Spatial analysis of sampling size

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 D.I. Cooper et al. Agricultur al and F or est Meteor olo gy 114 2003 213–234 225 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 D.I. Cooper et al. Agricultur al and F or est Meteor olo gy 114 2003 213–234 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 D.I. Cooper et al. Agricultur al and F or est Meteor olo gy 114 2003 213–234 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