Sensitivity of Displaced-Beam Scintillometer Measurements

of Area-Average Heat Fluxes to Uncertainties in Topographic

Heights

Matthew Gruber (matthewgruber@gi.alaska.edu) and JavierFochesatto (foch@gi.alaska.edu)Department of Atmospheric Science, College of Natural Sciences and Mathematics,Geophysical Institute, University of Alaska Fairbanks

Oscar Hartogensis (oscar.hartogensis@wur.nl)Meteorology and Air Quality Group, Wageningen University, the Netherlands

Abstract. Displaced-beam scintillometer measurements of the turbulence inner-scale length lo and refractive index structure function C2

n resolve area-average turbu-lent fluxes of heat and momentum through the Monin-Obukhov similarity equations.Sensitivity studies have been produced for the use of displaced-beam scintillometersover flat terrain. Many real field sites feature variable topography. We develop herean analysis of the sensitivity of displaced-beam scintillometer derived sensible heatfluxes to uncertainties in spacially distributed topographic measurements. Sensitiv-ity is shown to be concentrated in areas near the center of the beam and where theunderlying topography is closest to the beam height. Uncertainty may be decreasedby taking precise topographic measurements in these areas.

Keywords: Displaced-beam scintillometer, Effective beam height, Scintillometererror, Scintillometer uncertainty, Turbulent fluxes

1. Introduction

Displaced-beam scintillometers are useful to make measurements ofturbulent heat fluxes since their footprint is much larger than othermeasurement devices such as sonic anemometers. Their source mea-surements are the index of refraction structure parameter C2

n and theturbulence inner scale length lo, which are used to infer the turbulentheat fluxes (Hill, 1988; Andreas, 1992). This inference follows fromequations from the Monin-Obukhov similarity hypothesis, which is amodel for turbulent fluxes of heat and momentum in the atmosphericsurface layer (Sorbjan, 1989). Studying the sensitivity of the derivedturbulent heat flux to the uncertainties in the source measurements isimportant, and several studies have been published exploring the case ofmeasurements over flat terrain (Andreas, 1989; Moroni et al., 1990; An-dreas, 1992; Solignac et al., 2009). There is debate over whether theMonin-Obukhov similarity hypothesis can be applied over variable ter-rain since the canonical model is for flat terrain, however many scin-

c© 2021 Kluwer Academic Publishers. Printed in the Netherlands.

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tillometer field campaigns take place over variable terrain (Hartogensiset al., 2002; Meijninger et al., 2002). The turbulent fluxes are derivedthrough equations which extend the Monin-Obukhov equations fromflat terrain to variable terrain via the application of an effective beamheight (Hartogensis et al., 2003; Kleissl et al., 2008; Evans and deBruin,2011; Geli et al., 2012). The effective beam height is calculated assum-ing that the Monin-Obukhov surface layer profiles in height are validover the slowly varying terrain. The research question to be asked is, ifone can safely extend the Monin-Obukhov equations from flat terrainto variable terrain, then in what manner is the error propagated fromthe uncertainty in topography to the derived turbulent heat flux?

Through the assumptions of the Monin-Obukhov similarity hypoth-esis, the sensible heat flux in the atmospheric surface layer is givenby

HS = −ρcpu?T?, (1)

where HS is the sensible heat flux, ρ is the density of air, cp is theheat capacity, u? is the friction velocity, and T? is the temperaturescale. Resolving u? and T? from C2

n and lo starts at another similarityequation relating ζ ≡ zeff/L to T? and u?:

ζ =κGT?zeffu2?T

, (2)

where zeff is the effective beam height of the scintillometer beam,L is the Obukhov length, κ is the Von Karman constant, G is theacceleration due to gravity, and T is the temperature (Hartogensis etal., 2003). In equation 2 we omit the term due to humidity seen inAndreas (1992) for simplicity as it has no effect on the results here.

When ζ > 0 we are in a stable atmospheric surface layer. In thestable case we have

C2T z

2/3eff

T 2?

= a(1 + cζ2/3)→ ±T? =

√C2T

a

z1/3eff√

1 + cζ2/3, (3)

zeff =

(∫ 1

0z(u)−2/3G(u)du

)−3/2

, (4)

u3? =

κzeff ε

(1 + hζ3/5)3/2→ u2

? =κ2/3z

2/3eff ε

2/3

(1 + hζ3/5), (5)

where a, c and h are empirical constants (Wyngaard et al., 1971;Wyngaard and Cote, 1971), z(u) is the height profile of the beam above

BLM2db1.tex; 18/06/2021; 0:04; p.2

3

the ground where u is the normalized distance along the beam, G(u) isthe path weighting function , C2

T is the temperature structure param-eter which is inferred from C2

n, and zeff is the effective beam heightassuming that the C2

T profile satisfies equation 3 for any z along z(u)(Andreas, 1992; Hartogensis et al., 2003). The turbulent dissipationrate ε is directly related to lo by

lo =(9Γ(1/3)KD(ρ, T ))3/4

ε1/4, (6)

where Γ is the Gamma function, K is the Obukhov-Corrsin constantand D(ρ, T ) is the thermal diffusivity of air (Andreas, 1992).

When ζ < 0 we are in an unstable atmospheric surface layer. In theunstable case we have

C2T z

2/3eff

T 2?

=a

(1− bζ)2/3→ ±T? =

√C2T

az

1/3eff (1− bζ)1/3,(7)

zeff =zeff2bζ

1−

√√√√1− 4bζ

zeff

[∫ 1

0z(u)−2/3

(1− bζ z(u)

zeff

)−2/3

G(u)du

]−3/2 ,(8)

u3? =

κzeff ε

(1 + d(−ζ)2/3)3/2→ u2

? =κ2/3z

2/3eff ε

2/3

(1 + d(−ζ)2/3),(9)

where b and d are empirical constants, and zeff is the effective beamheight assuming that the C2

T profile satisfies equation 7 for any z alongz(u) (Hartogensis et al., 2003).

Error is propagated from the source measurements to the derivedvariable HS via the error propagation equation

σf =N∑i=1

(∂f

∂xi

)σxsi +

√√√√ N∑i=1

(∂f

∂xi

)2

σ2xri

+ σfc , (10)

where the general derived variable f is a function of general sourcemeasurement variables x1, x2, ..., xN with respective systematic errorσxs1 , σxs2 , ..., σxsN and with respective independent Gaussian distributeduncertainties with standard deviations σxr1 , σxr2 , ..., σxrN as seen inTaylor (1997). Computational error is given by σfc . The first and lastterms represent an offset from the true solution, whereas the centralterm is a measure of the width of the error bars.

Error propagation can be studied with Monte-Carlo methods as inMoroni et al. (1990), however this is best reserved for cases where aclosed-form error propagation equation cannot be derived. We will seek

BLM2db1.tex; 18/06/2021; 0:04; p.3

4

a closed-form equation. As a start, it is practical for the purpose of asensitivity study to rewrite Eq. 10 as

σff

=

N∑i=1

Sf,xiσxsixsi

+

√√√√ N∑i=1

S2f,xi

σ2xri

xri2

+σfcf, (11)

where Sf,x are unitless sensitivity functions defined by

Sf,x ≡x

f

(∂f

∂x

). (12)

The sensitivity functions are each a measure of the portion of theerror in the derived variable f resulting from error on each individ-ual source measurement x (Andreas, 1992). Our goal is to evaluateSHS ,z(u), where the height profile z(u) is distributed, hence functionalderivatives will be used as in Gruber et al. (2014). We have

SHS ,z(u) =z(u)

HS

(δHS

δz(u)

). (13)

The sensitivity function SHS ,z for this measurement strategy overflat terrain has been evaluated in Gruber and Fochesatto (2013).

We will solve for the sensitivity function in equation 13 for stableconditions in section 2. In section 3 we will solve for the sensitivityfunction in equation 13 for unstable conditions. In section 4 we willapply the sensitivity function to the topography of a real field site. Weconclude in section 5.

2. Stable case (ζ > 0)

Here we can combine equations 2, 3 and 5 to arrive at

ζ2 + cζ8/3 = A(1 + hζ3/5)2z4/3eff , (14)

where

A ≡κ2/3g2C2

T

T 2aε4/3. (15)

BLM2db1.tex; 18/06/2021; 0:04; p.4

5

Fig

ure

1.

Vari

able

inte

r-dep

enden

cytr

eedia

gra

mfo

rHS

under

stable

condit

ions

corr

esp

ondin

gto

ζ>

0.

The

tree

dia

gra

mfo

rth

edep

enden

ceofε

onP

,T

andl o

isom

itte

d.

Sin

cez eff

isd

eter

min

edin

dep

end

entl

yfr

omeq

uat

ion

4,eq

uat

ion

14

isa

sin

gle

equ

ati

onin

the

sin

gle

un

kn

own

ζ,

wh

ere

all

oth

erva

riab

les

inth

eeq

uat

ion

are

mea

sure

d.

Fro

mth

iseq

uati

onw

eca

nd

eter

min

eth

eva

riab

lein

ter-

dep

end

ency

as

illu

stra

ted

inth

etr

eed

iagr

amse

enin

figu

re1.

Fro

mth

etr

eed

iagr

am

seen

infi

gure

1,w

eh

ave

( δHS

δz(u

)) =∂HS

∂T?

[ ( ∂T?

∂z eff

) ζ

+

( ∂T?

∂ζ

)(∂ζ

∂z eff

)] (δzeff

δz(u

)) +∂HS

∂u?

[ ( ∂u?

∂z eff

) ζ

+

( ∂u?

∂ζ

)(∂ζ

∂z eff

)] (δzeff

δz(u

)) .(16

)

BLM2db1.tex; 18/06/2021; 0:04; p.5

6

Man

yof

thes

ed

eriv

ativ

esfo

llow

dir

ectl

yfr

omth

ed

efin

itio

ns.

For( ∂ζ ∂

z eff

) we

mu

stim

pli

citl

yd

iffer

enti

ate

bot

h

sid

esof

equ

atio

n14

toar

rive

at ( ∂ζ

∂z eff

) =4(ζ

2+cζ

8/3)

3zeff

( 2ζ+

8 3cζ

5/3−

6 5h( ζ8/5

−cζ

34/15

1+hζ

3/5

)) .(1

7)

We

then

achie

ve

z(u

)

HS

( δHS

δz(u

)) =

2 3−(

cζ2/3

3(1

+cζ

2/3)

+3hζ3/5

10(1

+hζ3/5)

) 4 3(1

+cζ

2/3)

2+

8 3cζ

2/3−

6 5h( ζ3/5

−cζ

19/15

1+hζ

3/5

) z(u

)−2/3G

(u)

∫ 1 0z(u

)−2/3G

(u)du.

(18)

3.

Unstable

case

(ζ<

0)

Her

ew

eco

mb

ine

equ

atio

ns

2,7

and

9;w

eac

hie

ve

ζ

z eff

=−A

Φ(ζ

),(1

9)

BLM2db1.tex; 18/06/2021; 0:04; p.6

7

where

A =

√κg3/2(C2

T )3/4

T 3/2εa3/4, (20)

Φ(ζ) =

√(1− bζ)(1 + d(−ζ)2/3)3

(−ζ). (21)

In the unstable case, zeff is coupled to ζ through equation 8. Weinput equation 19 into equation 8 to arrive at

ζ =1

2b

1 −

√1 + 4bAΦ(ζ)

[∫ 1

0

(z(u) + bz(u)2AΦ(ζ)

)−2/3

G(u)du

]−3/2 .(22)

This is a single equation in the single unknown ζ, where all othervariables are measured. This equation is in fixed point form and canbe solved numerically via fixed point recursion as seen in Traub (1964)and in Agarwal et al. (2001). The variable interdependency is mappedout in the tree diagram seen in figure 2.

BLM2db1.tex; 18/06/2021; 0:04; p.7

8

Fig

ure

2.

Vari

able

inte

r-dep

enden

cytr

eedia

gra

mfo

rHS

under

unst

able

condit

ions

corr

esp

ondin

gto

ζ<

0.

The

tree

dia

gra

mfo

rth

edep

enden

ceofε

onP

,T

andl o

isom

itte

d.

Fro

mth

etr

eed

iagr

amse

enin

figu

re2

we

hav

e

( δHS

δz(u

)) =∂HS

∂T?

[ ( ∂T?

∂ζ

) zeff

+

( ∂T?

∂z eff

)( ∂z eff

∂ζ

)] (δζ δz

(u)) +

∂HS

∂u?

[ ( ∂u?

∂ζ

) zeff

+

( ∂u?

∂z eff

)( ∂z eff

∂ζ

)] (δζ δz

(u)) .

(23)

Fro

meq

uat

ion

19w

eh

ave

BLM2db1.tex; 18/06/2021; 0:04; p.8

9

∂z eff

∂ζ

=z eff

1 ζ−

( ∂Φ ∂ζ

)Φ

(ζ)

,(2

4)

an

dfr

om

equat

ion

21w

eh

ave

( ∂Φ ∂ζ

) =1

2Φ(ζ

)

[ (1+d(−ζ)2/3)3

+2dζ(1−bζ

)(1

+d(−ζ)2/3)2

(−ζ)−

1/3

ζ2

] .(2

5)

Fro

meq

uat

ion

22w

eh

ave

( δζ δz(u

)) =−( δf δ

z(u

)) ζ( ∂f ∂

ζ

) +4b(

1−

2bζ),

(26)

wh

ere

f( A

,z(u

),ζ(A,z

(u))) ≡

1+

4bA

Φ(ζ

)

[ ∫ 1 0

( z(u

)+bA

Φ(ζ

)z(u

)2) −2/

3G

(u)du

] −3/2

.(2

7)

We

thu

sh

ave

z(u

)

HS

( δHS

δz(u

)) =z(u

)

2 3

1 ζ−

( ∂Φ ∂ζ

)Φ

(ζ)

+d

3(1

+d(−ζ)2/3)(−ζ)1/3−

b

3(1−bζ

) ( δζ δz(u

)) ,(2

8)

BLM2db1.tex; 18/06/2021; 0:04; p.9

10

from

equ

atio

n23

,w

her

e( δζ δ

z(u

)) isso

lved

thro

ugh

equ

atio

n26

as

( δζ δz

(u)) =

−A

Φ(ζ

)( z(u

)+bA

Φ(ζ

)z(u

)2) −5

/3( 1

+2bAz(u

)Φ(ζ

)) G(u

)

A( ∂Φ ∂ζ

)[ ∫1 0

( z(u)

+bA

Φ(ζ

)z(u

)2) −2

/3G

(u)du

] +bA

2Φ

(ζ)( ∂Φ ∂ζ

)[ ∫1 0

( z(u)

+bA

Φ(ζ

)z(u

)2) −5

/3z(u

)2G

(u)du

] +(1

−2bζ)

[ ∫ 1 0

( z(u)

+bA

Φ(ζ

)z(u

)2) −2

/3G

(u)du

] 5/2.(29)

4.

Resu

lts

We

hav

eso

lved

the

sen

siti

vit

yfu

nct

ionSHS,z

(u)

wh

ich

dep

end

son

z(u

)an

dζ

for

bot

hst

able

con

dit

ion

sw

her

eζ>

0an

dfo

ru

nst

ab

leco

nd

itio

ns

wh

ereζ<

0.In

ord

erto

vis

ual

ize

the

sen

siti

vit

yfu

nct

ion

,w

ew

ill

dem

onst

rate

itw

ith

afi

eld

site

bea

mp

ath

hei

ght

pro

filez(u

).W

eu

seth

eIm

nav

ait

bas

infi

eld

site

seen

infi

gure

3.

BLM2db1.tex; 18/06/2021; 0:04; p.10

11

Easting UTM 5N 2011 (m)

Nor

thin

g U

TM

5N

201

1 (m

)

6.5025 6.503 6.5035 6.504 6.5045 6.505

x 105

7.6152

7.6154

7.6156

7.6158

7.616

7.6162

x 106

927

928

929

930

931

932

933

934

935

936

937

938Elevation Above Sea Level (m)Beam PathTransmitter and Receiver Position

Figure 3. Imnavait basin topography and beam path. Imnavait basin is located onthe North Slope of Alaska. The beam emitter and receiver are raised on tripods1.8m high.

BLM2db1.tex; 18/06/2021; 0:04; p.11

12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

Normalized Beam Distance u (unit−less)

Path Weighting Function G(u) (unit−less)Imnavait Path Beam Height Above Topography z(u) (m)

Figure 4. Imnavait basin beam path heights above topography. Imnavait basin islocated on the North Slope of Alaska. The beam emitter and receiver are raised ontripods 1.8m high.

This field site has a beam height z(u) as seen in figure 4. A randomerror component of 0.5m is used in figure 4 since the difference betweenground truth GPS measurements and the digital elevation map usedhas a standard deviation of approximately 0.5m as seen in figure 5.Note that the magnitude of the error does not influence the results ofthis study.

BLM2db1.tex; 18/06/2021; 0:04; p.12

13

Figure 5. Imnavait basin ground truth GPS measurements. Overlaid is a histogramof the difference between the GPS measurements and the digital elevation map usedin figures 3 and 4. The standard deviation is approximately 0.5m.

For the Imnavait basin path z(u) seen in figure 4, taking equation18 for the stable case, and equations 28 and 29 for the unstable case,we arrive at the sensitivity function seen in figure 6. Note that valuesof A for a given ζ in equation 29 follow from the numerical solution toequation 22.

BLM2db1.tex; 18/06/2021; 0:04; p.13

14

u

ζ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10

1

−100

−10−1

−10−2

u

ζ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

−2

10−1

100

101

0

1

2

0

1

2

Figure 6. Sensitivity function SHS ,z(u) over the Imnavait basin field site seen infigure 3 and in figure 4.

The sensitivity gets higher from stable conditions to unstable con-ditions, and it is focused in areas near the center of the beam path,as well as in areas of topographic protrusion. Note the clear spikes insensitivity at u = 0.38 and at u = 0.5 in figure 6. These points inthe path correspond to local minima in z(u) as seen in figure 4. Thismakes sense since C2

T decreases nonlinearly in height above the ground,most rapidly near the surface. In areas where the beam approaches theground, the gradient in C2

T is higher as seen in equations 3 and 7,therefore uncertainties in the actual height of the beam in those areaswill translate into high uncertainties in the derived variables.

Note that if we consider a constant ratio of σz(u)z(u) , the term in, for

example Eq. (11), can be re-written as

1∫0

σz(u)

z(u)SHS ,z(u)du =

σz(u)

z(u)

1∫0

SHS ,z(u)du

, (30)

where the term in square brackets is plotted in figure 7.

BLM2db1.tex; 18/06/2021; 0:04; p.14

15

−101

−100

−10−1

−10−2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

∫ 01 SH

S,z

(u)

du

10−2

10−1

100

101

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

∫ 01 SH

S,z

(u)

du

Figure 7. Sensitivity function SHS ,z(u) integrated over u for any field site.

This result is the average value of the sensitivity along the wholelength of the path, and it converges to the same result for all paths. Thesensitivity function in figure 7 is compatible with the one-dimensionalsensitivity function for flat terrain seen in Gruber and Fochesatto (2013).

5. Conclusion

Using the effective beam height extension of the Monin-Obukhov sim-ilarity equations to variable terrain, we have solved for how uncer-tainty in topographic heights propagates to displaced-beam scintillome-ter measurements of turbulent heat fluxes over any field site. We havefound that uncertainty is concentrated in areas around topographicprotuberances, as well as near the center of the beam path as seen infigures 4 and 6. The local sensitivity can easily approach values of 200%in unstable conditions as seen in figure 6, but the average sensitivityover the beam path never exceeds 100% as seen in figure 7. These

BLM2db1.tex; 18/06/2021; 0:04; p.15

16

results carry important ramifications in the selection of beam paths, inthe calculation of uncertainty, and in the type of topographic data used.It may be that for many scintillometer beam paths, in order to achievereasonable uncertainty we must use high precision LIDAR topographicdata in order to reduce what is likely the greatest contributor to overalluncertainty.

It is interesting that the average value of the sensitivity functionSHS ,z(u) over the beam path reduces to the identical sensitivity func-tion SHS ,z for diplaced-beam scintillometers over flat terrain as seenin figure 7 and in Gruber and Fochesatto (2013). We have essentiallyexpanded the flat terrain sensitivity function first explored in Andreas(1992) from one dimension to two, and then we averaged through onedimension to arrive back at the original one-dimensional sensitivityfunction. Future work should perhaps focus on applying this type ofsensitivity analysis on large aperture scintillometers using the Businger-Dyer relation to obtain path averaged u?, T? and HS measurements.Additional work should be performed for scintillometer paths whichare below the blending height over heterogeneous terrain (Wieringa,1986; Mason, 1987; Claussen, 1990; Claussen, 1995; Meijninger et al.,2002; Hartogensis et al., 2003; Lu et al., 2009).

Acknowledgements

Matthew Gruber thanks the Geophysical Institute for its support dur-ing his Master’s degree program in Atmospheric Sciences at the Univer-sity of Alaska Fairbanks. We thank Jason Stuckey and Randy Fulweberat ToolikGIS, Chad Diesinger at Toolik Research Station, and MattNolan at the Institute for Northern Engineering for the digital elevationmap of Imnavait, data support, field site GPS measurements and figure5.

References

Agarwal RP,Meehan M,O’Regan D (2001) Fixed Point Theory and Applications.Cambridge University Press, Cambridge, 184 pp

Andreas EL (1989) Two-Wavelength Method of Measuring Path-Averaged Turbu-lent Surface Heat Fluxes. J Atmos Oceanic Tech 6:280–292.

Andreas EL (1992) Uncertainty in a Path Averaged Measurement of the FrictionVelocity u?. J Appl Meteorol 31:1312–1321

Claussen M (1990) Area-Averaging of Surface Fluxes in a Neutrally Stratified,Horizontally Inhomogeneous Atmospheric Boundary Layer. Atmos Eviron24a:1349–1360

BLM2db1.tex; 18/06/2021; 0:04; p.16

17

Claussen M (1995) Flux Aggregation at Large Scales: On the Limits of Validity ofthe Concept of Blending Height. J Hyrol 166:371–382

deBruin HAR, Meijninger WML, Smedman A-S, Magnusson M (2002) Dispaced-Beam Small Aperture Scintillometer Test. Part I: The WINTEX Data-Set.Boundary-Layer Meteorol 105:129–148

Evans J, deBruin HAR (2011) The Effective Height of a Two-WavelengthScintillometer System. Boundary-Layer Meteorol 141:165–177

Geli HME, Neale CMU, Watts D, Osterberg J, deBruin HAR, Kohsiek W, PackRT, Hipps LE (2012) Scintillometer-Based Estimates of Sensible Heat FluxUsing Lidar-Derived Surface Roughness. J Hydrometeorol 13:1317–1331

Gruber M, Fochesatto GJ (2013) A New Sensitivity Analysis and Solution Methodfor Scintillometer Measurements of Area-Averaged Turbulent Fluxes. Boundary-Layer Meteorol, DOI 10.1007/s10546-013-9835-9

Gruber M, Fochesatto GJ, Hartogensis OK (2014) Sensitivity of Large-ApertureScintillometer Measurements of Area-Averaged Heat Fluxes to Uncertainties inTopographic Heights. Atmos Meas Tech, 7:33–68, doi:10.5194/amtd-7-33-2014

Hartogensis OK, deBruin HAR, Van de Wiel BJH (2002) Dispaced-Beam SmallAperture Scintillometer Test. Part II: CASES-99 Stable Boundary-Layer Exper-iment. Boundary-Layer Meteorol 105:149–176

Hartogensis OK, Watts CJ, Rodriguez J-C, deBruin HAR (2003) Derivation of anEffective Height for Scintillometers: La Poza Experiment in Northwest Mexico.J Hydrometeorol 4:915–928

Hill RJ (1988) Comparison of Scintillation Methods for Measuring the Inner Scaleof Turbulence. Appl Optics 27(11):2187–2193

Kleissl J, Gomez J, Hong S-H, Hendrickx JMH, Rahn T, Defoor WL (2008) LargeAperture Scintillometer Intercomparison Study. Boundary-Layer Meteorol128:133–150

Lu L, Liu S, Xu Z, Yang K, Cai X, Jia L, Wang J (2009) The Characteristicsand Parameterization of Aerodynamic Roughness Length over HeterogeneousSurfaces. Adv Atm Sci 26:180–190

Mason PF (1987) The Formation of Areally-Averaged Roughness Lengths. QuartJ Roy Met Soc 114:339–420

Meijninger WML, Hartogensis OK, Kohsiek W, Hoedjes JCB, Zuurbier RM, deBruinHAR (2002) Determination of Area-Averaged Sensible Heat Fluxes With ALarge Aperture Scintillometer Over a Heterogeneous Surface - Flevoland FieldExperiment. Boundary-Layer Meteorol 105:37–62

Moroni C, Navarra A, Guzzi R (1990) Estimation of the Turbulent Fluxes inthe Surface Layer Using the Inertial Dissipative Method: a Monte Carlo ErrorAnalysis. Appl Optics 6:2187–2193

Solignac PA, Brut A, Selves J-L, Beteille J-P, Gastellu-Etchegorry J-P, Keravec P,Beziat P, Ceschia E (2009) Uncertainty Analysis of Computational Methods forDeriving Sensible Heat Flux Values from Scintillometer Measurements. AtmosMeas Tech 2:741–753

Sorbjan Z (1989) Structure of the Atmospheric Boundary Layer. Prentice-Hall,Englewood Cliffs, New Jersey, 317 pp

Taylor J (1997) An Introduction to Error Analysis: The Study of Uncertaintiesin Physical Measurements, 2nd edition. University Science Books, Sausalito,California, 327 pp

Traub JF (1964) Iterative Methods for the Solution of Equations. Prentice-Hall,Englewood Cliffs, New Jersey, 310 pp

BLM2db1.tex; 18/06/2021; 0:04; p.17

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Wieringa J (1986) Roughness-Dependent Geographical Interpolation of SurfaceWind Speed Averages. Quart J Roy Met Soc 112:867–889

Wyngaard JC, Izumi Y, Collins Jr. SA (1971) Behavior of the Refractive IndexStructure Parameter Near the Ground. J Opt Soc Amer 61:1646–1650

Wyngaard JC, Cote OR (1971) The Budgets of Turbulent Kinetic Energy andTemperature Variance in the Atmospheric Surface Layer. J Atmos Sci 28:190–201

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