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Using Lidar Remote Sensing for Spatially Resolved Measurements of Evaporation and Other Meteorological Parameters

^a Univ. of Iowa, Iowa City, IA 52242
^b Los Alamos National Lab., Los Alamos, NM 87545

^* Corresponding author (william-eichinger{at}uiowa.edu)

Received for publication April 18, 2005.

	ABSTRACT

TOP ABSTRACT INTRODUCTION RESULTS SUMMARY AND FUTURE RESEARCH REFERENCES

 
Remote sensors are useful tools for making measurements that are not possible with conventional point instruments. We developed methods to obtain spatially resolved latent energy fluxes from Raman water vapor data and the regional virtual potential heat flux from elastic lidar data. The evaporation method is based on Monin–Obukhov similarity theory applied to spatially and temporally averaged data. Latent heat flux estimates were found to be well correlated (R² = 0.84, slope = 0.98) compared with eddy correlation measurements. The standard error of the flux estimates was 36.5 W m^–2 (a 14% root mean square [RMS] difference), which is close to the predicted uncertainty of 15%. A vertically staring elastic lidar was used to obtain a continuous record of the boundary layer height and thickness of the entrainment zone between a soybean [Glycine max (L.) Merr.] and a corn (Zea mays L.) field. The surface heat flux was calculated using the Batchvarova–Gryning boundary layer model. The virtual potential heat flux estimates were found to be well correlated (R² = 0.79, slope = 0.95) compared with eddy correlation measurements. The standard error of the flux estimates was 21.4 Wm^–2 (31% RMS difference between estimates and surface measurements), higher than the predicted uncertainty of 16%. Other parameters such as the Monin–Obukhov length, the stability correction functions, and integral scale can be obtained from lidar data.

Abbreviations: LANL, Los Alamos National Laboratory

	INTRODUCTION

TOP ABSTRACT INTRODUCTION RESULTS SUMMARY AND FUTURE RESEARCH REFERENCES

 
MODERN SCIENCE has come a long way toward understanding the atmosphere. Much is known about the basic processes that govern climate, so that large-scale weather prediction is fairly reliable. At smaller scales, the problem is more complex. Historically, researchers have assumed homogeneity across large areas. It is only recently that we have had the ability to make spatially resolved measurements and a concomitant interest to investigate how surface heterogeneity affects our measurements, models, and the local climate. Serious issues also exist in our understanding of small-scale flows, particularly in complex environments and stable boundary layers. Traditional techniques of measuring atmospheric variables rely on point sensors to collect information and nearly always in close proximity to the surface; however, the data from individual point sensors near the surface are of limited value. This is due to their relatively small footprint, the necessarily limited number of sensors that can be used to make the measurements, and our current inability to extend the values measured at a point (or series of points) to an understanding of the details of the processes that occur on larger scales. Part of the problem is that the bulk of the earth's surface is not horizontally homogeneous with respect to topography, soil moisture availability, soil type, or canopy. More highly resolved information across large areas is necessary to separate the contributions of each of these variables. It is in this area that remote sensing can be particularly helpful. Remote sensors that can make spatially resolved measurements across large areas can be particularly valuable, not just in providing spatially resolved data, but also in allowing visualization of complex processes and measurements of large-scale motion. Data in the region between 30 m and 3 km above the surface is especially needed.

Eddy correlation has been successfully used from aircraft to cover large areas, but the usefulness of the data has been called into question for use in mixed canopies where spatially resolved fluxes are desired (e.g., Mahrt, 1998). The ability of the method to spatially resolve fluxes and locate their source on the surface is limited. It is possible to obtain spatially resolved evapotranspiration estimates from surface temperature measurements made by aircraft (Neale et al., 2000) or satellite (Moran et al., 1989; Anderson et al., 1997; Bastiaanssen et al., 1998a, 1998b; Allen et al., 2005; Caparrini et al., 2004). Measurements from aircraft and satellites (along with the current lidar method), however, require supporting meteorological information, surface roughness characteristics, and assumptions of local homogeneity to obtain surface flux information.

The three-dimensional scanning Raman lidar built by Los Alamos National Laboratory (Eichinger et al., 1999) can provide detailed maps of the water vapor concentration in three dimensions with high spatial (approximately 1.5 m) and temporal resolution (approximately 15 min to cover a field). Using this information, we have developed a method to estimate spatially resolved evaporative fluxes over the scanned area. A description of the method used to measure the water vapor concentrations is followed by the details of the method used to determine the evapotranspiration rates from the water vapor concentration. Other information can be obtained from a miniature elastic lidar built at the University of Iowa. This lidar can map aerosols out to several kilometers, which provides a tracer for the motion of large structures. From measurements of the evolution of the top of the boundary layer, the virtual potential heat flux can be estimated. The data from several experiments were used.

Raman Lidar Evapotranspiration Estimates
Raman Lidar Method
Raman lidars use a technique originally pioneered by Fiocco and Smullins (1963), Cooney et al. (1969), Cooney (1970), and Melfi et al. (1969). A Raman lidar operates by emitting a pulsed laser beam, usually in the ultraviolet or near ultraviolet, into the atmosphere. Atmospheric gases, such as N₂, O₂, and water vapor interact with this light via the Raman scattering process, causing light of longer wavelengths to be scattered. The amount of the wavelength shift is unique to each molecule. This enables the measurement of different atmospheric gas species. Figure 1 is a plot of the spectrum of light returning from an emitted 248-nm pulse, showing the peaks from various atmospheric constituents.

View larger version (18K): [in this window] [in a new window]   Fig. 1. A plot of the spectrum of light returning from a 248-nm Raman lidar showing the Raman scattering peaks from the major atmospheric constituents.

View larger version (18K): [in this window] [in a new window]   Fig. 2. A diagram showing the layout of the Los Alamos scanning Raman lidar. With the exception of the scanning mirror, the arrangement is typical of Raman lidars.

View larger version (109K): [in this window] [in a new window]   Fig. 3. The Los Alamos scanning Raman lidar.

[1]

where _N2,R and _H2O,R are the Raman N₂ and H₂O scattered wavelengths, _N2 and _H2O are the Raman backscatter cross sections for the laser wavelength, M_N2 and M_H2O are the molecular weights of N₂ and H₂O molecules, _t(r',_N2,R) and _t(r',_H2O,R) are the total attenuation coefficients at the Raman-shifted wavelengths of N₂ and water vapor molecules, fr_N2 is the fractional N₂ content of the atmosphere (0.78084), and C_N2 and C_H2O are the system coefficients that take into account the effective area of the telescope, the transmission efficiency of the optical train, and the detector quantum efficiency at the Raman-shifted wavelengths. Thus the water vapor mixing ratio at any distance is given by the ratio of the magnitude of the signal in the water vapor channel to the magnitude of the signal in the N₂ channel, a multiplicative constant (the first quantity in parentheses on the right side of Eq. [1]), and a small exponential correction due to the difference in extinction between the N₂-shifted and H₂O-shifted wavelengths. Comparison of the lidar signals to conventional hygrometers can be used to determine the multiplicative constant. Because the signal-to-noise ratio decreases with distance, the uncertainty in the mixing ratio values is a function of distance from the lidar. For a midrange distance (350 m), the estimated uncertainty is approximately 3.6% This is consistent with the comparisons of lidar and calibrated references over land surfaces along horizontal paths. The standard deviation between the lidar and capacitance hygrometer data taken at concurrent times and locations was found by regression to be ±0.34 g kg^–1 (Eichinger et al., 2000; Cooper et al., 1996).

When the Raman lidar aims along a given line of sight, data are obtained every 1.5 m along that line. By aiming the lidar in a series of different directions, a two- or three-dimensional map of the water vapor concentrations can be assembled. Figure 4 is a conceptual drawing showing how the various lidar lines of sight are used to scan the area and produce an image such as the one shown in Fig. 5 . Figure 5 is a typical scan from the LANL Raman lidar showing the water vapor concentration in one vertical plane in a corn field in Iowa during the Soil Moisture- Atmospheric Coupling Experiment (SMACEX). The intense red color at the bottom is a result of the attenuation of the laser beam by the ground or canopy (in this case, corn). The change in the lidar signal as it reaches the canopy top enables determination of the shape and orientation of the surface.

View larger version (23K): [in this window] [in a new window]   Fig. 4. A conceptual drawing showing how different lines of sight from the lidar are combined to map the water vapor concentration in a vertical slice of the atmosphere. The water vapor concentration is determined every 1.5 m along each of the lines shown. The lines of sight in actual practice are 0.07 to 0.25° apart.

View larger version (19K): [in this window] [in a new window]   Fig. 5. A vertical scan showing the water vapor concentration in a vertical plane above the corn canopy during SMEX02. The day was strongly convective. Red colors represent areas of high water vapor concentration, while blue colors represent lower concentrations. The intense red color at the bottom is a result of the attenuation of the laser beam by the ground or canopy (in this case, corn). The change in the lidar signal as it reaches the canopy top enables determination of the shape and orientation of the surface.

[2]

where the Monin–Obukhov length, L, is defined as

[3]

z_0v is the roughness length for water vapor, q_s is the surface specific humidity, T is the atmospheric temperature, q(z) is the specific humidity at height z, H is the sensible heat flux, E is the latent energy flux (the evapotranspiration rate), is the density of the air, L_e is the latent heat of evaporation for water, u_* is the friction velocity (Brutsaert, 1982), k is the von Karman constant (taken as 0.40), d₀ is the displacement height, g is the acceleration due to gravity, and _v is the Monin–Obukhov stability correction function for water vapor and is calculated for unstable conditions as

[4]

where

[5]

and where x_0v represents the function x calculated for the value of z_0v. The roughness length is a site-specific free parameter that can be calculated from the lidar data. The Monin–Obukhov length, L, is negative when the atmosphere is unstably stratified. This kind of condition occurs when the soil or canopy is warmer than the air above and convective mixing will occur. The value is positive when the atmosphere is stably stratified. In this case, the potential temperature increases with altitude, suppressing vertical transport. The value of L is infinite in a neutral atmosphere, when the potential temperature is constant with altitude. This condition normally occurs only in the transition from day to night or night to day, but may occur when the winds are exceptionally still.

Heat and momentum fluxes are often determined from measurements of temperature, humidity, and wind speed at two or more heights. These relations are valid in the inner region of the boundary layer where the atmosphere reacts directly with the surface. This region is limited to an area between the roughness sublayer (the region directly above the surface roughness elements) and extending to a height of 5 to 30 m above the canopy top. The concentrations of passive scalars are logarithmic with height in this region. The vertical extent of this layer is highly dependent on the local conditions and wind velocity. The top of this region can be readily identified by a fairly strong departure from the logarithmic profile found near the surface. Figure 6 is an example of a water vapor profile with a logarithmic fit showing such a departure at approximately 5 m above the canopy top. It has been suggested that the atmosphere is logarithmic to higher levels and may integrate fluxes across large areas (e.g., Brutsaert, 1998) so that large-scale fluxes could potentially be obtained from profiles high into the boundary layer.

View larger version (13K): [in this window] [in a new window]   Fig. 6. An example of a lidar fitted vertical profile (solid line) and the data from which it was calculated. The data are from a 25-m section from a vertical scan. The variability in the data about the fitted line is due to the presence of discrete structures. If a large enough area is averaged, the mean value at each elevation converges to a logarithmic profile.

[6]

where M is the slope of the fitted function M = E/(L_eku_*), z' is a reduced height parameter (z' = ln(z – d₀) – _v[(z – d₀)/L]), and c is a regression constant [c = M ln(z₀) + q_s]. Measurements of the slope are made based on a least squares fit to several hundred measurements of water vapor concentration. Having determined M from the slope of the fitted line, the flux is then

[7]

where u_* and L are obtained from local measurements using three-dimensional sonic anemometers. Note that the surface roughness length, z₀ can be estimated from the value of c obtained from the fit.

The method is similar to other gradient methods for determining fluxes that are well established (Stull, 1988, Brutsaert, 1982). The lidar method is unique in that it uses a large number of measurements to determine the vertical water vapor gradient. The extension of the method to rough terrain presents issues relating to assumptions of horizontal homogeneity as well as the determination of the canopy top location (with respect to the lidar) and the direction of the normal to the surface.

A key capability of the lidar that is useful in estimating fluxes over complex terrain is the ability to determine the location of the canopy top. The lidar is sited so that it overlooks the experimental site and is thus able to determine the location of the canopy top for all of the canopy types. For the case of mixed terrain and canopy, the lidar is used to find the location of the canopy top in the range interval under investigation. Figure 7 is a conceptual drawing of how this is accomplished. The top of the canopy is found either from the abrupt change in the apparent water concentration or from the abrupt change in the elastic lidar signal, which is also recorded along each line of sight. The location of the top of the canopy as a function of distance is determined using multiple lines of sight. A linear least squares fit is made to determine the elevation and slope of the top of the canopy within the range interval under consideration.

View larger version (19K): [in this window] [in a new window]   Fig. 7. A conceptual drawing of a 25-m region and all of the lidar lines of sight within it. The location of a line approximating the surface is determined and the distance from each measured value to this line along a perpendicular to the line is calculated. All of the measured values of water vapor concentration are used to create vertical profiles similar to that in Fig. 6.

For an individual scan, all of the measurements within the designated region are used to estimate the slope of the single line described by Eq. [2]. Figure 6 is an example of such a fit to a logarithmic profile. While there is spread in the measurements at each height above the ground, the slope can be determined to an accuracy of a few percentage points. The spread in the measurements is due to the existence of coherent structures containing high and low water vapor concentrations. These structures can be seen in the two-dimensional plot shown in Fig. 5.

A measured value of the Monin–Obukov length is used to further adjust for atmospheric stability. In practice, however, the use of this correction results in a small (usually <3%) change in the estimated flux. A distinct limitation of this method is the lack of a u_* measurement for each 25-m region. In the ideal case, we divide the region into surface types and use a measured u_* typical of that region. There is evidence that u_* is the parameter most likely to be uniform above a canopy (Katul et al., 1999).

When using this method to determine fluxes, the maximum height for which water vapor measurements are included must be determined. This corresponds to the height of the change in slope shown in Fig. 6. While the largest possible distance across which the measurements are made leads to the greatest accuracy, measurements too close to the canopy top (so that they are in the roughness sublayer) or so high that they are outside the inner region lead to erroneous estimates of the water concentration gradient. This height varies throughout the day so the method for determination must be dynamic and adjust to the existing conditions. At present, the maximum and minimum heights for each half-hour averaging period are determined manually by visual inspection of the vertical profiles, a time-consuming process.

The evapotranspiration estimate obtained from each of the 25-m squares can be assembled into a map of the evapotranspiration rates across the scanned area. Two examples are presented. Figure 8 has examples of evapotranspiration maps made from 0815 to 1215 h over corn and soybean fields on 27 June 2002 during the Soil Moisture Experiment (SMEX02). Each map represents a half-hour average (±15 min from the given time). The average height of the corn on 27 June was 1.37 m. The leaf area index was 2.75. The average height of the soybean on 27 June was 0.30 m. The leaf area index was 1.55. There is also a high-resolution (1.5-m pixel) multispectral digital image, presented as a false-color composite, of three spectral bands simulating the Landsat Thematic Mapper bands TM2 (green), TM3 (red), and TM4 (near infrared). The imagery was acquired with the Utah State University airborne multispectral digital system (Neale and Crowther, 1994; Cai and Neale, 1999) on 1 July at 1215 h. It can be seen that the corn canopy (north) had considerably more biomass than the soybean crop (south), which shows distinct patterns of low canopy cover and bare soil. There are correlations between the deep red areas in the aerial photograph (which are areas of high canopy biomass) and regions of relatively high evapotranspiration rate and conversely. The correlation is not perfect, or consistent in all cases. This is not totally surprising in that the lidar senses the water vapor content in the atmosphere and is thus sensitive to the wind direction and the intermittent nature of turbulence. There is a footprint associated with the lidar measurements. Early in the morning of 27 June, there was sufficient near-surface soil moisture so that evapotranspiration was at or near the potential rate. By 1000 h, the surface moisture was significantly depleted, but in small local areas it was possible to have evapotranspiration rates higher than the average. By 1200 h, the evapotranspiration in the soybean area was appreciably lower than that in the corn. Because the leaf area index of the soybean was small (0.3), it is reasonable to assume that the spatial evapotranspiration rates were dominated by soil moisture considerations.

View larger version (28K): [in this window] [in a new window]   Fig. 8. Three evapotranspiration maps of the area around the lidar at various times on 27 June 2002 during SMEX02. Red indicates areas of higher evapotranspiration and blues are lowest. To show the variability of the fields, the color scales are different for each time. Soybean was planted to the south of the lidar and corn to the north. The dividing line is the fence that can be seen in the photograph just below the lidar location. Also shown is an aerial three-band false color composite of canopy reflectance (near infrared [red color], red band [green color] and green band [blue color]) of the site, at the same scale, for comparison. Red colors are indicative of greater amounts of biomass.

View larger version (51K): [in this window] [in a new window]   Fig. 9. A comparison of an aerial thermal infrared photograph of an area populated with salt cedar in the Bosque del Apache riparian area in New Mexico (right) and a lidar-generated evapotranspiration map of the same area (left). Red areas of the photograph indicate hot areas (which should have low evapotranspiration rates) while blue areas are cool areas, which should have high evapotranspiration rates.

[8]

where E, u_*, M, , and q are the uncertainties in the latent heat flux, u_*, slope, air density, and water vapor concentration measurements, respectively (Bevington and Robinson, 1992). The last term on the right is a contribution from a systematic uncertainty (or bias error) in the lidar measurement of water vapor. While an individual measurement may be uncertain to the 3 to 4% level (a measure of the precision error), the determination of the mean concentration from a number of measurements (a measure of the bias error) is more accurate. The bias error in the mean value is taken to be <2%. The value of the slope, M, can be estimated with high certainty due to the large number of measurements used in fitting Eq. [2]. The nominal uncertainty in the value of the slope is 1 to 2%. The air density is obtained from local measurements of temperature and air pressure. The uncertainty in the value of the air density is <<2%. The value of the friction velocity, u_*, is normally the primary source of uncertainty. While there are no reported estimates of the absolute accuracy of u_* estimates from eddy correlation, it is reasonable to assume that the accuracy will be similar to those for eddy correlation flux measurements which normally range from 5 to 25% (Wilson et al., 2002). We have assumed that the typical uncertainty in u_* is 15%. The contribution of u_* to the total uncertainty is a function not only of the uncertainty in the measurements of u_* at a given point, but also a contribution from the assumption that a measurement at one point may be applied to a similar surface some distance away (the magnitude of this contribution is highly site specific). For a typical measurement of the evaporative flux, the total uncertainty is determined almost entirely by the uncertainty in u_* and leads us to estimate an overall uncertainty on the order of 15%. For areas far from u_* measurements, the uncertainty is potentially as much as twice as large. While we use different values of u_* for different canopy types, it is assumed that one value of u_* is representative for an entire area covered by that canopy. Using a limited number of tower measurements, Katul et al. (1999) showed that u_* is the parameter most likely to be uniform above a canopy. This lends support for the assumption that one value can be used across a relatively uniform crop canopy; however, the fractional uncertainty in the evapotranspiration estimate is directly proportional to the fractional uncertainty in u_*.

Figure 10 is a comparison of the latent heat flux estimates from eddy correlation measurements with lidar measurements made in the same 25-m region. These measurements represent half-hour averages. The differences between the two instruments are smaller when more data are available from the lidar for either temporal or spatial averaging. The lidar coordinate system is spherical, so that near the lidar, more data are available with which to determine the vertical profile. The smaller the area to be covered, the faster the lidar can revisit the same locations, providing more data for analysis. The calculated values of the latent heat flux were found to be well correlated (R² = 0.84, with a slope of 0.98) when compared with eddy correlation measurements in the area. The standard error of the flux estimates was 36.5 Wm^–2 (14% root mean squared difference between this method and surface measurements), in keeping with the predicted uncertainty of 15%.

View larger version (24K): [in this window] [in a new window]   Fig. 10. A comparison of eddy correlation evapotranspiration rates over a salt cedar canopy with lidar-estimated evapotranspiration rates made in the same 25-m lidar pixel. The agreement is generally good and along the 1:1 line. The six-point excursion was from an afternoon with exceptionally high winds in which the eddy correlation instruments may have been above the inner region.

The flux estimation method is not suitable for use near areas with sharp transitions in which moist areas upwind alter the water vapor concentration near the canopy top so that it is not logarithmic with height. When this occurs, the methodology described here cannot be used. When the concentrations are not logarithmic with height, the flux estimation method does not always produce evapotranspiration estimates that are unreasonable and thus this condition can be found only by visual examination of the vertical scans. In complex terrain, changes in the canopy and elevation lead to changes in the evapotranspiration rate, which lead to changes in the water vapor concentration along the surface. The conservation of water vapor equation can be used to estimate the magnitude of the effects:

[9]

Under most circumstances, the diffusion term (the last term on the right), and terms involving , v'q', and are small enough to be ignored. Because of advection, changes in evapotranspiration rates may result in flux divergence, particularly in the vertical direction, 'q'/z. The size of this term can be estimated from the (/x) and /t) terms in Eq. [9]. It is not uncommon to find horizontal gradients in water vapor concentration on the order of 8 x 10^–6 kg water kg^–1 air m^–1 between adjacent 25- by 25-m analysis regions downwind of the riparian area. This can lead to a divergence of about 50 W m^–2 m^–1 height above ground. At this time, the effect of advection on the Monin–Obukov flux method is unknown, but is a subject of current research. We have found that the corrections due to nonstationarity are, in general, small. The changes in water vapor concentration duirng a 10-m period are on the order of 3 to 4 x 10^–4 kg water kg^–1 air or less. This results in a correction of <5 W m^–2.

Related to the question of advection is the question of the location of the source (also known as the footprint) for a measurement at a given height. As noted in the discussion of Fig. 9, there is often a systematic offset between the lidar evapotranspiration estimates and probable location of the source. This is a subject of considerable current interest (e.g., Leclerc and Thurtell, 1990; Horst and Weil, 1994; Finn et al., 1996; Horst, 1999). More than two-thirds of the measurements used in any given profile are below 8 m. On more than half of the profiles, the maximum height used is 8 m or less. The greatest curvature in the profile is found at heights <4 m, and it is those measurements from <4 m above the canopy that play the greatest role in determining the slope of the line. Using the methodology outlined by Horst and Weil (1994), one can estimate the upwind distance contributing to the flux at a given height. For a height of 8 m and an L value of –20 m, the upwind distance past which <20% of the flux is generated is a factor of approximately five to 10 times the measurement height, or about 40 to 80 m. This would tend to indicate that the bulk of the flux in a given 25-m section is generated inside that section and the section immediately upwind. Thus it would be prudent to recognize that the flux locations as given by the methodology are not exact, but rather are somewhat diffuse in the upwind direction.

There is also the question of how well the measurements averaged across distances as short as 25 m and <1 min in time can represent the average conditions. It is noted that in a half-hour estimate, as many as six scans may contribute to the estimate. An individual scan will often show structure near the canopy top. In most cases, this will produce deviations both above and below the average value of the water vapor concentration, but which average to profiles that are logarithmic. Occasionally there are plumes that contain water vapor concentrations that are significantly higher than normal. In such cases, the profiles are significantly altered and may no longer be logarithmic. At present, when such events are found, the evapotranspiration estimate from that 25-m section is discarded. No analysis method has been found that can incorporate such structures to produce an evapotranspiration estimate. A more detailed analysis of the structure of the water vapor concentrations and fluxes is presented in Cooper et al. (2000).

While limitations of the method exist, the amount and type of data provided by the lidar allows one to visually determine what is happening at a particular location that causes the estimates to be anomalous. The existence of visual two-dimensional information that allows correction for unusual circumstances is a very powerful asset and offers the potential for improved algorithms that may overcome the deficiencies in the current formulation. At present, these conditions require the intervention of a human analyst to determine the proper method of analysis to be made. Because of the sheer volume of data that must be processed to use this methodology, however, it must be highly automated if it is to be truly efficacious.

Fully Independent Lidar Estimates
The utility of the scanning Raman lidar to derive maps of water vapor flux is presently dependent on tower-based three-dimensional sonic anemometers for critical variables including the Obukhov length and friction velocity (Cooper et al., 2000). An L value used for the entire mapped area, however, is usually derived from a single point as an averaged value across the lidar scanning region. The lidar-derived maps of the latent energy fluxes strongly suggest that the water vapor flux is variable across spatial scales on the order of several hundred meters (Cooper et al., 2000). Since the flux is variable across space, then it follows that L should also be variable. Thus, for estimating the flux, it would be advantageous to have spatially resolved L values along with the spatially resolved water vapor gradients. Furthermore, because Monin–Obukhov similarity theory is commonly used as a framework for characterizing the atmospheric boundary layer, a method to spatially resolve L may be useful for a better understanding of surface–atmosphere exchange processes.

The Monin–Obukhov length can be obtained from the integral length scale, , as

[10]

where the a, ß₁, and C₁ are empirically derived constants. The value of is estimated from the integral of the autocorrelation function of water vapor integrated to its first zero crossing (Kaimal and Finnigan, 1994; Pope, 2000):

[11]

where the autocorrelation function for water vapor is

[12]

where the subscript i represents an individual measurement of water vapor q at some location, and is the lag in distance between the two points. Details of the calculation of the integral scale can be found in Stull (1988).

Equation [10] requires only knowledge of the integral length scale and the height above the canopy to determine L, albeit by an iterative process. Equation [10] can be compared to an empirically derived equation developed by Wilson et al. (1981):

[13]

A comparison of lidar-derived values of L and those from this method is shown in Fig. 11 . Also shown is the line predicted from Eq. [13]. A total of 78 vertical scans acquired during five separate days (three in 1998 and two in 1999 across the riparian area at the Bosque del Apache) representing a wide range of stability conditions were processed using standard lidar analysis techniques (Eichinger et al., 1999, 2000) and then used to calculate the integral length scale and values of L.

View larger version (20K): [in this window] [in a new window]   Fig. 11. Relationship between lidar- and eddy-covariance-derived integral length scales and the Obukhov length (L) estimated from two field experiments. The similarity model based on a stability function (Eq. [13]) is shown as a dotted line and the ±10% uncertainty functions are shown as dashed lines; the equations used to compute the similarity based estimates based on z/L and friction velocity are also shown.

[14]

with ß₁ = 3, ß₂ = 6, a = 1/3, and b = 1/4, respectively (Stull, 1988), C₁ is a constant, usually taken to be 1.25 and has been modified to 1.25 + 1.5/ln(z/z₀). Details of the method and derivation can found in Cooper et al. (2007). A comparison of lidar-derived values of u_* and those from this method is shown in Fig. 12 .

View larger version (15K): [in this window] [in a new window]   Fig. 12. Relationship between sonic-anemometer-measured and lidar-estimated friction velocity (u*). The dashed line is the least-squares regression fit, and the solid line is the 1:1 line. The dotted lines are the 95% confidence intervals for the regression line.

View larger version (120K): [in this window] [in a new window]   Fig. 13. The vertically staring elastic lidar. This system is highly compact and portable and requires no operator input once started.

A high-bandwidth (60-MHz) amplifier is located inside the detector housing. The signal is amplified as part of the detector system and fed to a 100-MHz, 12-bit digitizer on an IBM PC-compatible data bus. A computer is used to control the system and to take the data. The computer controls the system using high-speed data transfer to various cards mounted on the PC bus. The same multipurpose card is used to both set and measure the high voltage applied to the APD. The digitizers on the PC data bus are set up for data collection by the host computer and start data collection on receipt of the start pulse.

Virtual Potential Heat Fluxes from Boundary Layer Heights
The height of the boundary layer is a strong function of the heat flux at the surface. The Batchvarova–Gryning model (Batchvarova and Gryning, 1991, 1994; Gryning and Batchvarova, 1994) is based on a one-dimensional approach for the growth of an inversion-capped atmospheric boundary layer originally developed by Betts (1973), Carson (1973), Tennekes (1973) and Zilitinkevich (1974) (Fig. 14 ). This model and its more simplified forms have been the basis for nearly all of the boundary layer height models that have been developed. The Batchvarova–Gryning model uses a parameterization of the turbulent kinetic energy budget within the mixed layer, and of the temperature jump at the top of the mixed layer. According to the model, the relationship between the virtual potential heat flux at the surface, _surface, the height of the boundary layer, h, and the other surface parameters is:

[15]

where dh/dt is the growth rate of the boundary layer, t is time, T is the temperature of the mixed layer, is the potential temperature gradient above the boundary layer, and w_s is the subsidence velocity. The parameters B and C are normally taken to be parameterization constants with commonly accepted values: B = 2.5 and C = 8 (Melas and Kambezidis, 1992; Gryning and Batchvarova, 1996; Källstrand and Smedman, 1997; Steyn et al., 1999). The parameter A is the ratio of the entrainment of virtual potential heat flux (from the entrainment of warm air above the inversion into the boundary layer) to the surface virtual potential heat flux (Stull, 1988):

[16]

View larger version (27K): [in this window] [in a new window]   Fig. 14. A traditional thermodynamic model of an unstable atmospheric boundary layer. A logarithmic layer near the surface blends into a constant-temperature mixed layer that extends to the top of the boundary layer. A stable atmosphere with a temperature inversion acts as a lid to the vertical motions of the air below. A lidar signal showing the height of the boundary layer with time is shown on the right. Reds are highest particulate concentrations and blues are lowest. The thermodynamic diagram is shown to the left, scaled to the lidar signal.

[17]

where h_bottom is the height of the bottom of the entrainment zone and EZT is the thickness. There have been many studies that have attempted to measure the entrainment parameter A. Laboratory experiments and some observational studies have shown the ratio to be approximately A 0.2 for thermally driven, dry, convective boundary layers (Stull, 1976, 1988; Nicholls and LeMone, 1980). More complicated models that include the mechanical generation of turbulence or forced convection (Mahrt and Lenschow, 1976; Zeman and Tennekes, 1977; Smeda, 1979; Driedonks, 1982) and more recent studies (Betts et al., 1990, 1992; Culf, 1992; Betts and Ball, 1994, 1995, 1998; Betts and Barr, 1996; Davies et al., 1997; Angevine, 1999; Margulis and Entekhabi, 2004) have found values of this entrainment parameter to be significantly larger (A 0.4) for windy conditions. The value of this parameter is dependent on the existing weather conditions. Because of the key role that the entrainment parameter plays, it is fortunate that it can be expressed in terms of the physical size and shape of the top of the boundary layer (Eq. [17]), parameters that can be measured directly by the lidar. Figure 16 is a plot of the value of A as a function of the potential temperature gradient, . Note that the often assumed value A 0.2 occurs only for relatively strong gradients.

View larger version (28K): [in this window] [in a new window]   Fig. 15. A plot showing the values of the ratio of the entrainment of virtual potential heat flux (from the entrainment of warm air above the inversion into the boundary layer) to the surface virtual potential heat flux, A, determined for 25, 27, and 29 June as a function of the potential temperature gradient with height, . Note that there appears to be a relationship between the values of A and the gradient.

Given the thermodynamic model in Eq. [15], ignoring the contribution of mechanical turbulence, and assuming a convective boundary layer (L is small), the following equation may be used to relate the virtual potential heat flux to the characteristics of the boundary layer:

[18]

where H_v is the virtual potential heat flux and c_p is the specific heat for air. The last term in parentheses, when multiplied by c_p, represents the amount of energy that must be added to the boundary layer in order for the boundary layer to grow 1 m. Thus the rate at which the boundary layer grows is a measure of the rate at which energy is added to the boundary layer. This also implies that for a given temperature profile, there is a 1:1 relationship between the height of the boundary layer and the total amount of energy that has been added to the boundary layer.

To use Eq. [18], some estimate of the subsidence velocity, w_s, must be found. Vertical motion at the top of the boundary layer is caused by convergence or divergence in the mesoscale flow and is difficult to determine from meteorological measurements. Fortunately, the boundary layer grows into the residual layer from the day before, whose height can be examined to obtain an estimate of the magnitude of the subsidence velocity. Figure 16 contains a plot of a typical lidar return during a convective morning period. The top of the residual layer in this case corresponds to the bottom of the cloud layer at 2150 m. For this day, the altitude of the residual layer or bottom of the cloud layer stayed approximately constant until later in the afternoon. Because the subsidence velocity must be zero at the surface, we assume that the subsidence velocity decreases linearly with height so that

[19]

View larger version (13K): [in this window] [in a new window]   Fig. 16. A plot of the lidar data for 27 June from 900 through 1230 h. The data is shown as altitude vs. time, with color showing the relative aerosol content (reds are highest concentrations with blues being the lowest). The blue color above the boundary layer is the residual layer from the previous day. White areas are regions of low aerosol content, generally indicating air from the free troposphere.

The one variable that cannot be obtained in its entirety from lidar data is the Monin–Obukhov length, L. To determine L requires values for the air density, , and temperature. The air temperature is easily measured and density can be obtained from local temperature and pressure measurements. The friction velocity, u_*, is normally obtained from high-speed sonic anemometers that are not normally colocated with lidar systems. This value could be estimated from measurements of the mean wind velocity that could be made from a simple cup anemometer. At worst, if no information is available, constant values for u_*, typical of values in the area, could be used (in essence, an educated guess). In such a case, the uncertainty of the estimates of the sensible heat flux would be increased, but since the term containing L is <15% of the (1 + 2A)h term, even large errors in u_* or L will not have a large effect on the determination of H_v.

Uncertainty Analysis
A conventional uncertainty analysis begins with the expression for the virtual potential heat flux expressed in Eq. [18]. Standard error propagation methods (Coleman and Steele, 1989; Taylor, 1982) allow estimation of the maximum probable uncertainty of the method as well as indicating those areas that most significantly affect the estimate. Using Eq. [18] to define the heat flux estimate and assuming the measurements are independent, the contributions to the fractional uncertainty in the heat flux from each of the input variables can be found (Table 2).

View this table: [in this window] [in a new window]   Table 2. Contribution to the fractional uncertainties in the estimated sensible heat flux at the surface (H/H) due to input variables.

[20]

This uncertainty estimate is substantially smaller than a root mean square comparison with the eddy correlation instruments (31%) shown in Fig. 17 . What is striking about this analysis is that most of the uncertainty comes from the estimate for , with the remaining coming from the uncertainties in dh/dt, w_s, and h, in that order. Using the variables in Table 3, the contribution to the uncertainty in the heat flux by ignoring the Zilitinkevich correction is <1%, but results in a consistent overestimate of H_v.

View larger version (37K): [in this window] [in a new window]   Fig. 17. A comparison of virtual potential heat flux estimates from the lidar with virtual potential heat flux estimates from eddy correlation instruments. The data are from five mornings during the SMEX02 experiment. The crosses are the data for 27 June and correspond to the lidar data shown in Fig. 16.

	RESULTS

TOP ABSTRACT INTRODUCTION RESULTS SUMMARY AND FUTURE RESEARCH REFERENCES

	SUMMARY AND FUTURE RESEARCH

TOP ABSTRACT INTRODUCTION RESULTS SUMMARY AND FUTURE RESEARCH REFERENCES

 
Maps of the spatial distribution of evapotranspiration can be produced using spatial water vapor concentration data from a scanning Raman lidar. The estimates of evapotranspiration rates from the lidar compare favorably with other estimates made using the eddy correlation method. The method described allows estimates of the evapotranspiration rate to be made with relatively small (25-m) spatial resolution across an area approaching a 1 km². Because of the amount of data (25–40 Mbytes) and time required to perform the analysis, methods and criteria are currently under development to automate the entire analysis process over all of the azimuthal angles for a given averaging time period. Criteria are being developed to flag and ignore data that do not converge to a logarithmic profile and to not include data above the internal boundary region in the analysis. Automation of the analysis will allow near-real-time determination of the evapotranspiration estimates.

While there are significant limitations to the method, it remains a relatively direct method of estimating the fluxes in situations where conventional methods fail or when the topography makes it difficult to site instruments or field enough of them to achieve an areal average. A major limitation is related to changes in the topography, and how much of the atmosphere above a given site can be considered to be influenced by the surface and thus used to estimate the flux in that region. An advantage of this method is that the spatial water vapor measurements themselves can be used to determine the regions in which these problems may occur. The large number of water vapor measurements used to determine the slope also makes it possible to determine where the slopes change and thus limit the maximum height of the water vapor measurements used to determine the slope.

Efforts are currently underway to improve the range of the lidar system by increasing the photon efficiency of the system. This will make it possible to scan faster so that more scans can be repeated across a larger area. Efforts are also being made to add the ability to measure spatially resolved temperature using a Raman technique (Nedeljkovic et al., 1993). The addition of temperature will allow determination of the partitioning of solar energy between sensible and latent heat fluxes using similar methods. The ability to estimate these fluxes in a spatial manner will enable progress in a number of fields that examine the role that the canopy plays in partitioning solar energy. Improvements to the inversion algorithms are being made so that auxiliary measurements are not required, which will eliminate issues relating to advection. The ideal inversion method would not require assumptions of homogeneity, would correctly identify the fluxes and the source locations, and would not require external meteorological measurements. Therefore, methods of inverting the advection–diffusion equations, Eq. [9], to obtain spatially resolved evaporation rates directly are being investigated.

The method used here can provide reliable estimates of the evapotranspiration rate across a relatively large area with relatively fine spatial resolution. The method is a more direct method of estimating the fluxes than most remote sensing techniques that estimate the evapotranspiration rate as a residual. It is, however, limited in that it assumes that a single measurement of the friction velocity is representative of the value across a large region. It also provides regular estimates throughout the day as opposed to intermittent satellite or aircraft measurements. This information can be used in a wide variety of ways to study the spatial variations in evapotranspiration caused by changes in soil type and moisture content, canopy type, and topography. Because of the extensive nature of the estimates in space and time, evaluation of the relative contributions of each of these can be determined. The type of measurements provided also provide the opportunity to span the scales between the footprint of point measurements and the kilometer-scale measurements made by satellites.

The method presented estimates the regional virtual potential heat flux from vertically staring, elastic lidar measurements using the Batchvarova–Gryning model for energy balance in the boundary layer. To use the method, auxiliary measurements of the potential temperature gradient with height from balloons or radiosondes and surface temperature are required. Three variables are obtained from the lidar data: the average boundary layer height, H; the time rate of change in this height, dh/dt; and the ratio of the downward sensible heat flux at the top of the boundary layer to the surface sensible heat flux, A. All three variables are derived from measurements of instantaneous boundary layer heights (here 1 s apart) during some averaging period.

Comparisons of the sensible heat fluxes from the lidar with eddy correlation measurements are favorable, but with larger errors than expected. A portion of this discrepancy is probably the result of the point nature of the measurements. Both an analysis of the causes of large errors in the estimates and the uncertainty analysis show the importance of accurate measures of , the vertical potential temperature gradient. It is important not only to measure the gradient accurately, but also to accurately measure the altitude at which the measured changes occur.

The goal of this effort is to provide information of use to meteorologists and agronomists that is less restrictive than that from conventional point sensors. The use of data from point se