## 1. Introduction

Horizontal homogeneity of the air temperature field is considered a prerequisite for tower-based micrometeorological techniques to accurately measure the turbulent heat exchange between the surface and the atmosphere (Kaimal and Finnigan 1994). In reality, such conditions are probably rarely met. Measuring the spatial distribution of air temperature allows checking whether spatial differences are small enough that this basic requirement can be assumed. If not, it allows quantifying additional fluxes, such as horizontal advection, for example. Flux contributions that originate from spatially stationary structures also cannot be captured with single-tower measurements and can therefore only be measured through spatial Reynolds averaging (Mahrt 1998; Steinfeld et al. 2007). The resulting underestimation of turbulent fluxes is often referred to as the energy balance closure problem (e.g., Culf et al. 2004; Mauder and Foken 2006; Foken 2007; Oncley et al. 2007).

*H*is the sensible heat flux,

*N*is the number of samples,

*w*is the vertical wind velocity,

*T*is air temperature, and [ · ] denote a spatial average.

Temperature fluctuations were measured by a sonic anemometer–thermometer. Since this instrument is based on a completely different measurement principle, great care had to be taken in correcting systematic errors of the spatially averaged temperature measurement. The dominant error source of a temperature sensor unit in a multiplate radiation shield is radiative solar heating.

According to Steinfeld et al. (2007), about 25 measurement sites are necessary to obtain a good estimate of the spatially averaged air temperature if they are equally distributed over a 10 km × 10 km area at a measurement height of 20 m. Although the size of the observation area and the measurement height were different for our setup, we decided to follow this recommendation and deployed 25 HOBO 12-bit smart sensors (Onset Computer Corp., Bourne, Massachusetts, part S-TMB-M002) in combination with HOBO solar radiation shields (Onset, part M-RSA) as a practical and economical solution. These HOBO systems are an improved version of the temperature datalogger HOBO H8 Pro, which has been evaluated by Whiteman et al. (2000). It has been shown that these sensors are durable, compact, easy to maintain, and very low in power consumption. Nakamura and Mahrt (2005, hereafter NM05) have developed a correction for the radiative error of the HOBO M-RSA radiation shield. However, this correction is not sufficient for the purpose of this experiment, that is, determining the sensible heat flux by replacing the temporal average with a spatial average when computing turbulent air temperature fluctuations. Turbulent temperature fluctuations are usually on the order of 1.0°C (standard deviation of sonic temperature), and the measurement error of the HOBO sensors should preferably be one order of magnitude less.

Therefore, our goal is to reduce the measurement error of the HOBO systems as much as possible. In this study, an improved correction for the radiative heating error of naturally ventilated HOBO radiation shields is presented. This correction is able to provide better results than the approach of NM05 because it considers the shield’s geometry. The approach of NM05 uses wind speed and the shortwave radiation normal to the earth’s surface only. It does not consider the shield’s area normal to the sun and is therefore not as effective. The method presented in this paper uses an adaptation of the correction developed by Anderson and Baumgartner (1998, hereafter AB98) for cylindrical shields to the cuboidal HOBO shields.

## 2. Model for the radiative heating error

Since our correction for radiative heating is based on the work by AB98, we will briefly summarize this method and point out necessary modifications for its application to the HOBO shield. Sensor and shield are considered as one unit. The three major effects that influence the temperature of this system are

- radiative heating, meaning primarily the incoming shortwave solar radiation that hits the shield’s surface;
- natural convection, meaning the heat transfer by fluid circulation or movement solely due to the natural forces of buoyancy at zero wind speed (this process is called “conductive cooling” by AB98); and
- forced convection, meaning the heat transfer due to movement of the fluid with the mean wind (called “convective cooling” by AB98).

*α*, the shortwave flux density

_{s}*R*, and the area normal to the incident solar radiation

_{s}*A*;

_{s}*L*and

*S*are the heat transfer terms for forced and natural convection. In accordance with AB98,

*R*can be described as where

_{s}*θ*is the sun’s elevation angle, and

*SW*↓ is the flux density of shortwave radiation normal to the surface. Here, a constant ratio

*r*of diffuse radiation to total downward radiation of 0.1 is assumed. The justification is that the largest radiative heating occurs under clear skies rather than under cloudy conditions (AB98).

_{d}*A*has to be adapted for the HOBO shield. Its geometry can be described with a cuboid of 0.213-m length

_{s}*l*, 0.188-m depth

*d*, and 0.152-m height

*h*. The contribution of the top area

*A*

_{top}of the cuboid to

*A*is a function of

_{s}*θ:*The contribution of the sides of the cuboid

*A*

_{side}

*to the total area*

_{s}*A*is a function of

_{s}*θ,*the sun’s azimuth

*α*, and the orientation of the HOBO shield

*ϕ*. It can be written as the sum of the contribution from the longer side

*A*and the contribution from the shorter side

_{l}*A*: The sun’s angles

_{d}*θ*and

*α*were computed according to a parameterization (available online at http://www.jgiesen.de/SME/tk/index.htm). The orientation angle

*ϕ*is defined as the direction toward which the mounting plate on the longer side of the cuboid is facing.

*L*and

*S*is a function of the shield surface temperature

*T*and the ambient air temperature

*T*. It is written as where

_{a}*A*is the surface area of the shield, which is the actual heat exchange area;

_{c}*h*and

_{u}*h*are the heat transfer coefficients for forced convective and natural convective cooling, respectively. The coefficient

_{o}*h*is a constant, because natural convection is defined for zero wind and is solely based on buoyancy. The unit of

_{o}*h*is W m

_{o}^{−2}°C

^{−1}. The heat transfer coefficient for forced convection

*h*is usually described by an empirical model (Incropera and DeWitt 1985). In accordance with AB98,

_{u}*h*can be expressed as a function of wind speed

_{u}*V*and two empirical parameters

*C*and

*m*: where

*m*is a nondimensional constant, and the coefficient

*C*incorporates another nondimensional constant

*C*′, the thermal conductivity

*k*, the viscosity of the air

*ν*, and the Prandtl number Pr: It is assumed that all the parameters in Eq. (8) are constant over the observed air temperature range. Thus, the unit of

*C*is J m

^{−3}°C

^{−1}. Combining Eqs. (1), (5), and (6) and dividing by

*α*, the sum of both heat transfer coefficients,

_{s}*δ*, is where

*T*can be substituted by the reference (aspirated) temperature measurements, while

_{a}*T*is the shielded HOBO temperature. Here, we assume that the shield surface temperature can be substituted by the temperature measured inside the shield by the HOBO sensor. All the variables on the right-hand side of Eq. (7) and wind speed

*V*are available, which allows determining the three empirical constants

*C*,

*m*, and

*h*from a regression analysis as described in AB98. The corrected ambient air temperature measurement in the naturally ventilated shield then becomes

_{o}## 3. Experimental setup

The HOBO 12-bit smart sensor is a thermistor with a built-in A/D converter. Its digital output was recorded on HOBO micro station dataloggers (Onset, part H21-002). The sensor was placed inside the HOBO solar radiation shield, which was mounted on a 3-meter tripod (Onset, part M-TPB). The measurement height of the temperature sensor was 2.60 m above ground level. The accuracy of the sensor itself is ±0.2°C according to the manufacturer (Table 1). The response time specification of <2 min is approximately confirmed by the findings of Whiteman et al. (2000) and NM05. This is adequate for measuring horizontal temperature gradients to calculate advective fluxes. We also do not expect the spatial mean to change at a faster rate over our study area of 3.5 km × 3.5 km, because the spatial averaging acts as a low-pass filter.

The measurements were carried out over farmland in southwest Ottawa, Ontario, Canada (45°18′13″N, 75°46′12″W, 88 m MSL). The observation period was from 17 May to 20 June 2007. Wheat, corn, soybean, grass, and alfalfa were cultivated on this land. The 25 HOBO sensors were distributed over the 3.5 km × 3.5 km area in a regular 5 × 5 grid. The orientation of the HOBO shields *ϕ* was 54° for all 25 sensors in our setup. The central site of that grid was located on a 700 m × 300 m large grassland area. In addition to the HOBO sensor–datalogger combination, this site was equipped with a high-precision temperature sensor of type 063 (MetOne Instruments, Inc., Grants Pass, Oregon) in a radiation shield of type 076B, also produced by MetOne. This system served as reference for determining the correction coefficients *h _{o}*,

*C*, and

*m*, since its fan-aspirated shield reduces the radiative error to <0.03°C according to the manufacturer (MetOne Instruments 1997). This extremely low error is safeguarded though the shield’s construction, including a large umbrella-shaped cover plate of 51 cm in diameter, triple-sided walls, double-cross mask bottoms toward the ground, and a relatively large flow rate of 37.7 L min

^{−1}.

Further, a CSAT3 sonic anemometer (Campbell Scientific, Inc., Logan, Utah) and a CMA6 first-class albedometer (Kipp&Zonen, Delft, Netherlands) were deployed at this site. All these sensors were collocated in an area of 20 m × 20 m. The sonic data were recorded on a Campbell CR23X datalogger at a sampling rate of 20 Hz. All slow-response sensors were recorded at 30-s intervals.

## 4. Field intercomparison

To estimate the precision of the HOBO systems under field conditions, a side-by-side intercomparison was conducted before the actual deployment in the main experiment. All 25 HOBO sensors were set up together with the aspirated MetOne sensor over grassland on the Central Experimental Farm of Agriculture and Agri-Food Canada in Ottawa, Ontario, Canada. The 25 HOBO sensors were set up in a 5 × 5 grid in an area of 10 m × 10 m. The MetOne sensor was located 2 m beside this grid (Fig. 1). To ensure comparable conditions to the later deployment in the main experiment, all sensors were mounted at a height of 2.60 m above ground level facing the same direction, and the sampling interval was set to 30 s.

The systems recorded air temperature on 2 and 3 May 2007. Temperatures ranged from 9.5° to 16°C during this period. Root-mean-square errors (rmses) were calculated for all HOBO sensors, one of them chosen randomly as a reference. The rmse values ranged between 0.05° and 0.14°C, with a median of 0.09°C. However, the maximum deviation of the HOBO measurements from the fan-aspirated MetOne temperature was +1.54°C. This clearly shows the need for a radiation correction. Fortunately, it is possible to use the same radiation correction for all 25 sensors because of the good precision of the HOBO units and because all shields were oriented in the same direction. However, it has to be assumed that the incoming shortwave radiation and the wind speed were similar over the entire study area, and that possible differences cancel out when calculating the spatial average from the 25 sensors. The topography of the measurement domain is generally flat, with elevation changes of only a few meters over the entire area. The assumption of a constant *V* is therefore reasonable.

## 5. Results and discussion

Wind speeds *V* and *δ* values were calculated as 30-min averages in order to determine the empirical coefficients for the radiation correction. A dependency of *δ* on either wind direction or the sun’s azimuth, as found by AB98, could not be detected for our measurements. Therefore, it was not necessary to apply wind-blocking or shading adjustments. From an analysis of nighttime data, an offset of +0.32°C was found for the HOBO sensor compared to the MetOne system. This was probably related to the characteristics of the sensor/data acquisition system and was therefore generally subtracted from the HOBO readings. After discarding data from periods with rain, 593 pairs of *V* and *δ* values remained from the 6-week observation period. These samples were classified into 30 equal sample size bins of ascending wind speed. The wind speed range was 0.04–6.36 m s^{−1}. A nonlinear regression analysis was conducted, which resulted in the coefficients *h _{o}* = 242.24 W m

^{−2}°C

^{−1},

*C*= 44.87 J m

^{−3}°C

^{−1}, and

*m*= 2.05 (nondimensional).

Figure 2 shows examples of magnitude of the correction term (*T _{c}* −

*T*) as a function of wind speed

*V*for various combinations of

*R*and

_{s}*A*. For calm wind, the correction term reaches up to −1.2°C for

_{s}*R*= 1000 W m

_{s}^{−2}and

*A*= 0.06 m

_{s}^{2}, which are both at the upper limit of the range of values that occurred during this experiment. The correction term decreases rapidly for wind speeds larger than 1 m s

^{−1}. At 3 m s

^{−1}the correction term is 0.5°C for the highest product of

*R*and

_{s}*A*values. For wind speeds larger than 5 m s

_{s}^{−1}the correction term is generally less than 0.2°C. For a radiative heating term of 500 W m

^{−2}, which corresponds to the midday conditions on an overcast day in summer, the maximum correction term is 0.6°C for calm winds.

In theory, the parameters *h _{o}*,

*C*, and

*m*can be universally applied for this type of HOBO radiation shield over similar surfaces. However, the exposure of the sensors during this experiment in May and June 2007 was limited, and only a selection of all possible meteorological conditions was covered. Thus, these parameters are only valid for daytime conditions as long as wind speed, solar radiation, and solar angles are within the range of the observations during this experiment (i.e.,

*V*< 6.36 m s

^{−1},

*SW*↓ < 1008 W m

^{−2},

*θ*< 68.1°). This correction cannot be applied for periods of precipitation or when the shield’s surface characteristics are altered, for example, through adherent water or snow. A site with a different albedo than grass (≈0.2), which formed the reflecting surface during this experiment, would also require a recalibration of the correction parameters. Regular weather stations mostly use measurement heights of 1.5 or 2.0 m. The HOBO sensors in our setup were located at a height of 2.6 m. We do not expect that the radiative error of the HOBO shields would be significantly different between these heights.

The effect of the consideration of the cuboidal HOBO geometry on the solar heating term is shown in Fig. 3 for three very different consecutive days. The first day, 19 May 2007, was cloudless, with a maximum incoming solar radiation of 970 W m^{−2} and low wind speeds between 0.1 and 2.5 m s^{−1}. The second day was overcast, and the maximum incoming solar radiation was 444 W m^{−2}. Wind speeds were much higher, ranging between 2.3 and 6.0 m s^{−1}. The third day was cloudless in the morning before some cumulus clouds developed in the afternoon. Incoming shortwave radiation was undisturbed from clouds at solar noon and reached 984 W m^{−2}. Wind speeds were between 3.1 and 6.0 m s^{−1}.

The shield’s area normal to the sun *A _{s}* had a trimodal diurnal course. On the clear sky day, 19 May 2007, the highest maximum was at 0842 local solar time (LST), with a secondary maximum at 1615 LST and a tertiary maximum at 1220 LST. In comparison, the diurnal course of

*R*for a cylindrical shield is bimodal, with a distinct minimum at solar noon (AB98). The geometry factor

_{s}A_{s}*A*modulated the incoming shortwave radiation in such a way that the largest radiative heating of the HOBO shield

_{s}*R*occurred at 0855 LST. The maximum radiation correction on 19 May 2007 was 1.17°C. This maximum was reached twice on that day, first between 0830 and 0900 LST, when the

_{s}A_{s}*R*term had its absolute maximum, and second between 1130 and 1200 LST, when wind speeds were lowest. On 20 May 2007, the maximum temperature correction was only 0.30°C at maximum. The radiative heating was lower, and wind speeds were higher compared to the previous day. On 21 May 2007, the radiation conditions were similar to 19 May 2007 in the morning, when the radiative heating is usually largest. However, due to higher wind speeds, the maximum temperature correction was only 0.37°C.

_{s}A_{s}The overall improvement of the HOBO temperature measurements through the radiation correction is shown in Fig. 4. Without this correction, the HOBO temperature measurements were sometimes up to 1.2°C higher. The differences between the uncorrected HOBO temperature *T* and the aspirated temperature *T _{a}* scattered around an average of +0.42°C, with a standard deviation of 0.27°C. The median was +0.38°C. After applying the radiation correction, the HOBO temperatures

*T*and

_{c}*T*were almost identical on average, with significantly less scatter. The mean difference between

_{a}*T*and

_{c}*T*was +0.01°C, with a standard deviation of 0.15°C and a median of +0.02°C. The root-mean-square difference between the HOBO temperature measurements and the fan-aspirated reference measurements was reduced from 0.49° to 0.15°C. The remaining error is only slightly larger than the precision of the HOBO sensor–shield combinations themselves.

_{a}## 6. Conclusions

The principle of AB98’s correction for radiative heating of naturally ventilated shields was successfully adapted to the cuboidal geometry of the HOBO solar radiation shields. This modified correction method makes it possible to deploy a network of many relatively low-cost and low-maintenance temperature sensors in naturally ventilated shields with a data quality close to aspirated systems, provided wind and radiation measurements are available for at least one of the sites over the entire measurement period. The study area also must be small enough that similar conditions for incoming shortwave radiation and wind can be assumed. At many weather stations, shortwave radiation and wind speed belong to the standard set of measured variables, so this radiation correction can be applied for the temperature measurements in HOBO solar radiation shields. An aspirated temperature sensor was deployed over a 35-day period in May and June 2007 to obtain estimates for the three empirical correction coefficients required. These parameters may be generally applied for clean and dry HOBO solar radiation shields of type M-RSA during daytime periods over surfaces with similar radiative properties. Further measurements are needed to validate these parameters for meteorological conditions that are not covered by the field observations presented here.

We are grateful to Doug Glowenlock, manager of the Canadian Food Inspection Agency research farm in Ottawa, and Bert Moore, manager of the Greenbelt research farm in Ottawa, for their cooperation during the realization of the presented measurements.

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Selected specifications of the HOBO 12-bit smart sensor.