Estimating Ground Heat Flux from Net Radiation

Abstract

Ground heat flux may play an important role in surface energy balance. In this study we evaluate the performance of the objective hysteresis model (OHM) for estimating ground heat flux from net radiation and compare it with the linear regression model. The experimental sites include residential roofs (concrete), campus grassland, agricultural grassland, and peat bog. Our field measurements show that the mean partition coefficient from net radiation to ground heat flux varied from 0.47 (concrete roof) to 0.079 (agricultural grassland). The mean hysteresis (lag) factors for residential roof, campus grassland, and peat bog were 0.55, 0.26, and −0.11 h, respectively; and the hysteresis factor at the agricultural site was only 0.032 h. However, the partition and hysteresis coefficients in the OHM were found to vary with time for the same surface. Our measurements and analysis show that when the hysteresis factor is larger than 0.11 h, ground heat flux estimates from net radiation can be improved (17–37% reduction in the root mean square error) by using OHM instead of a simple linear regression model.

1. Introduction

In the surface energy balance, net radiation is usually divided into sensible heat, latent heat, and ground heat fluxes. Ground heat fluxes may play an important role in the energy budget of both agricultural and natural surfaces and affect evapotranspiration [1,2,3]. Furthermore, concrete and asphalt surfaces in the city absorb more net radiation than a vegetated surface; therefore, the ground heat flux is a key parameter in the urban surface energy balance [4,5] and is essential for predicting the surface temperature [6,7]. The ground heat flux is also an input parameter in models such as the Penman—Monteith equation [2] and non-parametric method [8,9] for estimating sensible and latent heat fluxes.

The direct measurement of the ground heat flux at the soil surface is difficult. Usually it is measured by using a soil heat flux plate buried at a certain depth in the soil plus the heat storage in this soil layer. The heat storage is then calculated from the soil temperature time series. When the ground heat flux is desired but not measured, estimating it from the net radiation is a convenient solution. Assuming a simple linear relationship between the ground heat flux (G) and net radiation (R_n) is the easiest and most common method. In other words, G = a₁R_n + a₃, where a₁ is the slope and a₃ is the intercept. It has been noticed that a₁ varies from 0.1 to 0.67 [4] for different surface types. On the other hand, Camuffo and Bernadi [10] observed a hysteresis phenomenon between net radiation and ground heat flux and proposed the objective hysteresis model (OHM) for estimating G from R_n. This relationship is a binary linear regression equation, and once the required coefficients (depends on surface characteristics) are determined [11], this method is attractive for estimating ground heat flux.

Anandakumar [12] studied the OHM coefficients of an asphalt surface (near Vienna, Austria) for an entire year and found that these coefficients vary with time throughout the year (e.g., a₁ ranged from 0.66 (August) to 0.91 (March) for different days in the year). Panagiotakis et al. [13] employed a surface urban energy and water balance scheme to validate the feasibility of natural based solutions in urban environments. They indicated that the OHM coefficients should be adjusted with different seasons. Dou et al. [14] studied the energy partitioning variability in the suburban area of Beijing. They also found the OHM coefficients vary with season. Cui and Chui [15] investigated the variations in surface energy fluxes and energy balance closure on complex urban surfaces in Hong Kong. However, their results show that the model simulation was not sensitive to different OHM coefficients for estimating heat storage.

The OHM coefficients are important for calculating and modeling the surface energy balance (e.g., [5,13,14,16,17,18,19,20]). Many studies [21,22,23,24,25,26,27,28,29,30] have also investigated the OHM coefficients for various surface types as listed in

Table 1. Coefficients in the objective hysteresis model (i.e., Equation (3)) for different surfaces. (Coefficients from studies before year 2009 were summarized in Meyn and Oke [21]; Grimmond and Oke [22]). a₁: slope (partition factor); a₂: hysteresis factor; a₃: intercept.

Surface Type	a1	a2 (h)	a3 (W/m2)	Reference Source
Bare and snow-covered soil (Antarctic area)	0.13	−0.07	8.3	Alves and Soares [23]
Long grass field, Oklahoma	0.20	0.13	−23.4	Marciotto [24]
Short grass, Oklahoma	0.16	0.04	−11.1	Marciotto [24]
Grass field small, Oklahoma	0.221	0.294	10.1	Marciotto [24]
Short grass, St. Louis	0.32	0.54	−27.4	Doll et al. [4]
Wheat in winter	0.07	−0.29	−11.8	Dou et al. [14]
Wheat in summer	0.23	−0.07	−17.2	Dou et al. [14]
Lakeshore wetland, Indiana, USA	0.537	0.215	−30.4	Souch et al. [30]
Campus grassland, Taiwan	0.31	0.55	−25.82	This study
Agricultural grassland, Ireland	0.08	0.03	−6.58	This study
Peat bog, Ireland	0.15	−0.11	−6.63	This study
Building roof, Mexico city	0.67	0.45	−52	Velasco et al. [25]
Cuiaba, Brazil (concrete)	0.21	−0.43	−13.9	Callejas et al. [26]
Athens, Greece	0.69	0.26	−57.54	Loupa et al. [27]
London, UK	0.553	0.303	−37.6	Ward et al. [28]
Swindon (residential rural), UK	0.417	0.319	−30.5	Ward et al. [28]
Sakai, Osaka, Japan	0.71	0.13	−51	Ando and Ueyama [29]
Residential roof (concrete),Taiwan	0.47	0.26	−32.69	This study

. It can be summarized that for vegetation surface: a₁ ranges from 0.07–0.31; for urban area: a₁ varies from 0.21 to 0.71. However, most of these studies’ experimental periods are only a few days (e.g., [27,30]) and the variability of these OHM coefficients by month have not yet been reported and are needed for better model simulations [13,28]. So far, only four studies [12,15,23,28] have reported monthly/seasonal values. In addition, the hysteresis effect for peat bog ecosystems has not yet been studied. Since urban roof and grassland are important factors for urban heat island study and agricultural grassland and peat bog fields are important for climate change research, we have selected a residential roof, a campus grassland, an agricultural grassland, and a peat bog site for this study. The objectives of this study are (1) to investigate the partition of net radiation into ground heat flux and the hysteresis effect over an extended long period for four different sites: residential roof, campus grassland, agricultural grassland, and peat bog; (2) to study the monthly variations of the OHM coefficients for these four sites; and (3) to assess the performance of the OHM in comparison with simple linear regression model.

2. Experiment

The experimental data for this study were collected from four different field sites: residential roof, campus grassland, agricultural grassland, and peat bog. Each experiment is described below.

2.1. Residential Roof—Taiwan

The experimental mast is situated on the roof of a three-story residential building in a T-shaped street canyon (24°16′30.1″ N 120°34′22.5″ E) within the suburban residential area of Qingshui Town, Taichung City, central Taiwan. The climate is subtropical and the mean annual precipitation, summer and winter temperatures, and humidity are 1750 mm, 29 and 18 °C, and 75%, respectively. The surrounding buildings near the experimental site are mostly three stories tall. The roof height of the experimental site is 9.3 m. The experiment was conducted from 2 May 2007 to 5 September 2007. The eddy covariance method was employed to measure surface fluxes. The instruments were mounted on a 4.3 m tall observation mast installed on the rooftop of the building. At a height of 13.62 m above the ground, an ultrasonic anemometer (Young 81000, R. M. Young, Traverse City, MI, USA) was installed to measure air temperature (°C) and three-dimensional wind speed (m s⁻¹). Additionally, an open-path infrared H₂O/CO₂ gas analyzer (Li-7500, Li-Cor, Lincoln, NE, USA) was mounted at the same height to measure the molar densities of water vapor and carbon dioxide (mmol m⁻³). At a slightly lower height, 12.57 m above the ground, a net radiometer (REBS Q7, Radiation and Energy Balance Systems, Inc., Seattle, WA, USA) was installed to measure the net radiation. Data from all instruments were collected at a rate of 10 Hz by a data logger (CR3000, Campbell Scientific, Logan, UT, USA) and transferred to a computer. The data collected were averaged into 30 min intervals. Surface fluxes calculations in this study followed the general standard procedures outlined in Hsieh et al. [31]. The site characteristics and instruments heights are also listed in

Table 2. Characteristics of experimental sites and instruments heights.

Experimental Site	Residential Roof	Campus Grassland	Agricultural Grassland	Peat Bog
Data period	May–September 2007	June 2012–August 2013	April–September 2013	February–November 2013
Altitude (m)	12	22	195	150
Latitude and Longitude	24°16′30″ N 120°34′22″ E	25°01′ N, 121°53′ E	51°59′ N, 8°45′ W	51°55′ N, 9°55′ W
Climate	Subtropical	Subtropical	Temperate	Temperate
Annual rainfall (mm)	1773	2405	1161	1800
Mean temperature (°C)	22	23.9	9	10
Canopy height (cm)	--	5–45	10–45	5–30
Measurement height (m)
Eddy-covariance	13.62	1.25	5	3
Soil heat flux plate	--	−0.08	−0.1	−0.1
Soil thermometer	--	−0.02, −0.06	−0.015, −0.05, −0.075	−0.1

. The geographical location of the residential roof field and a site photo are shown in Figure 1a,b.

For this site, the ground heat flux was calculated using the energy balance equation. In other words, ground heat flux was determined as the residual between the measured net radiation and the sensible and latent heat fluxes using the following equation:

$$\mathrm{G}={\mathrm{R}}_{\mathrm{n}}-\mathrm{H}-\mathrm{L}\mathrm{E}$$

(1)

where H and LE are sensible and latent heat fluxes (W m⁻²), respectively.

2.2. Campus Grassland—National Taiwan University

This campus grassland is situated at the Atmospheric Sciences Observation Field (25°00′52″ N, 121°32′20″ E) in National Taiwan University, Taipei, northern Taiwan. Taipei has a subtropical climate. The mean annual precipitation, summer and winter temperatures, and humidity are 2000 mm, 30 and 15 °C, and 75%, respectively. This site covers an area of approximately 1800 m² (about 40 m in length and 45 m in width) and the grass heights ranged from 5 to 45 cm. The predominant vegetation was Bermuda grass, also known as Cynodon dactylon. Figure 1a,c show the geographical location of this campus grassland field and the site photo. The experiment was conducted from 5 June 2012 to 18 September 2013. The eddy covariance system was employed to measure sensible heat and water vapor fluxes. An ultrasonic anemometer (Young 81000) was installed at a height of 1.25 m above the ground to measure air temperature (°C) and three-dimensional wind speed components (m s⁻¹). At a height of 1.05 m, an infrared H₂O/CO₂ analyzer (Li-7500A) was deployed to measure the molar density (mmol m⁻³) of water vapor and carbon dioxide. The eddy covariance system collected data at a frequency of 10 Hz and the averaging period was 30 min. At 1.6 m above the ground, a net radiometer (NR lite) was installed to measure the net radiation. At the same height, an infrared temperature sensor (IRTS) was set up to measure surface temperature (°C). A soil heat flux plate (hfp01) was placed at 8 cm below the ground to measure ground heat flux, and soil temperature probes were installed at 2 cm and 6 cm below the ground to measure soil temperature (°C). The data mentioned above were collected at one-minute intervals and averaged every 30 min. In this experiment, the ground heat flux was calculated using the ground heat flux equation: [32,33]

$$\mathrm{G}={\mathrm{G}}_{\mathrm{z}}+{\mathsf{\rho }}_{\mathrm{s}}{\mathrm{c}}_{\mathrm{s}}\frac{\partial \mathrm{T}}{\partial \mathrm{t}}\Delta \mathrm{z}$$

(2)

where G_z is the soil heat flux at depth z (W m⁻²), ρ_s is the soil density (kg m⁻³), c_s is the soil specific heat capacity (J kg⁻¹ K⁻¹), T is the soil temperature, t is time, and Δz is the soil depth.

2.3. Agricultural Grassland—Ireland

This grassland is an agricultural site in Dripsey, approximately 25 km northwest of Cork City in Southern Ireland (51°59′ N, 8°45′ W; elevation 195 m). The region experiences a temperate climate with a high level of rainfall. The 30-year average temperature is 9.4 °C, and the annual average precipitation is 1207 mm (based on climate norms from 1960 to 1990 at Cork Airport Meteorological Station). The Dripsey site is a high-quality pasture and grassland, with perennial ryegrass being the predominant plant species, interspersed with small quantities of fescue and clover. The grass height in the pasture typically ranged from 10 to 20 cm, while in the grassland, it could reach up to 45 cm height [34]. The soil type in this area is classified as Gleysol, characterized by fertile soil texture. The soil particle composition consists of 42% sand, 41% silt, and 17% clay [35]. In the top 20 cm of soil, the soil organic carbon concentration is 5.9% [36]. The geographical location of this agricultural grassland and a site photo are shown in Figure 2a,b.

An eddy covariance system consisting of a CO₂/H₂O analyzer (Li-7500) and an ultrasonic anemometer (CSAT3) was used to measure the fluxes of sensible heat, water vapor, and CO₂. The system was installed at 5 m above the ground. The flux data was collected at a frequency of 10 Hz with an average period of 30 min. Other meteorological parameters collected include net radiation (CNR1), soil heat flux (Hfp01), temperature, and relative humidity (HMP45C). The heights of these instruments were 4, −0.1, and 2.5 m, respectively. Soil temperature sensors were placed at depths of 1.5, 5, and 7.5 cm below the surface, while soil moisture sensor was located at a depth of 5 cm. Ground heat flux was obtained by equation (2). Meteorological data was collected every minute and averaged every 30 min [37]. Data used for this study was from April–September 2013. Further details regarding the experimental site can be found in Peichl et al. [37].

2.4. Peat Bog—Ireland

This peat bog site is located in Glencar, County Kerry, in the southwestern region of Ireland. The coordinates of the research station are (51°55′ N, 9°55′ W), with an elevation of 150 m above sea level. The region experiences a temperate climate with a high level of rainfall, with an average temperature of 10 °C and an annual precipitation around 1800 mm. In the center of the bog, the upper peat layer primarily consists of reed sedge peat. The soil bulk density is approximately 0.05 g cm⁻³, and the porosity is 95%. The depth of the peat layer ranges from about 2 to 5 m deep [38]. During the summer months, vascular plants cover approximately 30% of the bog’s surface area. This type of bog is characterized by significant spatial heterogeneity at the surface level. In different relative elevations, the proportion of vegetation and standing water varies [39]. Figure 2a,c show the geographical location of this peat bog field and the site photo.

Located at the center of the bog, a 3 m tall flux tower was equipped with an eddy covariance system. The system consisted of an ultrasonic anemometer (CSAT3) to measure three-dimensional wind speed and virtual potential temperature. In addition, the water vapor and carbon dioxide concentrations were measured using a CO₂/H₂O analyzer (Li-7500). These instruments collect data at a frequency of 10 Hz, and average every 30 min. Other meteorological and environmental parameters were also measured. Net radiation was measured at 2 m above the ground using a CNR1 net radiometer, while soil heat flux was measured at 10 cm below the surface using an Hfp01 soil heat flux plate. Soil temperature sensors were positioned 10 cm below the ground. Ground heat flux was then calculated using Equation (2). A temperature and humidity sensor (HMP45C) was installed at 3 m to measure air temperature and relative humidity. Meteorological measurements were collected once a minute and averaged every 30 min [40,41]. Data adopted for this study was from February–November 2013. Further experimental details can be found in McVeigh et al. [41]

3. Method

Camuffo and Bernardi [12] observed that the scatterplot of net radiation and ground heat flux exhibits a hysteresis phenomenon and forms a loop, as shown in Figure 3. They proposed a formula to express the relationship between net radiation and ground heat flux, known as the objective hysteresis model (OHM):

$$\mathrm{G}={\mathrm{a}}_1{\mathrm{R}}_{\mathrm{n}}+{\mathrm{a}}_2\frac{\partial {\mathrm{R}}_{\mathrm{n}}}{\partial \mathrm{t}}+{\mathrm{a}}_3$$

(3)

In Equation (4), on the right hand side, the first and third terms are essentially the linear regression slope and intercept terms, respectively. The second term accounts for deviations from the best-fitting straight line, indicating how the actual values differ from the linear fit. The coefficients a₁ and a₂ are influenced by soil properties, soil moisture, and are related to both net radiation and its time derivative. The coefficient a₁ represents the proportion of net radiation that is transformed into ground heat flux. The coefficient a₂ (hour) plays a crucial role in determining the level and direction of the hysteresis phenomenon. When a₂ is positive (Figure 3a), it implies that the peak of ground heat flux arrives earlier than the peak of net radiation, and the hysteresis pattern in the scatterplot of net radiation and ground heat flux is clockwise. On the contrary, when a₂ is negative (Figure 3b), it indicates that the peak of net radiation precedes the peak of ground heat flux. In this case, the hysteresis pattern in the scatterplot of net radiation and surface heat flux is counterclockwise. When a₂ is zero, it suggests the absence of a hysteresis phenomenon. In addition to land surfaces, the OHM is also applied to study the heat storage in wetlands (e.g., Soucha et al. [30]). Coefficient a₃ (W m⁻²) is the intercept term and may be related to the temperature difference between the surface and the air or the release of latent heat from the surface, as suggested by Camuffo and Bernardi [12]. When R_n is zero, a₃ represents the ground heat flux resulting from the temperature difference between the surface and the soil. A negative value for a₃ indicates heat being released into the atmosphere during night time [14].

Each of these coefficients can be determined through a binary linear regression analysis. In this study, the term ∂Rn/∂t is calculated using the following formula:

$$\frac{\partial {\mathrm{R}}_{\mathrm{n}}}{\partial \mathrm{t}}=\frac{{\mathrm{R}}_{\mathrm{n}(\mathrm{t}+1)}-{\mathrm{R}}_{\mathrm{n}(\mathrm{t}-1)}}{2\Delta \mathrm{t}}$$

(4)

where Δt is the time interval (=0.5 h for this study).

4. Results and Discussion

4.1. Diurnal Variations of Net Radiation and Ground Heat Flux

The diurnal variations of the average net radiation and ground heat flux for all data during the experimental period at the four sites are shown in Figure 4. For the residential roof, camps grassland, and agricultural grassland, the peak of ground heat flux occurred before the peak of net radiation (Figure 4a,c); while at the peat bog site, the peak of ground heat flux occurred after that of net radiation (Figure 4d). Specifically, for the residential roof, the peaks of ground heat flux and net radiation occurred at 10:30 and 11:30, respectively. In the case of the campus grassland, the peaks for G and R_n were at 10:30 and 12:00, respectively. For the agricultural grassland, the peaks of ground heat flux and net radiation occurred at 13:30 and 14:00, respectively. In the peat bog site, the peak of G occurred at 14:00 which was 2 h after the peak of R_n.

As net radiation increases with the sunrise, ground heat flux also starts to rise. However, the residential roof, and the two grassland surfaces were heated faster and resulted in an early peak of ground heat flux. On the other hand, the peat bog had a larger volumetric heat capacity and was heated slower, and then resulted in a later peak of ground heat flux.

4.2. Temporal Variations of Coefficients in the Objective hysteresis Model

The monthly variations of the OHM coefficients for the four sites are shown in Figure 5. It is observed that all the coefficients (a₁, a₂, and a₃) may vary month by month significantly. Hence, it is important to notice the difference between the average value of a site and its monthly value. The average values of these coefficients are also summarized in Table 1. Detailed monthly coefficients for each site are listed in the Appendix A:

Table A1. Objective hysteresis model (Equation (3)) coefficients for the residential roof, May 2007–July 2007. Values within the parentheses represent regression values for the entire data. a₁: slope (partition factor); a₂: hysteresis factor; a₃: intercept.

Month	a1	a2 (h)	a3 (W m−2)
5	0.46	0.23	−32.88
6	0.46	0.25	−27.60
7	0.49	0.29	−37.60
Average	0.47 (0.46)	0.26 (0.25)	−32.69 (−31.71)
Standard deviation	0.014	0.024	4.08

,

Table A2. Objective hysteresis model coefficients for the campus grassland, June 2012–August 2013. Values within the parentheses represent regression values for the entire data.

Month	a1	a2 (h)	a3 (W m−2)
6 (year 2012)	0.26	0.37	−21.76
7	0.29	0.66	−37.93
8	0.21	0.67	−30.42
9	0.25	0.53	−28.11
10	0.27	0.69	−22.35
11	0.30	0.62	−19.57
12	0.37	0.69	−14.29
1 (year 2013)	0.46	0.56	−20.45
2	0.46	0.70	−20.40
3	0.43	0.79	−29.47
4	0.50	0.32	−43.19
5	0.29	0.49	−34.56
6	0.22	0.33	−23.93
7	0.18	0.41	−21.89
8	0.17	0.49	−19.05
Average	0.31 (0.26)	0.55 (0.55)	−25.82 (−24.67)
Standard deviation	0.10	0.15	7.70

,

Table A3. Objective hysteresis model coefficients for the agricultural grassland, April–September 2013. Values within the parentheses represent regression values for the entire data.

Month	a1	a2 (h)	a3 (W m−2)
4	0.11	0.051	−7.57
5	0.081	0.044	−6.76
6	0.080	0.032	−6.81
7	0.074	0.018	−6.21
8	0.066	0.020	−6.85
9	0.063	0.025	−5.30
Average	0.079 (0.079)	0.032 (0.035)	−6.58 (−6.67)
Standard deviation	0.015	0.012	0.70

and

Table A4. Objective hysteresis model coefficients for the peat bog, February–November 2013. Values within the parentheses represent regression values for the entire data.

Month	a1	a2 (h)	a3 (W m−2)
2	0.18	−0.23	−4.61
3	0.14	−0.082	−6.25
4	0.12	−0.17	−4.60
5	0.13	−0.066	−12.14
6	0.15	−0.067	−11.43
7	0.14	−0.063	−11.55
8	0.12	−0.057	−9.22
9	0.16	−0.080	−0.79
10	0.20	−0.14	−5.80
11	0.19	−0.17	0.11
Average	0.15 (0.13)	−0.11 (−0.077)	−6.63 (−8.25)
Standard deviation	0.029	0.057	4.16

. We also list the average values of the summer period (May–July) for these OHM coefficients at the four sites in

Table A5. Mean objective hysteresis model coefficients for the summer period (May-July) at the four sites.

Site	a1	a2 (h)	a3 (W m−2)
Residential roof	0.47	0.26	−32.69
Campus grassland	0.25	0.45	−28.01
Agricultural grassland	0.078	0.031	−6.59
Peat bog	0.14	−0.065	−11.7

. For each site, there might be a small difference between the averages of the whole experimental period and summer period, but the patterns of partition and hysteresis factors are the same.

The partition and hysteresis factors (0.47 and 0.26) for the residential roof in Taichung, Taiwan are similar to Swindon, UK, but smaller than those dense cities like Mexico City and London (Table 1). For the campus grassland, all the three coefficients are close to the short grass site at St. Louis, MO, USA [4], but different from the short and long grass sites at Oklahoma, USA [24].

For the peat bog site, the partition and hysteresis factors (0.15 and −0.11) are unique (no peat bog literature values have been published yet). For the agricultural grassland, the hysteresis between net radiation and ground heat flux was small, only 0.03 h. This is similar to the short grass in Oklahoma, OK, USA [24]. Nevertheless, the partition factor is small (0.08) and is half of the value of the short grass in Oklahoma, OK, USA [24].

An interesting finding for a₁ at the campus grassland is that lower values in the summer and higher values in the winter. This phenomenon, in this subtropical campus grassland, can be attributed to the increased evapotranspiration during the summer months when photosynthesis is more active, and then less energy is distributed to ground heat flux. The same trend for a₁ at the temperate agricultural grassland in this study is also noticed (Figure 5c).

The values of coefficient a₃ are all negative across all experimental sites. This shows that when R_n is zero, the soil temperature is higher than the surface temperature at all the four sites. Figure 5a,b (Table A1 and Table A2) show that this heat flux release from the ground is less than about 40 W m⁻² for both the two subtropical sites (residential roof and campus grassland); on the contrary, this value is even smaller (less than 10 W m⁻²) for both the two temperate sites (agricultural grassland and peat bog, Figure 5c,d).

4.3. Comparisons between Linear Regression Model and OHM

Comparisons of estimated ground heat fluxes using linear regression model (G_linear) against measurements (G) for each experimental site are shown in Figure 6. It is evident that linear regression model performs reasonably well in estimating ground heat fluxes for the residential roof, agricultural grassland, and peat bog (Figure 6a,e,g) where the R² values all larger than 0.8. However, the linear regression model was not suitable for the campus grassland, where R² is only 0.39. Detailed statistics between the estimated and measured ground heat fluxes for the four sites are summarized in

Table 3. Summary of regression analysis between estimated and measured ground heat fluxes. Y = aX + b; Y is estimation; X is measurement. G_linear is the estimation from linear regression model. G_OHM is the estimation from objective hysteresis model. Data within the parentheses represent the statistical results obtained using the average values of a₁ a₂, a₃.

Site	Model	Slope *	Intercept	R2	RMSE (W m−2)
Residential roof	Glinear	0.89 (0.89)	5.23 (5.27)	0.89 (0.89)	41.45 (41.61)
	GOHM	0.92 (0.92)	3.62 (3.66)	0.92 (0.92)	34.48 (34.79)
Campus grassland	Glinear	0.39 (0.35)	2.48 (2.65)	0.39 (0.35)	70.18 (72.55)
	GOHM	0.72 (0.65)	1.16 (1.42)	0.72 (0.65)	48.07 (53.07)
Agricultural grassland	Glinear	0.85 (0.76)	0.58 (0.62)	0.82 (0.76)	6.13 (7.05)
	GOHM	0.83 (0.76)	0.45 (0.62)	0.83 (0.77)	5.96 (7.04)
Peat bog	Glinear	0.81 (0.78)	0.80 (0.89)	0.81 (0.78)	13.28 (14.01)
	GOHM	0.87 (0.84)	0.52 (0.66)	0.87 (0.84)	10.68 (12.05)

* The p-values for all slopes are all less than 0.05 and are very small (close to zero).

.

Figure 6 also presents the comparisons between objective hysteresis model estimated (G_OHM) and measured ground heat fluxes for each site. These plots reveal that, regardless of the experimental site, the performance of estimating ground heat flux using OHM is better than the linear regression model. Detailed comparison statistics are also summarized in Table 3. From Table 3, it is evident that the performance of OHM is much better than the linear regression model for the campus grassland site (a₂ = 0.55 h) where R² is improved from 0.39 to 0.72 and the root mean square error (RMSE) reduced by 37% (from 70 to 48 W m⁻²). For the residential roof site (a₂ = 0.26 h), if the OHM is adopted then R² is slightly improved from 0.89 to 0.92 and RMSE is reduced by 17% (from 41 to 34 W m⁻²). For the peat bog (a₂ = −0.11 h), R² is slightly improved by 7% (from 0.81 to 0.87) and RMSE is reduced by 22% (from 13.3 to 10.7 W m⁻²) if the OHM is applied. For the agricultural grassland site, the estimations by the linear model and OHM are almost the same, since the hysteresis factor is small, only 0.032 h.

In addition, data in the parentheses in Table 3 represent the statistical values of OHM estimations resulting from the average values of coefficients a₁, a₂, and a₃; in other words, in this case, when the simulation is performed for each month, the values of a₁, a_2, and a₃ do not change from month to month, but are substituted by the average of all months. From Table 3, it is observed that when using the average values of the coefficients for estimating G by OHM, except for the residential roof site (where the values of coefficients do not change with month), the model performance is worse than that when using monthly coefficients. Nevertheless, the model performance of OHM in conjunction with mean values of coefficients is still better than the linear regression model if the hysteresis factor is not small (larger than 0.11 h).

5. Conclusions

This study evaluates the performance of objective hysteresis model for estimating ground heat flux across various surfaces, including residential roofs, campus grassland, agricultural grassland, and peat bog. Our measurements and analyses demonstrate the following:

(1)

Although the OHM coefficients (a₁, a₂, and a₃) may change from month to month, taking the average of these coefficients still yields better ground heat flux estimates than the linear regression model.

(2)

If the hysteresis factor is greater than 0.11 h, then the OHM is recommended over the linear regression model (at least 17% reduction in the root mean square error). For the agricultural grassland site, the hysteresis effect is only 0.03 h; hence, the linear regression model performs as good as OHM.

(3)

The partition coefficient of net radiation into ground heat flux at the above mentioned subtropical grassland was found to be smaller in the summer and larger in the winter. This is attributed to the higher evapotranspiration in the summer than in the winter.

(4)

The hysteresis factor (a₂) for the residential roof in Taichung, Taiwan was 0.26 h; this value is similar to Swindon, UK, but smaller than dense cities (e.g., London). For the campus grassland, a₂ = 0.55; this value is high and the same as the short grass site in St. Louis, USA. For the peat bog, a₂ = −0.11 h.

(5)

For the agricultural grassland, the hysteresis between net radiation and ground heat flux was small, only 0.03 h. This is similar to the short grass in Oklahoma, USA.

(6)

During the summer period, for a tropical grassland, ground heat flux could occupy 25% of the net radiation which is important in the surface energy balance budget. For a concrete residential roof, this partition is even up to 47%.