224 D.I. Cooper et al. Agricultural and Forest Meteorology 114 2003 213–234
4. Spatial analysis of sampling size
The standard sampling size for lidar derived LE flux estimates previously used an empirical 25 m × 25 m
region. However that sample size was found to under- estimate the flux; a 75 m × 75 m footprint was found
to produce better LE estimates Cooper et al., 2000
. To quantitatively determine the spatial extent of the
average horizontal flux source area, four techniques were employed using lidar data, point-sensors derived
time series data, and a model, 1 an analysis of lidar derived LE flux as a function of source-area size; 2
an analysis of w and q co-spectra from local eddy co- variance measurements; 3 analysis from a footprint
model; and 4 integral length scales.
4.1. Normalized flux and horizontal average length To evaluate the individual weighted contributions
from sources at different positions throughout the lidar field-of-view and to infer the total spatial flux of that
region, we truncated the range-height scans for differ- ent lengths and examined the fractional contribution
to the flux as a function of the horizontal sampling length.
The gradient-based M–O latent energy estimation method for lidar requires input of u
∗
, L, air density, and a segment of lidar data extracted from a series
of vertical scans averaged and interpolated over space and time see
Section 3.1 . From these data, a profile of
water vapor is generated and fitted with a M–O-derived water vapor distribution estimate, allowing latent en-
ergy and q
∗
to be calculated, as well as a coeffi- cient of determination r
2
for the best-fit between the lidar-derived moisture profile and the M–O-predicted
profile. Examples of vertical profiles fitted with an M–O
water vapor distribution are shown in Fig. 6
. These plots show the lidar data as points, a 20-point
smoothed profile as large filled circles, and the model as a thick line. It appears from the two examples that
the M–O-derived profiles are somewhat drier above 4 m than the lidar derived moisture distributions. This
dry bias is clear in a comparison of the 20-point smoothed profile and the M–O-derived distribution,
as they significantly diverge differences ≥0.25 gkg above 5 m. The for the best-fit for the two examples
is shown in the upper right-hand corner of each plot. The upper panel has an r
2
of 0.93 for a 50 m diameter footprint; in contrast, the r
2
for a 90 m diameter foot- print is considerably lower at 0.77. These statistics
are almost counter-intuitive, as the expected result would be that the larger the number of data points
averaged together, the better the fit. The effect of spatial sample size can also be seen in the smoothed
profile, as the large horizontal sample length pro- file shows considerable variability when compared
with the 50 m diameter sampling length profile. The degraded fit for the larger footprint indicates that
over-sampling 1 is reflected in the spatial variability of the vertical profile moisture by 50 larger dy-
namic range of mixing ratio; 2 does not alleviate the dry bias of M–O derived profiles when compared
with lidar-derived profiles and 3 does not appear to improve flux estimates relative to those made from
tower-based measurements. Following
McNaughton and Laubach 2000
, the dry bias in the M–O-derived profile appears to result from moist microscale con-
vective eddies rising from the surface and mixing with advected dry air that is representative of the arid
regions surrounding the Bosque see Section 3.2
. To evaluate optimal horizontal source-area averages
for lidar-based latent energy estimates, we performed spatial analyses using a ratio of lidar derived fluxes
to tower-based flues normalized flux.
A normalized LE flux LE
N
was calculated as LE
N
L
HL
= LE
L
LE
T
r
2
6 where LE
L
is the M–O-derived LE flux for a given horizontal lidar flux sampling length L
HL
, LE
T
is the tower-based eddy covariance-derived flux, and r
2
is the coefficient of determination derived from the M–O
best-fit to the lidar-derived moisture profile. We note that different L
HL
’s can in some cases produce the same lidar derived flux estimate. However, because the
quality of the fit between the theoretical M–O profile and the actual data varies substantially See
Fig. 4 , r
2
can be used as a weighting factor to separate similar flux values with differing fits. The set of L
HL
’s used in this study was from 5 to 105 m in 10 m increments
or lags. The hypothesis is that a properly sampled lidar
derived flux estimate will cover the same source re- gion as the tower instruments. In contrast, spatially
under- or over-sampled flux estimates should show
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225 Fig. 6. Vertical profiles acquired on day of year 254 at 1341 LST showing the distribution of water vapor as a function of footprint size. Small dots represent lidar data,
while the large filled circles represent a 20 point average. A Monin–Obukhov estimated water vapor distribution was fitted to the profiles thick line and the coefficient of determination r
2
is shown for the region between the top of the canopy and 4.2 m.
226
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Fig. 7. Lidar derived normalized fluxes as a function of horizontal lags representing source-area diameters from 5 to 105 m at three different stability conditions.
D.I. Cooper et al. Agricultural and Forest Meteorology 114 2003 213–234 227
diverging values from the tower-based eddy covari- ance ‘truth-set’.
Using the normalized flux, we evaluated the hori- zontal average sample length of LE over two orders
of magnitude, from between 5 and 100 m, and over various stability conditions
Fig. 7 . We found that for
relatively unstable z − dL values −0.48, the LE source area appears to be relatively broad. In contrast,
small z−dL values −0.13 influence the horizontal averaging length, exhibiting a more pronounced and
narrow normalized-flux region. Further, under stable conditions, when z − dL = +0.16, the horizontal
averaging length appears, to be both broader and flat- ter then the curves derived under unstable conditions.
From this analysis, we conclude that as the convective activity decreases, surface–atmosphere coupling de-
creases, leading to lower normalized fluxes at larger horizontal average lengths 80 m. Furthermore, as
convection decreases, it appears that the source region increases in size. At more modest lengths, 30–60 m,
the source area was better matched to the rough- ness elements and wind speed thus optimizing the
surface–atmosphere coupling, which in turn leads to high-normalized flux values. If the lags are too small,
there will not be enough data to generate a statistically reliable flux value. These analyses suggest that 40 m ×
40 m 1600 m
2
to 60 m × 60 m 3600 m
2
sample size at 30 min averaging periods is reasonable for flux map-
ping in uniform terrain conditions. These empirical, lidar-data-based results are comparable to those from
footprint models, as described in the next section.
4.2. Theoretical footprint analysis We used a Lagrangian model described extensively
in the literature to predict the water vapor mixing ratio profile, latent energy flux, and footprints for a
crosswind line source within the surface layer of the ABL
Leclerc et al., 1997 . These flux footprints were
estimated from the upwind sources of moisture that would be measured at a point in space with the cross-
wind integrated probability density function of mois- ture transported from the surface
Baldocchi, 1997; Leclerc et al., 1997
. The vertical eddy flux density, F is defined as
F x, z =
x
Qx
′
f x
′
, z dx
′
7 where x is the stream-wise distance, z the observa-
tion or measurement height, x
′
the upwind distance, Q
x
′
the upwind surface source density, and fx
′
, z
is the crosswind-integrated footprint probability density function. We note that the general footprint
calculation used here Eq. 7
also includes specific corrections for atmospheric stability. Specific numer-
ical solutions to fx
′
, z are found in Leclerc et al.
1997; Leclerc and Thurtell 1990 and
Schuepp et al. 1990
describing the original work on the sub- ject. A plot of the relationship between z − dL
and the cumulative flux is shown in Fig. 8
, where each symbol represents a selected downwind distance
or lag as a function of z − dL and cumulative flux, and the lines connecting the points are sec-
ond order polynomial best-fits. As the atmosphere becomes more convectively active, the footprints be-
come smaller; at z − dL = −0.6, 90 of the cumulative LE flux is within a 60 m diameter source
region. Conversely, at z − dL = −0.1, 81 of the cumulative LE flux is accounted for within a 90 m
diameter footprint. Further, under stable conditions the footprints become considerably larger and the cu-
mulative flux for a given footprint becomes relatively small when compared with unstable conditions in the
surface layer. By extrapolating the family of curves shown in
Fig. 8 to increasingly unstable conditions,
the cumulative flux reaches an asymptote at 93 when z − dL = −1.1 suggesting that there is a
limit to the effect of vertical coupling on footprint size.
Flux footprints were calculated for a series of L, u
∗
and wind–field values measured at the Bosque. In modestly convective conditions, such as when
L = − 10 and z − dL = −0.48, the footprint
lag length analogous to the L
HD
was between 40 and 60 m and represented cumulative fluxes between
82 and 88, respectively, while peak fractional flux occurred at 10 m
Fig. 8 , Inset A. Under stable
conditions L = +13 the footprints were consid- erably longer with cumulative fluxes only 31–41
defined by the 40–60 m lengths which was less than half of the values calculated for unstable con-
ditions, although the fractional fluxes also peaked under 10 m. When L was highly negative −79
the footprint lag length between 40 and 60 m rep- resented cumulative fluxes between 65 and 74,
respectively.
228
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Fig. 8. Relationship between the scaled Monin–Obukhov length and the cumulative flux from the footprint model, lines between the data points are 2nd order fitted polynomials. Inset A is a plot of the cumulative and fractional flux as a function of downwind distance for a selected Monin–Obukhov length. Note that most of the fractional flux
footprint is under 60 m.
D.I. Cooper et al. Agricultural and Forest Meteorology 114 2003 213–234 229
4.3. Co-spectra analysis for temporal and spatial scales
For the construction of relevant co-spectra that was coincident with lidar data, we used standard microme-
teorological techniques to process selected time-series values from the three-dimensional sonic anemome-
ter and Krypton hygrometer located above the canopy
Stull, 1988; Kaimal and Finnigan, 1994 . The raw
data were screened for continuity and stationarity, lin- ear trends were removed, and finally the time-series
data were multiplied by a tapered window to create a conditioned time series data set. The conditioned data
set consisted of 20 Hz sampled values of u, v, w, and q
. The data were transformed from time-domain to frequency-domain by discrete Fourier analysis. Spec-
tral density distributions for water vapor and verti- cal wind were computed, and periods of interest were
further processed into the co-spectra. The co-spectra were normalized by the covariance of w and q for
the comparison of frequency between various times
Prueger et al., 2000 . Frequency information from the
co-spectra were transformed into the spatial domain by invoking Taylor’s hypothesis by assuming that the
temporal turbulence statistics are equivalent in space as:
1 f ¯
u 2π
8 where f is the co-spectra frequency, ¯
u is mean wind,
and 2π is used to convert from angular to linear di- mensions.
The co-spectra results were used to evaluate the temporal and spatial length scales associated with
the fractional contribution to latent energy flux. We assumed that spatial and temporal length scales as-
sociated with co-spectra are directly related to the source-area contribution to the flux integrated in time
over 30 min.
Two sample co-spectra with z − dL of −0.20 Fig. 8A
and +0.16 Fig. 8B
illustrate some of the surface–atmosphere processes occurring at the
Bosque. The results shown in Fig. 9A
are typical of the unstable conditions reflected by the co-spectra pro-
cessed from the Bosque data set. The peak frequency is approximately 0.04 s
− 1
; when translated into tem- poral and spatial scales, the peak represents 25 s and
191 m, respectively. The bulk of the energy-containing
Fig. 9. Co-spectra of w and q acquired from the north tower eddy covariance-time series during selected unstable A and stable B
periods.
length scales were between 50 and 200 m, with little contribution above 225 m. Spectra from locally stable
periods shown in Fig. 9B
were somewhat different from the unstable periods. The frequency for peak
power density was lower at 0.0263 s
− 1
or 38 s and 207 m for the time and space scales. Our analyses
suggest that latent energy fluxes are dominated by rel- atively small microscale processes, with eddy lengths
less than 200 m under convective conditions at the Bosque. The source area under stable conditions ap-
pears to be longer than that under unstable conditions.
Our analyses also indicate that 10–15 min sampling periods are adequate for measuring both eddy covari-
ance and lidar-based LE over the Bosque. Eddies mov- ing at mid-range time scales of between 5 and 40 s
contribute most of the energy involved in the verti- cal transport of latent energy. Fundamental turbulence
simulations at 2 m and 8 Hz resolution also support these results, indicating that lidar scan rates, while
slow, are adequate for mapping the moisture field. Using the HIGRAD model as a lidar simulator,
Kao
230 D.I. Cooper et al. Agricultural and Forest Meteorology 114 2003 213–234
et al. 2000 found that the lidar can be used to re-
solve plumes and eddies with reasonable statistical and physical accuracy even though the scan rate is rel-
atively slow. From the co-spectra shown in Fig. 8A
and B , it appears that eddy lengths between 190 and
274 m were responsible for peak fractional fluxes. The dominant eddies contributing to the flux were on the
order of 10–200 m in diameter, while eddies range from 500 to 1000 m in diameter made minor contribu-
tions to the total flux. The time-series analysis of wq co-spectra suggests that the source areas for LE are
microscale, supporting the results of the footprint and lidar-based analyses of sample size region.
4.4. Integral length scales Integral time and spatial scales were estimated
from moisture observations with both tower-based time series and lidar-based spatial series to determine
the length and time of eddy and plume coherency. The Eulerian integral length scale Λ is traditionally
derived from the Lagrangian integral time scale τ and multiplied by the mean wind to estimate a spatial
scale
Hanna, 1981 . Because lidar data are acquired
by a spatially fixed instrument that are range-resolved, the mean wind is not required, allowing the spatial
scale to be computed directly. The τ can be esti- mated from the autocorrelation function of a variable
when the lag is equal to le, if the time-series PDF is near-Gaussian and the autocorrelation function can
be integrated to infinity
Kaimal and Finnigan, 1994; Pope, 2000
. Thus, the Eulerian integral length scale for water vapor is
Λ
q
= ¯ uτ
q
= ¯ u
∞
ρ
q
ξ 9
where the autocorrelation function for water vapor is ρ
q
ξ = q
′ i
q
′ i+ξ
σ
2 q
10 and the conditional water vapor estimates are
q
′ i
= q
i
− ¯ q
i
q
′ i+ξ
= q
i+ξ
− ¯ q
i+ξ
11 where i in this case is an individual water vapor value
q , at some range for lidar data or time for Krypton
hygrometer data and ξ is the lag in range or time. The integration of autocorrelation functions for water
vapor over infinite spatial or temporal lags is not pos- sible, and for convenience we assumed that the finite
sampled population is adequate. Furthermore, the le integrating limit does not work for high-frequency
variables such as 20 Hz measured q values, since the autocorrelation curves tend to have complex shapes,
instead the first zero crossing of the autocorrelation function was chosen as the integrating limit
Kaimal and Finnigan, 1994
. We calculated autocorrelation functions from wa-
ter vapor measurements from the lidar and Krypton hygrometer for both unstable and stable conditions.
Lidar range-resolved q transects 500 m long were ex- tracted from horizontal lines-of-sight 2.7 m above the
canopy where the tower-based sensors were located. Further, 10-min-long time-series of q values were ex-
tracted from the Krypton hygrometer data taken at the same time as the lidar data so we could also estimate
the integrated length scale. The spatial and temporal series were normalized by first detrending the data
using a least-squares linear fit and then smoothed using a Savitzky–Golay filter
Press et al., 1989 ; the
normalized data were then input into Eq. 9
. The results of the autocorrelation processing are
shown in Fig. 10
, where panels A and C are from the unstable case from the Krypton hygrometer and
the lidar, respectively, while panels B and D are for the stable case. The unstable cases panels A and C
show that the integral length scales are under 20 m. In contrast, the stable cases panel B and D show
that the length scales are larger, somewhat over 25 m. The integral time scales show that the time scale for
unstable periods 11.7 s is substantially shorter than for stable cases 21.7 s, since the unstable regime is
better mixed, reducing the lifetime of the eddies.
The integral temporal and length scales are a char- acteristic property of the water vapor eddy structure
and should be related to source properties such as roughness, stability, and spatial extent. The footprint
analyses described in Section 4.2
indicate that peak fractional fluxes occur under 10 m regardless of sta-
bility Fig. 8
, while the cumulative flux analyses show that stable footprints are considerably larger
than unstable cases, which is also supported by the Λ for the unstable and stable cases. The autocorrelation
analysis suggests that the eddies and plumes maintain their coherency within approximately 12–25 of the
optimal horizontal averaged source area.
D.I. Cooper et al. Agricultural and Forest Meteorology 114 2003 213–234 231
Fig. 10. Autocorrelation functions for Krypton hygrometer T subscript time series A and B and lidar spatial L subscript series C and D at 2.7 m above the canopy for unstable and stable periods during day of year 256. Also listed on the panels are the integral spatial
scale, the integral time scale panels A and B and the Monin–Obukhov lengths.
232 D.I. Cooper et al. Agricultural and Forest Meteorology 114 2003 213–234
Fig. 10. Continued .
D.I. Cooper et al. Agricultural and Forest Meteorology 114 2003 213–234 233
5. Conclusions