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SERRA: 西北干旱内陆河流域实际蒸散发量估算成果发表
热度 3 yaozi824 2016-6-13 14:19
西北干旱内陆河流域实际蒸散发量估算 流域实际蒸散发的精确计算,是流域水量平衡分析的关键问题,也是难点问题。 Budyko 水热耦合平衡假设为解决这一问题提供了基础,认为降水量是干旱区实际蒸散发量的主要控制性因素,在年尺度上干旱内陆地区实际蒸散发与潜在蒸散发之间呈互补关系。傅抱璞公式是具有水文气象的物理意义的 Budyko 假设解析表达式,而与流域下垫面特征相关的参数w 值是傅抱璞公式中的唯一参数,因此,参数 w 值的估算是关键点。 本研究选取 68 个中国西北干旱内陆河流域,验证了 Budyko 水热耦合平衡假设在我国干旱内陆河流域的适用性,以傅抱璞公式为基础,提出了植被-土壤相对蓄水能力 S max /ET 、平均坡度 tan β 和集水面积 A 为参数的计算w 值的半经验公式,构建了干旱内陆河流域实际蒸散发量的估算模型,并计算了各流域逐年实际蒸散发量。与水量平衡结果进行验证,发现傅抱璞公式及本研究提出的参数w 的半经验公式一起,可以准确的估算干旱内陆河流域实际蒸散发量,并发现该公式在中小流域估算更加准确,而在大流域有较大的误差。 值得一提的是,由于内陆河流域无产流的独特性,在整个流域面积上应用 Budyko 假设有理论上的局限,而应该以出山口以上的集水面积作为流域面积来考虑。 研究成 果以 Annual actual evapotranspiration in inland river catchments__of China based on t.pdf 为题发表在Stoch Environ Res Risk Assess上( Yao J Q , Mao W Y, Yang Q, et al., Annual actual evapotranspiration in inland rivercatchments of China based on the Budyko framework, Stoch Environ Res RiskAssess, 2016 (30), DOI 10.1007/s00477-016-1271-1 ),见附件。
个人分类: 气象万千|6469 次阅读|7 个评论
综述论文:基于遥感的农业用水效率评价方法研究进展
shangsh 2015-10-23 17:30
知网链接: http://www.cnki.net/KCMS/detail/detail.aspx?QueryID=23CurRec=6recid=filename=NYJX201510013dbname=CJFDTEMPdbcode=CJFQpr=urlid=yx=v=MjgxNDVyRzRIOVROcjQ5RVo0UjhlWDFMdXhZUzdEaDFUM3FUcldNMUZyQ1VSTCtlWnVSdUZ5SGdXNy9BS3pUQmQ= 农业机械学报期刊链接: http://www.j-csam.org/ch/reader/view_abstract.aspx?file_no=20151013flag=1 尚松浩, 蒋磊, 杨雨亭. 基于遥感的农业用水效率评价方法研究进展. 农业机械学报, 2015, 46 (10): 81-92. Shang Songhao, Jiang Lei and Yang Yuting. Review of Remote Sensing-based Assessment Method for Irrigation and Crop Water Use Efficiency .Transactions of the Chinese Society for Agricultural Machinery, 2015, 46(10): 81-92. DOI: 10.6041/j.issn.1000-1298.2015.10.013 基于遥感的农业用水效率评价方法研究进展 投稿时间: 2015-06-29 DOI: 10.6041/j.issn.1000-1298.2015.10.013 中文关键词 : 灌溉水利用效率 作物水分利用效率 遥感 蒸散发 作物分布 作物估产 基金项目 : 国家自然科学基金资助项目(51479090)和 “十二五”国家科技支撑计划资助项目(2013BAB05B03) 作者 单位 尚松浩 清华大学 蒋磊 清华大学 杨雨亭 澳大利亚联邦科学与工业研究组织水土研究所 中文摘要 : 遥感技术的发展为区域尺度蒸散发计算、作物分布识别及估产提供了一条有效途径,为基于遥感信息的灌区灌溉水利用效率及作物水分利用效率定量评价奠定了基础。回顾总结了遥感蒸散发模型、瞬时蒸散发升尺度方法、日蒸散发插值方法、作物分布识别方法及作物估产模型的研究进展,评述了遥感蒸散发及作物估产结果在灌区灌溉水利用效率及作物水分利用效率评价中的应用情况。提出了相关领域需要进一步研究的问题,包括适合非均匀下垫面特点且具有较强物理基础的灌区遥感蒸散发模型、日蒸散发插值中灌溉或降雨引起土壤含水量突变情况的处理、农田蒸散发中灌溉水有效消耗量的准确估算、能适应复杂种植结构并且适用于多年的作物分布遥感识别模型以及精度较高且可操作性强的遥感估产模型等。 Review of Remote Sensing-based Assessment Method for Irrigation and Crop Water Use EfficiencyIrrigation and Crop Water Use Efficiency Shang Songhao Jiang Lei Yang Yuting Tsinghua University,Tsinghua University and CSIRO Land and Water, Canberra ACT 2601, Australia) Key Words : Irrigation water use efficiency Crop water use efficiency Remote sensing Evapotranspiration Crop classification Crop yield estimation Abstract : Advances in remote sensing (RS) technology provide an effective approach for the estimation of regional evapotranspiration, crop classification and yield, which lays the foundation for the quantitative assessment of irrigation efficiency and crop water use efficiency. Relevant models and methods were reviewed, including RS-based evapotranspiration models, upscaling methods of instantaneous evapotranspiration to daily evapotranspiration, interpolation methods of available daily evapotranspiration, RS-based crop classification methods and crop yield estimation models, and assessment methods for irrigation efficiency and crop water use efficiency based on evapotranspiration and crop yield estimations. Furthermore, key issues that require further investigation were proposed, including RS-based evapotranspiration model with strong physical bases and applicable to heterogeneous land surfaces, proper interpolation method of daily evapotranspiration considering the abrupt change of soil water content due to irrigation and/or precipitation, precise estimation of irrigation water consumption in farmland evapotranspiration, RS-based crop classification model applicable to complex cropping pattern and multi-years, and operational RS-based crop yield estimation model with higher precision. With further developments of RS technology and relevant models, the quantitative assessment results of irrigation water use efficiency and crop water use efficiency will provide sound foundation for agricultural water management.
个人分类: 论著|3807 次阅读|0 个评论
论文:基于混合双源模式和梯形状态空间的遥感蒸散发模型HTEM
shangsh 2013-3-6 19:44
http://onlinelibrary.wiley.com/doi/10.1002/jgrd.50259/abstract A Hybrid Dual Source Scheme and Trapezoid Framework Based Evapotranspiration Model (HTEM) Using Satellite Images: Algorithm and Model Test † Yuting Yang, Songhao Shang * DOI:10.1002/jgrd.50259 2013 American Geophysical Union. All Rights Reserved. Issue Journal of Geophysical Research: Atmospheres Accepted Article (Accepted, unedited articles published online and citable. The final edited and typeset version of record will appear in future.) Additional Information (Hide All) Author Information Publication History Author Information State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing, China * Corresponding Author: Songhao Shang, Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China Email: shangsh@tsinghua.edu.cn Telephone: 86-10-62796674 Fax: 86-10-62796971 † This article has been accepted for publication and undergone full peer review but has not been through the copyediting, typesetting, pagination and proofreading process, which may lead to differences between this version and the Version of Record. Please cite this article as doi: 10.1002/jgrd.50259 Publication History Accepted manuscript online: 4 MAR 2013 09:43AM EST Manuscript Accepted: 7 FEB 2013 Manuscript Revised: 30 JAN 2013 Manuscript Received: 19 JUL 2012 AGU EASI Search Browse by Index Term ARTICLE TOOLS Get PDF (10570K) Save to My Profile E-mail Link to this Article Export Citation for this Article Get Citation Alerts Request Permissions More Sharing Services Share | Share on citeulike Share on connotea Share on delicious Share on twitter Abstract Cited By Get PDF (10570K) Keywords: Satellite remote sensing; Evaporation; Hybrid dual source scheme; Trapezoid framework; Transpiration Abstract Satellite remote sensing techniques are widely considered as the most promising way to estimate evapotranspiration (ET) over large geographic extents. In this study, a hybrid dual source scheme and trapezoid framework based evapotranspiration model (HTEM) is developed to map evapotranspiration from satellite imagery. It adopts a theoretically-determined vegetation index/Land surface temperature trapezoidal space to decompose bulk radiative surface temperature into component temperatures (soil canopy), and uses a hybrid dual source scheme of the layer approach and patch approach to partition net radiation and estimate sensible and latent fluxes separately from the soil and canopy. The proposed model was tested at the Soil Moisture-Atmosphere Coupling Experiment (SMACEX) site in central Iowa, U.S., on three days during the campaign in 2002 using Landsat TM/ETM + data, and at the Weishan flux site on North China Plain during the main growing season of 2007 with MODIS Terra images. Results indicate that HTEM is capable in estimating latent heat flux ( LE ) with the mean absolute percentage error of 6.4% and 11.2% for the SMACEX and the Weishan sites, respectively. Besides, the model was found to be able to give reasonable evaporation and transpiration partitioning at both sites. Compared with other models, the HTEM generally produced better sensible and latent flux estimates at the two sites and had comparable ability to estimate net radiation and ground heat flux. Sensitivity analysis suggests that HTEM is most sensitive to temperature variables and less sensitive to other meteorological observations and parameters. Get PDF (10570K) More content like this Find more content: like this article Find more content written by: Yuting Yang Songhao Shang All Authors Index Terms: Energy budgets Evapotranspiration Remote sensing
个人分类: 论著|6146 次阅读|0 个评论
论文:双源蒸散发模型估算潜在蒸散发量的对比
热度 1 shangsh 2012-12-22 18:34
期刊网址: http://www.tcsae.org/ch/reader/view_abstract.aspx?file_no=20122413flag=1 中国知网: http://www.cnki.net/KCMS/detail/detail.aspx?QueryID=15CurRec=5recid=filename=NYGU201224017dbname=CJFDTEMPdbcode=CJFQpr=urlid=yx 万方数据: http://d.g.wanfangdata.com.cn/Periodical_nygcxb201224015.aspx 杨雨亭,尚松浩.双源蒸散发模型估算潜在蒸散发量的对比 .农业工程学报,2012,28(24):85-91 双源蒸散发模型估算潜在蒸散发量的对比 Comparison of dual-source evapotranspiration models in estimating potential evaporation and transpiration 中文关键词 : 蒸散发,植被,模型,潜在蒸发,潜在蒸腾,层状模型,块状模型,混合型模型 英文关键词 : evapotranspiration , vegetation , models , potential evaporation , potential transpiration , layer model , patch model , hybrid dual-source model 作者 单位 杨雨亭 清华大学水沙科学与水利水电工程国家重点实验室,北京 100084 尚松浩 清华大学水沙科学与水利水电工程国家重点实验室,北京 100084 中文摘要 : 潜在蒸散发取决于气候条件与下垫面植被覆盖状况,是农田及流域水循环研究的重要内容。该文针对现有的3种估算潜在蒸散发的双源模型(层状模型、块状模型、混合型模型),设定不同的下垫面植被覆盖状况,较为详细地比较了3种模型在不同植被覆盖条件下的潜在蒸散发估算能力,并对其区分土壤表面潜在蒸发与植被冠层潜在蒸腾的合理性进行了评价。结果表明,混合双源模型较层状模型和块状模型具有更好的模拟效果,能够适用于更广的下垫面植被覆盖状况。 英文摘要 : Potential evapotranspiration depends greatly on local climate and vegetation distribution conditions, and is important in studying the cropland and basin hydrological circles. The current study provides a detailed comparison of the performances among three dual-source evapotranspiration models, including the layer model, the patch model and the hybrid model, in estimating and partitioning potential evaporation and potential transpiration under different hypothetical vegetation distribution conditions. The layer model ignores the difference of energy fluxes between under- and inter-canopy soil; while the patch approach assumes a full radiation loading for both the canopy and inter-canopy soil and ignores the evaporation from under-canopy soil surfaces; The hybrid model is a combination of the layer and patch models, and adopts the layer approach to partition available energy between canopy and soil and uses the patch approach to calculate energy fluxes. As a result, both under- and inter- canopy soil evaporation were estimated and distinguished in the hybrid model. In simulation scenarios, the height of vegetation was assumed to be 5 m with canopy leaf area index of 2 and minimum stomatal resistance of 170 s/m. The bulk surface leaf area index (LAI) varied from 0.5 to 5, and fractional vegetation coverage (Fr) varied from 10% to 100%. The vegetation clumpy patterns were quantified by fixing LAI while varying Fr. The climate data was obtained from the Linhe meteorological station located in an arid region in central Inner Mongolia of North China. The results indicated that both the patch and hybrid model performed reasonably well in estimating potential evapotranspiration under homogeneous vegetation distribution conditions. However, the layer model tended to overestimate potential evapotranspiration, as it generally gave higher potential evaporation estimates. The overestimation in potential evapotranspiration by the layer model was increased with the increase of Fr and the decrease of LAI. On the contrary, the patch model had a tendency to underestimate potential evaporation, especially with high Fr and low LAI. For heterogeneous vegetation distribution conditions, potential evapotranspiration estimated from the layer model was generally higher than that given by the patch and hybrid model, particularly with low Fr. Potential evaporation (potential transpiration) from the layer model increases (decreases) with the increase of LAI. However, both variables from the layer model did not change with changes of Fr. In contrast, potential transpiration estimated from the patch and hybrid model was increased with the increase of both LAI and Fr. Potential evaporation from the patch model was increased with the increase of Fr, but kept relative constant under various LAI conditions, while potential evaporation from the hybrid model was increased with the decrease of both Fr and LAI. The above results suggest that the layer model may give reasonable potential transpiration estimates over homogeneous vegetated surfaces, while the patch model is more suitable for surfaces with lower fractional and clumped vegetation cover. By contrast, the hybrid model performs better than both the layer model and the patch model, which can be used to estimate potential evaporation and potential transpiration partitioning for a wide range of surfaces with different vegetation distribution patterns. 查看全文
个人分类: 论著|6465 次阅读|1 个评论
我们在土壤水动态模拟模型方面的研究工作——主要论文汇总
热度 2 shangsh 2012-11-9 21:16
今天“土壤水研究”群组获批开张,总结一下我们在 过去 10余 年中开展的土壤水分动态模型模型研究工作,与各位群友交流。也特别邀请同行加入土壤水研究群组( http://bbs.sciencenet.cn/forum.php?mod=groupfid=620 ),共同探讨土壤水问题,交流研究心得与成果。 1. 指数消退模型: (1) 简单指数模型:尚松浩,雷志栋,杨诗秀 . 冬小麦田间墒情预报的经验模型.农业工程学报, 2000 , 16(5): 31 – 33. ( http://www.tcsae.org/ch/reader/view_abstract.aspx?flag=1file_no=20000508journal_id=nygcxb ) (2) 扩展指数模型: Shang Songhao, Mao Xiaomin. A physicoempirical model for soil water simulation in crop root zone. Pedosphere, 2011, 21(4): 512 – 521. ( http://blog.sciencenet.cn/home.php?mod=attachmentid=7364 ) (3) 模拟农田土壤水分动态的二参数指数消退模型 (2014.3.14 on line ): Shang Songhao, Mao Xiaomin. A two-parameter exponential recession model for simulating cropland soil moisture dynamics. Chinese Geographical Science , 2014. doi: 10.1007/s11769-014-0676-3. http://link.springer.com/article/10.1007/s11769-014-0676-3?sa_campaign=email/event/articleAuthor/onlineFirst 2. 人工神经网络模型:尚松浩,毛晓敏,雷志栋,杨诗秀.冬小麦田间墒情预报的 BP 神经网络模型.水利学报, 2002 , (4): 60 – 63. ( http://jhe.ches.org.cn/jhe/ch/reader/view_abstract.aspx?flag=1file_no=020412journal_id=jhe ) 3. 水量平衡模型及其应用: Shang Songhao, Mao Xiaomin. Application of a simulation based optimization model for winter wheat irrigation scheduling in North China. Agricultural Water Management, 2006, 85(3): 314 – 322. ( http://dx.doi.org/10.1016/j.agwat.2006.05.015 ) 下载: 200610AGWAT_Application of a simulation based optimization model for winter whea.pdf 4. 冻土水热耦合模型及其应用: (1) 冻土水热耦合迁移模型:尚松浩,雷志栋,杨诗秀.冻结条件下水热耦合迁移数值模拟的改进.清华大学学报, 1997 , 37(8): 62 – 64 ( 下载: 1997qhdxxb冻结条件下土壤水热耦合迁移数值模拟的改进.pdf ) (2) 模型应用 1 ——土壤冻结过程中潜水蒸发:雷志栋,尚松浩,杨诗秀,王 义,赵冬梅.土壤冻结过程中潜水蒸发规律的模拟研究.水利学报, 1999 , (6): 6 – 10. ( http://jhe.ches.org.cn/jhe/ch/reader/view_abstract.aspx?flag=1file_no=990602ljournal_id=jhe ) (3) 模型应用 2 ——土壤冻融过程中地下水位变化:尚松浩,雷志栋,杨诗秀,王 义,赵冬梅.冻融期地下水位变化情况下土壤水分运动的初步研究.农业工程学报, 1999 , 15(2): 64 – 68. ( http://www.tcsae.org/ch/reader/view_abstract.aspx?flag=1file_no=19990214journal_id=nygcxb ) (4) 冻土—大气系统水热传输模型:尚松浩,雷志栋,杨诗秀.冬灌对越冬期土壤水分状况影响的数值模拟.农业工程学报, 1997 , 13(3): 65 – 70. ( http://www.tcsae.org/ch/reader/view_abstract.aspx?flag=1file_no=19970313journal_id=nygcxb ) 5. 土壤—植物—大气连续体( SPAC )水分传输模型: Shang Songhao, Li Xichun , Mao Xiaomin, Lei Zhidong . Simulation of water dynamics and irrigation scheduling for winter wheat and maize in seasonal frost areas . Agricultural Water Management , 2004 , 68(2): 117 – 133. ( http://dx.doi.org/10.1016/j.agwat.2004.03.009 ) 下载: 200610AGWAT_Application of a simulation based optimization model for winter whea.pdf 6. SPAC 水热耦合传输模型: Yang Yuting, Shang Songhao, Guan Huade. Development of a soil-plant-atmosphere continuum model (HDS-SPAC) based on hybrid dual-source approach and its verification in wheat field. Science China Technological Sciences, 2012, 55(10): 2671 – 2685. DOI: 10.1007/s11431-012-4974-7. ( http://tech.scichina.com:8082/sciEe/EN/abstract/abstract508574.shtml ) ( 2009 年以前的相关研究工作已汇集在:尚松浩,毛晓敏,雷志栋,杨诗秀. 土壤水分动态模拟模型及其应用. 科学出版社, 2009 . http://bbs.sciencenet.cn/blog-39095-241519.html )
个人分类: 论著|6188 次阅读|7 个评论
基于混合双源模式的土壤-植物-大气连续体模型及其在小麦田的检验
shangsh 2012-8-9 08:40
SCIENCE CHINA Technological Sciences 2012, DOI: 10.1007/s11431-012-4974-7 Development of a soil-plant-atmosphere continuum model (HDS-SPAC) based on hybrid dual-source approach and its verification in wheat field 摘要及全文下载 : 201210-SC-TS-Development of a soil-plant-atmosphere continuum model (HDS-SPAC) b.pdf http://tech.scichina.com:8082/sciEe/EN/abstract/abstract508574.shtml YuTing Yang , SongHao Shang and Huade Guan Abstract HDS-SPAC, a new soil-plant-atmosphere continuum (SPAC) model, is developed for simulating water and heat transfer in SPAC. The model adopts a recently proposed hybrid dual source approach for soil evaporation and plant transpiration partitioning. For the above-ground part, a layer approach is used to partition available energy and calculate aerodynamic resistances, while a patch approach is used to derive sensible heat and latent heat fluxes from the two sources (soil and vegetation). For the below-ground part, soil water and heat dynamics are described by the mixed form of Richards equation, and the soil heat conductivity equation, respectively. These two parts are coupled through ground heat flux for energy transfer, root-zone water potential-dependent stomatal resistance, and surface soil water potential-dependent evaporation for water transfer. Evaporation is calculated from the water potential gradient at soil-atmosphere interface and aerodynamic resistance, and transpiration is determined using a Jarvis-type function linking soil water availability and atmospheric conditions. Some other processes, such as canopy interception and deep percolation, are also considered in the HDS-SPAC model. The hybrid dual-source approach allows HDS-SPAC to simulate heat and water transfer in an ecosystem with a large range of vegetation cover change temporally or spatially. The model was tested with observations at a wheat field in North China Plain over a time of three months covering both wet and dry conditions. The fractional crop covers change from 30% to over 90%. The results indicated that the HDS-SPAC model can estimate actual evaporation and transpiration partitioning and soil water content and temperature over the whole range of tested vegetation coverage.
个人分类: 论著|4451 次阅读|0 个评论
论文:水分胁迫条件下实际蒸散发量的计算方法与时间步长
热度 2 shangsh 2012-1-21 09:10
201210-HP-Calculating actual crop evapotranspiration under soil water stress con.pdf http://onlinelibrary.wiley.com/doi/10.1002/hyp.8405/abstract Calculating actual crop evapotranspiration under soil water stress conditions with appropriate numerical methods and time step(水分胁迫条件下实际蒸散发量的计算方法与时间步长) Songhao Shang * Author Information State Key Laboratory of Hydroscience and Engineering, Department of Hydraulic Engineering, Tsinghua University, Beijing, China Email: Songhao Shang (shangsh@tsinghua.edu.cn) * Songhao Shang, Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China. E-mail: shangsh@tsinghua.edu.cn Article first published online: 20 JAN 2012 DOI:10.1002/hyp.8405 Copyright 2011 John Wiley Sons, Ltd. Issue Hydrological Processes How to Cite: Shang, S. (2012), Calculating actual crop evapotranspiration under soil water stress conditions with appropriate numerical methods and time step. Hydrological Processes. doi:10.1002/hyp.8405 Publication History Article first published online: 20 JAN 2012 Accepted manuscript online: 13 NOV 2011 10:15PM EST Manuscript Accepted: 1 NOV 2011 Manuscript Received: 30 APR 2011 Keywords(关键词): evapotranspiration(蒸散发); water stress(水分胁迫); time step(时间步长); explicit Euler method(显式欧拉法); midpoint method(中点法); Heun's third-order method(Heun三阶方法) 摘要:水分胁迫条件下实际蒸散发的计算是水文模拟与灌溉用水管理中的基本问题。实际蒸散发的计算取决于根据根系层含水量估算水分胁迫系数,其结果与采用的数值计算方法和时间步长。在无降水及灌溉的土壤水分消退期,实际蒸散发量可根据解析法及数值法计算。以解析法计算结果为基础,比较了几种常用数值计算方法(包括显式、隐式及修正欧拉法、中点法及Heun三阶方法)的计算精度。结果表明实际蒸散发量的一步计算相对误差与初始土壤含水量无关,且相对误差绝对值随数值计算方法的阶数而减小。 相对误差绝对值也随着计算步数而减小,可以保证连续模拟过程中计算结果的稳定性。考虑不同时间步长及最大蒸散发量情况下的计算误差及计算复杂性,显式欧拉法可用于时间步长1d或2d(最大蒸散发量5 mm/d)时的时间蒸散发量计算;中点法或修正欧拉法可用于时间步长7d或10d(最大蒸散发量5 mm/d)的情况;而Heun三阶方法可用于时间步长15d的情况。 Abstract: Calculation of actual crop evapotranspiration under soil water stress conditions is crucial for hydrological modeling and irrigation water management. Results of actual evapotranspiration depend on the estimation of water stress coefficient from soil water storage in the root zone, which varies with numerical methods and time step used. During soil water depletion periods without irrigation or precipitation, the actual crop evapotranspiration can be calculated by an analytical method and various numerical methods. We compared the results from several commonly used numerical methods, including the explicit, implicit and modified Euler methods, the midpoint method, and the Heun's third-order method, with results of the analytical method as the bench mark. Results indicate that relative errors of actual crop evapotranspiration calculated with numerical methods in one time step are independent of the initial soil water storage in the range of soil water stress. Absolute values of relative error decrease with the order of numerical methods. They also decrease with the number of time step, which can ensure the numerical stability of successive simulation of soil water balance. Considering the calculation complexity and calculation errors caused by numerical approximation for different time step and maximum crop evapotranspiration, the explicit Euler method is recommended for the time step of 1 day (d) or 2 d for maximum crop evapotranspiration less than 5 mm/d, the midpoint method or the modified Euler method for the time step of up to one week or 10 d for maximum crop evapotranspiration less than 5 mm/d, and the Heun's third-order method for the time step of up to 15 d.
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