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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 10, NO. 5, SEPTEMBER 2013
Modeling of Day-to-Day Temporal Progression of Clear-Sky Land Surface Temperature Si-Bo Duan, Zhao-Liang Li, Hua Wu, Bo-Hui Tang, Xiaoguang Jiang, and Guoqing Zhou
Abstract—This letter presents a method to calculate the width ω over the half-period of the cosine term in a diurnal temperature cycle (DTC) model. ω deduced from the thermal diffusion equation (TDE) is compared with ω obtained from solar geometry. The results demonstrate that ω deduced from the TDE describes the shape of the DTC model more adequately around sunrise and the time of maximum temperature than ω obtained from solar geometry. Additionally, taking into account the physical continuity of land surface temperature (LST) variation, a day-to-day temporal progression (DDTP) model of LST is developed to model several days of DTCs. The results indicate that the DDTP model fits in situ [or Spinning Enhanced Visible and Infrared Imager (SEVIRI)] LST well with a root-mean-square error (RMSE) less than 1 K. Compared with the DTC model, the DDTP model slightly increases the quality of LST fits around sunrise. Assuming that only six LST measurements corresponding to the NOAA/AVHRR and MODIS overpass times for each day are available, several days of DTCs can be predicted by the DDTP model with an RMSE less than 1.5 K. Index Terms—Day-to-day temporal progression (DDTP), diurnal temperature cycle (DTC), land surface temperature (LST), modeling.
I. I NTRODUCTION
L
AND surface temperature (LST) is a key parameter at the land–atmosphere interface [1]. Land surface diurnal
Manuscript received June 22, 2012; revised October 19, 2012; accepted November 11, 2012. This work was supported by the National Natural Science Foundation of China under Grant 41071231 and Grant 41231170 and by the State Key Laboratory of Resources and Environment Information System under Grant 088RA800KA. The work of S.-B. Duan was supported by the China Scholarship Council for his stay in LSIIT, France. S.-B. Duan is with the State Key Laboratory of Resources and Environment Information System, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China; with the University of Chinese Academy of Sciences, Beijing 100049, China; and also with the Laboratoire des Sciences de l’Image, de l’Informatique et de la Télédétection, Université de Strasbourg, Centre National de la Recherche Scientifique, 67412 Illkirch, France (e-mail:
[email protected]). Z.-L. Li is with the State Key Laboratory of Resources and Environment Information System, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China, and also with the LSIIT, UdS, CNRS, 67412 Illkirch, France (e-mail:
[email protected]). H. Wu and B.-H. Tang are with the State Key Laboratory of Resources and Environment Information System, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China (e-mail:
[email protected];
[email protected]). X. Jiang is with the University of Chinese Academy of Sciences, Beijing 100049, China (e-mail:
[email protected]). G. Zhou is with the Guangxi Key Laboratory of Spatial Information and Geomatics, Guilin University of Technology, Guangxi 541004, China (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LGRS.2012.2228465
temperature cycle (DTC) is an important element in a wide range of applications within climatology and meteorology [2]. For instance, information on DTC can be used to infer thermal inertia and soil moisture [3]. Due to the intrinsic scanning characteristics of the sensors onboard polar orbiting satellites (e.g., NOAA/AVHRR or MODIS), the differences of local solar time for pixels on the same day or the same pixel on different days in a revisit period may reach up to 2 h [4]. Moreover, because LST changes with local solar time, it is not possible to directly compare LSTs of different pixels or of the same pixel at different days. DTC models have the potential to be used to interpolate LSTs to the same local solar time with a priori knowledge [5]. The performance of six DTC models was evaluated in terms of clear-sky in situ and satellite data [6]. All six models performed with similar accuracy at any time of the day except around sunrise and the time of maximum temperature. Nevertheless, none of the six DTC models were concerned with day-to-day temporal progression (DDTP) of LST, resulting in a physical discontinuity in the DTC models around sunrise. If no additional parameters are introduced, DTC models with the width over the half-period of the cosine term calculated from solar geometry cannot reproduce the slow and smooth increase in LST around sunrise well [6]. To improve the quality of LST fits around sunrise and the time of maximum temperature, a method is presented to calculate the width over the half-period of the cosine term in the DTC models. This width is deduced from the thermal diffusion equation (TDE). Taking into account the physical continuity of LST variation, a DDTP model of LST is developed to model several days of DTCs. The DDTP model can be used to further improve the quality of LST fits around sunrise. This letter is organized as follows: Section II introduces the DTC and DDTP models. Section III describes the data used in this letter. Results and discussion are presented in Section IV. Conclusion is drawn in the last section. II. M ETHODOLOGY The DTC model proposed by Inamdar et al. [7] is used in this letter. There are two reasons for the choice of this model. One reason is that this model is deduced from the TDE, which will be used in the following section. The other reason is that this model uses a hyperbolic function to more accurately describe the decay of LST at night. This model is referred to as the “INA08 model” in this letter and can be described as follows: π (t − tm ) , t < ts Tday (t) = T0 + Ta cos ω
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DUAN et al.: MODELING OF DDTP OF CLEAR-SKY LST
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π (ts − tm ) − δT Tnight (t) = (T0 + δT ) + Ta cos ω k , t ≥ ts (1) × (k + t − ts ) with k=
TABLE I D ESCRIPTION OF IN SITU AND SEVIRI LST DATA
π π δT ω (ts −tm ) − (ts − tm ) sin−1 tan−1 (2) π ω Ta ω
where Tday and Tnight are the LSTs of the daytime and nighttime parts, respectively; t is the time; T0 is the residual temperature around sunrise (tsr ); Ta is the temperature amplitude; ω is the width over the half-period of the cosine term; tm is the time at which temperature reaches its maximum; ts is the starting time of free attenuation; δT is the temperature difference between T0 and T (t → ∞); and k is the attenuation constant. More detailed description of these parameters can be found in [6]. A. INA08_1 Model The width ω (in hours) in the INA08 model can be determined by the duration of daytime [8], i.e., ω=
2 arccos (− tan φ tan δ) 15
(3)
where φ is the latitude, and δ is the solar declination. The INA08 model with ω estimated from (3) is called the “INA08_1 model” in this letter. There are five free parameters in the INA08_1 model (i.e., T0 , Ta , δT , tm , and ts ). B. INA08_2 Model Assuming the 1-D periodic heating of a uniform half-space of constant thermal properties, temperature obeys the TDE [9] K ∂ 2 T (z, t) ∂T (z, t) = ρc ∂z 2 ∂t
(4)
where K is the thermal conductivity, ρ is the density, c is the specific heat, and T (z, t) is the temperature at depth z below the surface and time t. A solution of the cosine function for (4) is π z z (t − tm ) − T (z, t) = a + b cos exp − (5) ω D D where D is the damping depth of the diurnal temperature wave, i.e., D = (2ωK/πρc )1/2 , and a and b are unknown coefficients. Under these conditions, the heat flux at the surface G(0, t), following the convention with positive sign in the downward direction, can be derived from (5), i.e., π Kb π ∂T cos (t−t . (6) = − )+ G(0, t) = −K m ∂z D ω 4 z=0
Assuming that G(0, t) at time tsr is equal to zero, i.e., G(0, tsr ) = 0, the width ω can be obtained from (6) by requiring an argument of −π/2 for the cosine, i.e., ω=
4 (tm − tsr ). 3
(7)
The INA08 model with ω calculated from (7) is referred to as the “INA08_2 model” in this letter. Free parameters in the INA08_2 model are the same as those in the INA08_1 model (i.e., T0 , Ta , δT , tm , and ts ). C. DDTP Model The INA08_2 model is used in the DDTP model to model several days of DTCs. Taking into account the physical continuity of LST variation, the nighttime part at day n and the daytime part at day n + 1 in the DDTP model are taken to be continuous at the time of minimum temperature tmin at day n + 1. By equating the end (lowest) temperature at day n with the starting temperature (around sunrise) at day n + 1, the value of T0 at day n + 1 can be obtained in terms of the values of the other parameters at day n or n + 1, i.e., π n n n (t − t ) − δT T0n+1 = T0n + δT n + Tan cos s m ωn π n
k n+1 n+1 n+1 × n t − T cos − t a m min n+1 ω n+1 k + tmin − tns (8) where the superscript n or n + 1 denotes day n or n + 1, respectively. The total number of free parameters in the DDTP model for n (n ≥ 2) days is 5n (i.e., Ta , tm , ts , and δT for each day; tmin for each day except for the first day; and T0 on the first day). To further reduce the number of free parameters in the DDTP model, it is assumed that the values of tmin for each day are equal within the range of several days. Therefore, the total number of free parameters in the DDTP model for n (n ≥ 2) days is 4n + 2 (i.e., Ta , tm , ts , and δT for each day; T0 on the first day; and tmin ). Except for the first day, the values of T0 for the other days are calculated from (8). In addition, the values of ω for each day are calculated from (7). III. DATA Both in situ measurements and geostationary satellite (e.g., Meteosat Second Generation (MSG) or GOES) observations can provide DTC. To evaluate the performance of the DTC and DDTP models at different spatial scales, a group of in situ LSTs and three groups of SEVIRI LSTs were collected with different geographical coordinates and land covers. Because several days of DTCs are needed to test the performance of the DDTP model, only a group of in situ LSTs is available to us.
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Fig. 1. LST differences (modeled LSTs minus in situ (or SEVIRI) LSTs) for the INA08_1 and INA08_2 models from sunrise to around ts on October 7, 1992 at Site A and on August 5, 2008 at Sites B–D. “INA08_1” denotes the INA08 model with the width ω calculated from solar geometry. “INA08_2” represents the INA08 model with ω deduced from the TDE.
The details of the collected data are presented in Table I. The selected four-day period is just taken as an example of several days continuously in time. The in situ LSTs were measured in the HAPEX-Sahel field experiment that was undertaken in western Niger, in the West African Sahel region. To reduce the influence of wind on the in situ LSTs, each data point was averaged at a 10-min interval. The SEVIRI LSTs were derived from MSG-SEVIRI data using the algorithm proposed by Jiang et al. [10] and Jiang and Li [11]. The three groups of the SEVIRI LSTs respectively represent different land cover types according to the Global Land Cover 2000 map. The selected date is based on the fact that more cloud-free days are available over the African and Iberian Peninsula area in August. The Levenberg–Marquardt minimization scheme is used to fit the DTC and DDTP models to the LST data sets. More detailed information on the initialization of free parameters in the models can be found in [6]. IV. R ESULTS AND D ISCUSSION A. Evaluation of the INA08_1 and INA08_2 Models The in situ LSTs on October 7, 1992 at Site A and the SEVIRI LSTs on August 5, 2008 at Sites B–D were taken as examples to evaluate the performance of the INA08_1 and INA08_2 models. The LST differences [modeled LSTs minus in situ (or SEVIRI) LSTs] for the two models at the four sites are shown in Fig. 1(a)–(d). Because there are no significant differences between the INA08_1 and INA08_2 models for the nighttime part, only the LST differences from sunrise to around ts are displayed in Fig. 1(a)–(d). The INA08_2 model shows a better performance around sunrise and tm than the INA08_1 model in terms of the LST differences and root-mean-square errors (RMSEs). The largest LST differences of the INA08_1 model around sunrise at Sites A–D are approximately −4.5, −3, −2.4, and −4 K, respectively, whereas those of the INA08_2 model are approximately −2.5, −1.5, −1.2, and −2 K,
Fig. 2. DDTP model fitting in situ (or SEVIRI) LSTs on October 4–7, 1992 at Site A and on August 2–5, 2008 at Sites B–D.
respectively. In addition, the RMSEs of the INA08_1 model at Sites A–D are 0.99, 0.64, 0.68, and 0.9 K, respectively, whereas those of the INA08_2 model are 0.62, 0.4, 0.4, and 0.43 K, respectively. The results can be largely explained by the fact that the width ω given by (7) is substantially smaller than that given by (3). The smaller width ω leads to the narrower shape of the cosine function in the INA08_2 model. The real DTC does not follow a pure cosine function controlled by solar geometry, which is narrowed by atmospheric attenuation of solar irradiation at large zenith angle (i.e., large air mass), as pointed out by Göttsche and Olesen [12]. Therefore, the INA08_2 model with smaller width ω fits the in situ (or SEVIRI) LSTs better than the INA08_1 model. However, the INA08_2 model still cannot completely reproduce the slow and smooth increase in LST around sunrise. The fast increase in LST around sunrise for this model is still unphysical. B. Modeling of DDTP of LST Fig. 2(a)–(d) displays the DDTP model fitting the in situ LSTs on October 4–7, 1992 at Site A and the SEVIRI LSTs on August 2–5, 2008 at Sites B–D. The selected four-day period is just taken as an example of several days continuously in time. The DDTP model shows a good performance to fit the in situ (or SEVIRI) LSTs with an RMSE less than 1 K. Fitting the DDTP model to several days of DTCs summarizes the thermal behavior of the land surface and yields representative and informative thermal surface parameters (TSPs) [8]. These TSPs depend on all modeled LSTs and are not influenced by small gaps [see Fig. 2(a)] due to the technical problems or brief cloud cover as well as by outliers [see Fig. 2(b)] due to the undetected clouds. Therefore, these TSPs can be used to interpolate missing data [8] or to improve cloud screening algorithms [7]. However, the performance of the DDTP model depends on the quality of LST as well as on atmospheric and surface wind conditions. A successful application of the DDTP model requires two or more nearly cloud-free DTCs. Such conditions can be met over arid and semiarid areas.
DUAN et al.: MODELING OF DDTP OF CLEAR-SKY LST
Fig. 3. LST differences (modeled LSTs minus in situ (or SEVIRI) LSTs) for the INA08_2 and DDTP models from sunrise to around 11 h on October 7, 1992 at Site A and on August 5, 2008 at Sites B–D. “INA08_2” represents the INA08 model with ω deduced from the TDE. “DDTP” represents the model of DDTP of LST.
Fig. 3(a)–(d) shows the LST differences (modeled LSTs minus in situ (or SEVIRI) LSTs) for the INA08_2 and DDTP models on October 7, 1992 at Site A and on August 5, 2008 at Sites B–D. Because of negligible differences between the INA08_2 and DDTP models after 11 h, only the LST differences from sunrise to around 11 h are displayed in Fig. 3(a)–(d). The DDTP model shows a slightly better performance around sunrise than the INA08_2 model at the four sites in terms of the LST differences and RMSEs. The largest LST differences of the INA08_2 model around sunrise at Sites A–D are approximately −2.5, −1.5, −1, and −2 K, respectively, whereas those of the DDTP model are approximately −1, −0.5, −0.5, and −1 K, respectively. Furthermore, the RMSEs of the INA08_2 model at Sites A–D are 0.96, 0.53, 0.51, and 0.57 K, respectively, whereas those of the DDTP model are 0.53, 0.37, 0.45, and 0.35 K, respectively. These results mainly come from the fact that the DDTP model takes into account the continuity of LSTs at time tmin . Comparing Fig. 3(a) with Fig. 3(b)–(d), the differences between the INA08_2 and DDTP models for the in situ LSTs are larger than those for the SEVIRI LSTs. By carefully analyzing the in situ LST data, we found that those data have relatively large fluctuations, which may be caused by the wind, leading to relatively poor data quality compared with the SEVIRI LSTs. One useful application of the DDTP model is to implement data interpolation in terms of limited satellite measurements. We assume that only six in situ (or SEVIRI) LSTs corresponding to the NOAA/AVHRR and MODIS overpass times (01:30, 07:30, 10:30, 13:30, 19:30, and 22:30) for each day are available. Several days of DTCs are predicted by the DDTP model by means of only six in situ (or SEVIRI) LSTs for each day on October 4–7, 1992 at Site A and on August 2–5, 2008 at Sites B–D. The results are shown in Fig. 4(a)–(d). The predicted LSTs are in good agreement with the in situ (or SEVIRI) LSTs with an RMSE less than 1.5 K. Comparing Fig. 4(a)–(d) with Fig. 2(a)–(d), the RMSEs using only six LSTs for each day
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Fig. 4. Several days of DTCs predicted by the DDTP model in terms of only six in situ (or SEVIRI) LSTs for each day on October 4–7, 1992 at Site A and on August 2–5, 2008 at Sites B–D. LST differences (modeled LSTs minus in situ (or SEVIRI) LSTs) are also shown. Filled squares represent the LST measurements corresponding to the NOAA/AVHRR and MODIS overpass times (01:30, 07:30, 10:30, 13:30, 19:30, and 22:30).
are approximately two times larger than the RMSEs using all LST measurements. Relatively larger LST differences can be observed between approximately 15 and 18 h for each day, due to the lack of LST measurements over this period. If one more LST measurement can be obtained over this period, the predicted LST accuracy can be improved. In addition, if six LST measurements more evenly distributed (e.g., 01:00, 05:00, 09:00, 13:00, 17:00, and 21:00), the predicted LST accuracy can be also improved. V. C ONCLUSION In this letter, we have presented a method to calculate the width ω over the half-period of the cosine term in the INA08 model. ω was deduced from the TDE and compared with ω calculated from solar geometry. The INA08_2 model shows a better performance around sunrise and tm than the INA08_1 model. However, for the INA08_2 model, it is still difficult to describe the slow and smooth increase in LST around sunrise. The INA08_2 model was used in the DDTP model to model several days of DTCs. The DDTP model shows a good performance in fitting the in situ (or SEVIRI) LSTs with an RMSE less than 1 K. Compared with the INA08_2 model, the DDTP model slightly improves the quality of LST fits around sunrise. Furthermore, the performance of data interpolation of the DDTP model was investigated, assuming that only six LST measurements corresponding to the NOAA/AVHRR and MODIS overpass times for each day are available. The results demonstrate that several days of DTCs can be predicted by the DDTP model with an RMSE less than 1.5 K. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their constructive and insightful comments, which helped to significantly improve this letter.
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