments include i
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References x horizontal position in
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Wagner R., Mikkelsen T., and Courtn
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5.2 End-to-end description of pulse
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Scanner Coherent lidar measure the
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Figure 62: Radial wind velocity ret
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transform in order to use data obta
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This wavelength is also the most fa
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Eq.(132)isadaptedforcollimatedsyste
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5.3.6 Existing systems and actual p
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(Gottshall et al., 2010; Albers et
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References Albers A., Janssen A. W.
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derived from fluctuations of the wi
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Acoustic received echo (ARE) method
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Figure 71: Sample time-height cross
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A gradient minimum is characterized
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Figure73:Bragg-relatedacoustic(belo
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stability (inversion strength) can
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Figure 76: Combined soundingwith a
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Figure 79: Favorite regions (shaded
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Direct detection of MLH from acoust
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Engelbart D.A.M.and Bange J. (2002)
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7 What can remote sensing contribut
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uyms 9.0 8.5 8.0 7.5 7.0 120 140 16
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Bottom of rotor Φ rotation r w u
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Height Height Hub 1.6 1.4 1.2 1.0 0
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PP rated PP rated 1.0 0.8 0.6 0.4 0
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KEprofileKEhub 1.2 1.1 1.0 0.9 0.8
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PP rated WS Lidarms 1.0 0.8 0.6 0.4
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8 Nacelle-based lidar systems Andre
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• Flexibletrajectories.Dependingo
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Figure 104: Sketch of simultaneous
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The normal wind direction vector nw
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Figure 109: Test site at DTU Wind E
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Figure 112: Power curve met mast an
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Notation C number of sent photons C
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9 Lidars and wind turbine control -
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for three unknowns, it is impossibl
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model of the blade pitch actuator,
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|GRL| [-] 1 0.8 0.6 0.4 0.2 10 k [r
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PSD(Ωg) [(rpm) 2 /Hz] PSD(Ωg) [
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0.04 0.03 ˆk [ rad m ] 0.02 0.63 0
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[%] 10 0 −10 −20 −30 MyT Moop
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PSD(θ1) [rad 2 /Hz] PSD(Moop1) [Nm
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Pel/Pel,max [-] 1 0.98 0.96 0.94 0.
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fL weighting function GRL transfer
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E. Hau, Windkraftanlagen, 4th ed. S
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¨¦¦§©¡§ ¥§¨¦¦§£ ¡¥
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vertical (m) 150 100 50 R d 0
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With feedback only, on the other ha
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Figure 136: Estimated preview requi
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Normalized C r = C r P Q 2 W (r) b
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Normalized C r = C r P Q 2 W (r) b
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Coherence 1 0.8 0.6 0.4 0.2 0 10
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Coherence 1 0.8 0.6 0.4 0.2 0 10
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Magnitude Squared 10 8 10 7 10 6 10
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Page 221 and 222: 11 Lidars and wind profiles Alfredo
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Page 225 and 226: z [−] zo 1 κ ln 40 38 36 34 32 3
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Page 239 and 240: Figure 163: Panahaiko: The scatter
Page 241 and 242: References Albers A. and Janssen A.
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Page 249 and 250: The theoretical systematic errors a
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Page 253 and 254: Height (m) Height (m) 160 140 120 1
Page 255 and 256: RMSPE (%) 70 60 50 40 30 20 10 0 vu
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Page 275 and 276: References s point in space Sν(s)
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Page 279 and 280: Christiansen et al., 2006). The res
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Page 283 and 284: Figure 188: Envisat ASAR wind field
Page 285 and 286: Figure 190: Envisat ASAR wind field
Page 287 and 288: Satellites in sun-synchronous polar
Page 289 and 290: 500 overlapping scenes for wind res
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Page 295 and 296: Ren Y. Z., Lehner S., Brusch S., Li
Page 297 and 298: tracking, climate studies, air-sea
Page 299 and 300: The modified logarithmic wind profi
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Page 303 and 304: QuikSCAT 25 20 15 10 5 Horns Rev Fi
Page 305 and 306: Figure 203:Example of an ASCAT coas
Page 307 and 308: References Bourassa M.A., Legler D.
Page 309: DTU Wind Energy Technical Universit
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