The effect of EGM2008-based normal, normal-orthometric and Helmert orthometric height systems

M. S. Filmer · W. E. Featherstone · M. Kuhn
Received: 16 November 2009 / Accepted: 18 May 2010 / Published online: 11 June 2010 © Springer-Verlag 2010
Abstract This paper investigates the normal-orthometric correction used in the definition of the Australian Height Datum, and also computes and evaluates normal and Helmert orthometric corrections for the Australian National Levelling Network (ANLN). Testing these corrections in Australia is important to establish which height system is most appropriate for any new Australian vertical datum. An approximate approach to assigning gravity values to ANLN benchmarks (BMs) is used, where the EGM2008-modelled gravity ﬁeld is used to ‘re-construct’ observed gravity at the BMs. Network loop closures (for ﬁrst- and second-order levelling) indicate reduced misclosures for all height corrections considered, particularly in the mountainous regions of south eastern Australia. Differences between Helmert orthometric and normal-orthometric heights reach 44 cm in the Australian Alps, and differences between Helmert orthometric and normal heights are about 26 cm in the same region. Normalorthometric heights differ from normal heights by up to 18 cm in mountainous regions >2,000 m. This indicates that the quasigeoid is not compatible with normal-orthometric heights in Australia. Keywords AHD Height systems · EGM2008 · Vertical datums ·
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502 Fig. 1 The Australian National Levelling Network (ANLN). First-order sections are in yellow, second-order sections in light green, third-order in thin grey, fourth-order in dark green, one-way (third-order) in reபைடு நூலகம் and two-way (order undeﬁned; Steed 2006, pers. comm.) in blue. Lambert projection. ANLN data courtesy of Geoscience Australia
J Geod (2010) 84:501–513 DOI 10.1007/s00190-010-0388-0
ORIGINAL ARTICLE
The effect of EGM2008-based normal, normal-orthometric and Helmert orthometric height systems on the Australian levelling network
M. S. Filmer (B) · W. E. Featherstone · M. Kuhn Western Australian Centre for Geodesy, The Institute for Geoscience Research, Curtin University of Technology, GPO Box U1987, Perth, WA 6845, Australia e-mail: M.Filmer@.au W. E. Featherstone e-mail: W.Featherstone@.au M. Kuhn e-mail: M.Kuhn@.au
1 Introduction The Australian Height Datum (AHD) is Australia’s ﬁrst and only ofﬁcial height datum. On the mainland, it was established in 1971 from the least-squares adjustment [ﬁxed to mean sea level (MSL) = zero at 30 tide gauges] of the (then named) Australian Levelling Survey (ALS) comprising ∼97,000 km of ‘primary’ levelling (Roelse et al. 1971). Due to the lack of computing power at that time, the primary network was adjusted on a State-by-State basis before the full national adjustment was conducted. Approximately 80,000 km of ‘supplementary’ levelling were subsequently least-squares adjusted onto the primary junction point (JP) heights (held ﬁxed), which were deﬁned in the full national adjustment (Roelse et al. 1971). The Australian National Levelling Network (ANLN) in Tasmania was adjusted in 1983 (NMC 1986), with MSL at two tide gauges held ﬁxed at zero in the adjustment. An offset between AHD(mainland) and AHD(Tas) thus exists which is estimated to be 10–20 cm (Rizos et al. 1991; Featherstone 2000). A number of issues have arisen with regard to the AHD, with numerous studies investigating different deﬁciencies in the datum (e.g., Featherstone 2001). The major areas of concern in the AHD include the ﬁxing of the levelling network (now referred to as ANLN; see Fig. 1) to 30 tide gauges using MSL as the zero reference (e.g., Hamon and Greig 1972; Mitchell 1973b; Coleman et al. 1979), the quality of the levelling (e.g., Morgan 1992; Kearsley et al. 1993; Filmer and Featherstone 2009) and the omission of gravity-based height corrections (e.g., Mitchell 1973a; Allister and Featherstone 2001). The levelling errors in the ANLN (data provided by Geoscience Australia; GA; G. Johnston pers. comm. 2007) are particularly problematic. The cause of the largest regional distortions in the AHD is gross levelling errors, chieﬂy in
M. S. Filmer et al.
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the interior of the continent (Filmer and Featherstone 2009). The magnitude of the above-tolerance loop misclosures often exceeds 0.5 m in central Australia, with the maximum reaching 0.93 m. Note that the perimeter length of some of these loops is >2,000 km (Fig. 1). When the AHD was deﬁned, insufﬁcient gravity observations were available to apply gravimetric height corrections. Instead, a truncated version of the so-called normal-orthometric correction of Rapp (1961) was applied; there is no requirement for observed gravity whatsoever in this correction. As such, the AHD should be considered a normalorthometric height system (Holloway 1988; Featherstone and Kuhn 2006). However, sufﬁcient gravity data now exist in Australia to retrospectively investigate the effects of gravimetric height corrections to the ANLN (cf. Mitchell 1973a). The problems with the AHD have become considerably more apparent with the emergence of GNSS heighting and regional gravimetric quasigeoid models (e.g., Featherstone 2008). Various methods of overcoming the incompatibility of GNSS-derived ellipsoid heights and AUSGeoid98 (Featherstone et al. 2001) with the AHD have been investigated (e.g., Featherstone 1998; Featherstone and Sproule 2006; Soltanpour et al. 2006). However, the long-term solution to this

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Two algorithms for extracting building models from raw laser altimetry data

ponding author. E-mail: h.-g.maas@geo.tudelft.nl
borne laser scanning has been shown by Weidner Ž1995.. and Forstner ¨ Due to its advantages as an active technique for reliable 3-D point determination without requirements towards surface reflectance variations and time consuming error-prone matching techniques, airborne laser scanning has meanwhile become a rather important source of information for the generation of 3-D city models. Although the point densities delivered by most systems in standard operation mode are still too small Žoften in the order of 1 pointr10 m2 ., sufficiently dense datasets can be acquired with several systems today. A number of authors has shown approaches for the generation of 3-D building models mainly or solely based on laser altimetry data: Haala and Brenner Ž1997. extract planar roof primitives from dense

EGM2008

EGM2008 - WGS 84 VersionIntroductionThe official Earth Gravitational Model EGM2008 has been publicly released by the U.S. National Geospatial-Intelligence Agency (NGA) EGM Development Team. This gravitational model is complete to spherical harmonic degree and order 2159, and contains additional coefficients extending to degree 2190 and order 2159. Full access to the model's coefficients and other descriptive files with additional details about EGM2008 are provided herein.Those wishing to use EGM2008 to compute geoid undulation values with respect to WGS 84,may do so using the self-contained suite of coefficient files, FORTRAN software, and pre-computed geoid grids provided on this web page. For other applications, the previous release of the full 'Geoscience' package for EGM2008 can be accessed through the link at the bottom of this web page.The WGS 84 constants used to define the reference ellipsoid, and the associated normal gravity field, to which the geoid undulations are referenced are:•a=6378137.00 m (semi-major axis of WGS 84 ellipsoid)•f=1/298.257223563 (flattening of WGS 84 ellipsoid)•GM=3.986004418 x 1014 m3s-2 (Product of the Earth's mass and the Gravitational Constant)•Ï‰=7292115 x 10-11 radians/sec (Earth's angular velocity)All synthesis software, coefficients, and pre-computed geoid grids listed below assume a Tide Free system, as far as permanent tide is concerned.Note that the harmonic synthesis software provided below applies a constant,zero-degree term of -41 cm to all geoid undulations computed using EGM2008 with the height_anomaly-to-geoid_undulation correction model (also provided). Similarly, all pre-computed geoid undulations incorporate this constant zero-degree term. This term converts geoid undulations that are intrinsically referenced to an ideal mean-earth ellipsoid into undulations that are referenced to WGS 84. The value of -41 cm derives from a mean-earth ellipsoid for which the estimated parameters in the Tide Free system are: a=6378136.58 m and 1/f=298.257686.Description of Software and DataTo compute point geoid undulations from spherical harmonic synthesis of the EGM2008 Tide Free Spherical Harmonic Coefficients and its associated Correction Model, at any WGS 84 latitude/longitude coordinate pair listed in a coordinate input file (such as INPUT.DAT), use the FORTRAN harmonic synthesis program, hsynth_WGS84.f.At present, we are also providing two grids of pre-computed geoid undulations: one at 1 x 1-minute resolution and one at 2.5 x 2.5-minute resolution. To interpolate geoid undulations from the 1 x 1-Minute Geoid Undulation Grid file, for any WGS 84 latitude/longitude coordinate pair listed in a coordinate input file (such as INPUT.DAT), use the FORTRAN interpolation program interp_1min.f. Similarly, the FORTRAN interpolation program interp_2p5min.f, will interpolate geoid undulations from the 2.5 x 2.5-Minute Geoid Undulation Grid file.Filenames containing a ".gz" in the suffix have been compressed using the Unix "gzip" command.All files that are SMALL ENDIAN format are highlighted in green. All files that are BIG ENDIAN format are highlighted in purple.Please carefully review the README_WGS84 file for a complete description of coefficient and data files for computing EGM2008 geoid undulations with respect to WGS 84. Also, before using any of the data files on this web page, please read the disclaimer.For GIS data formats, please visit the EGM2008 GIS Data Page.Software and Coefficients for WGS 84 Geoid Undulation Computations by Harmonic Synthesis•EGM2008 Harmonic Synthesis Program(hsynth_WGS84.f -184 KB) - Use this FORTRAN program to generate WGS 84 geoid undulations by spherical harmonic synthesis of EGM2008 and its associated height_anomaly-to-geoid_undulation correction model. This program requires that the coefficients for both EGM2008 and the correction model, and an INPUT.DAT file, all be located in the same directory as the hsynth_WGS84.f program.•Harmonic Synthesis Executable for Windows XP(hsynth_WGS84.exe - 696 KB) - Windows executable version of EGM2008 Harmonic Synthesis Program. Use as described above.•Spherical Harmonic Coefficients for Earth's Gravitational Potential (Tide Free System)-(EGM2008_to2190_TideFree.gz - 72 MB) - EGM2008 coefficients required by the harmonic synthesis program.•Correction Coefficients(Zeta-to-N_to2160_egm2008.gz - 50 MB) -Height_anomaly-to-geoid_undulation coefficients required by the harmonic synthesis program.•INPUT.DAT(4 KB) - sample input file of WGS 84 latitude/longitude coordinate pairs for testing the harmonic synthesis program.•OUTPUT.DAT(4 KB)- sample output file of geoid undulation values, generated by reading INPUT.DAT into the harmonic synthesis program (for test verification).*Users - Please verify results by comparing to OUTPUT.DAT file immediately above. Software and Grids for WGS 84 Geoid Undulation Computation by Interpolation*Note: The 1 x 1 minute interpolation program below requires a large PC RAM capacity.•Interpolation Program for 1 x 1-Minute Geoid Grid(interp_1min.f - 28 KB) - Use this FORTRAN program to interpolate geoid undulations from the 1 x 1-Minute Geoid Undulation Grid file, for any WGS 84 latitude/longitude coordinate pair listed in a coordinate input file (such as INPUT.DAT). When applied to the 1 x 1-Minute Geoid Undulation Grid file, this interp_1min.f program will generate geoid undulation values that match the corresponding values generated by harmonic synthesis (hsynth_WGS84.f or hsynth_WGS84.exe above) to within 1 mm. The interp_1min.f program requires that the 1 x 1-Minute Geoid Undulation Grid file and the INPUT.DAT are located in the same directory as interp_1min.f.• 1 x 1-Minute Grid Interpolation Executable for Windows XP(interp_1min.exe - 434 KB) - Windows executable version of the Interpolation Program for 1 x 1-Minute Geoid Grid. Use as described above. To be used with the SMALL ENDIAN Geoid Undulation Grid below.• 1 x 1-Minute Geoid Undulation Grid in WGS 84 - SMALL ENDIAN (Und_min1x1_egm2008_isw=82_WGS84_TideFree_SE.gz - 825 MB) - 1 x 1-minute global grid of pre-computed geoid undulations. This file has a SMALL ENDIAN internal binary representation.• 1 x 1-Minute Geoid Undulation Grid in WGS 84 - BIG ENDIAN(Und_min1x1_egm2008_isw=82_WGS84_TideFree.gz - 828 MB) - 1 x 1-minute global grid of pre-computed geoid undulations. This file has a BIG ENDIAN internal binary representation.•INPUT.DAT(4 KB) - sample input file of WGS 84 latitude/longitude coordinate pairs for testing the Interpolation Program for 1 x 1-Minute Geoid Grid.•OUTPUT.DAT(4 KB)- sample output file of geoid undulation values, generated by reading INPUT.DAT into the Interpolation Program for 1 x 1-Minute Geoid Grid above (for test verification).*Users - Please verify results by comparing to OUTPUT.DAT file immediately above.•Interpolation Program for 2.5 x 2.5 Minute Geoid Grid(interp_2p5min.f - 28 KB) - Use this FORTRAN program to interpolate geoid undulations from the 2.5 x2.5-Minute Geoid Undulation Grid file, for any WGS 84 latitude/longitude coordinatepair listed in a coordinate input file (such as INPUT.DAT). When applied to the 2.5 x2.5-Minute Geoid Undulation Grid file, this interp_2p5min.f program will generategeoid undulation values that match the corresponding values generated by harmonic synthesis (hsynth_WGS84.f or hsynth_WGS84.exe above) to within 1 cm. The interp_2p5min.f program requires that the 2.5 x 2.5-Minute Geoid Undulation Grid file and the INPUT.DAT are located in the same directory as interp_2p5min.f.• 2.5 x 2.5-Minute Interpolation Executable for Windows XP(interp_2p5min.exe - 434 KB) - Windows executable version of the Interpolation Program for 2.5 x 2.5 Minute Geoid Grid. Use as described above. To be used with the SMALL ENDIAN Geoid Undulation Grid below.• 2.5 x 2.5-Minute Geoid Undulation Grid in WGS 84 - SMALL ENDIAN (Und_min2.5x2.5_egm2008_isw=82_WGS84_TideFree_SE.gz - 135 MB) - 2.5 x2.5-minute global grid of pre-computed geoid undulations. This file has a SMALLENDIAN internal binary representation.• 2.5 x 2.5-Minute Geoid Undulation Grid in WGS 84 - BIG ENDIAN(Und_min2.5x2.5_egm2008_isw=82_WGS84_TideFree.gz - 135 MB) - 2.5 x2.5-minute global grid of pre-computed geoid undulations. This file has a BIG ENDIANinternal binary representation.•INPUT.DAT(4 KB) - sample input file of WGS 84 latitude/longitude coordinate pairs for testing the Interpolation Program for 2.5 x 2.5-Minute Geoid Grid.•OUTPUT.DAT(4 KB)- sample output file of geoid undulation values, generated by reading INPUT.DAT into the Interpolation Program for 2.5 x 2.5-Minute Geoid Grid above (for test verification).*Users - Please verify results by comparing to OUTPUT.DAT file immediately above. Software to Extract WGS 84 Geoid Undulations from Grid Files (No Interpolation)•Extract 1 x 1-Minute Grid Program(gridget_1min.f -20 KB) - Use this FORTRAN program to extract a user-defined sub-rectangle of geoid undulations from the 1 x 1-Minute Geoid Undulation Grid file. This program prompts the user for a rectangular area of interest and requires the 1 x 1-Minute Geoid Undulation Grid file to be located in the same directory as the gridget_1min.f program.•Extract 1 x 1-Minute Grid Executable for Windows XP(gridget_1min.exe - 401 KB) - Windows executable version of the Extract 1 x 1-Minute Grid Program. Use as described above. To be used with the SMALL ENDIAN Geoid Undulation Grid below.• 1 x 1-Minute Geoid Undulation Grid in WGS 84 - SMALL ENDIAN (Und_min1x1_egm2008_isw=82_WGS84_TideFree_SE.gz - 825 MB) - 1 x 1-minute global grid of pre-computed geoid undulations. This file has a SMALL ENDIAN internal binary representation.• 1 x 1-Minute Geoid Undulation Grid in WGS 84 - BIG ENDIAN(Und_min1x1_egm2008_isw=82_WGS84_TideFree.gz - 828 MB) - 1 x 1-minute global grid of pre-computed geoid undulations. This file has a BIG ENDIAN internal binary representation.•Extract 2.5 x 2.5-Minute Grid Program(gridget_2p5min.f - 20 KB) - Use thisFORTRAN program to extract a user-defined sub-rectangle of geoid undulations from the 2.5 x 2.5-Minute Geoid Undulation Grid file. This program prompts the user for a rectangular area of interest and requires the 2.5 x 2.5-Minute Geoid Undulation Grid file to be located in the same directory as the gridget_2p5min.f program.•Extract 2.5 x 2.5-Minute Grid Executable for Windows XP(gridget_2p5min.exe - 397 KB) - Windows executable version of the Extract 2.5 x 2.5-Minute Grid Program.Use as described above. To be used with the SMALL ENDIAN Geoid Undulation Grid below.• 2.5 x 2.5-Minute Geoid Undulation Grid in WGS 84 - SMALL ENDIAN (Und_min2.5x2.5_egm2008_isw=82_WGS84_TideFree_SE.gz - 135 MB) - 2.5 x2.5-minute global grid of pre-computed geoid undulations. This file has a SMALLENDIAN internal binary representation.• 2.5 x 2.5-Minute Geoid Undulation Grid in WGS 84 - BIG ENDIAN(Und_min2.5x2.5_egm2008_isw=82_WGS84_TideFree.gz - 135 MB) - 2.5 x2.5-minute global grid of pre-computed geoid undulations. This file has a BIG ENDIANinternal binary representation.Additional Information•Original Release of the EGM2008 Model Coefficients from EGU General Assembly, Vienna, Austria 2008: Geoscience Package•An Earth Gravitational Model to Degree 2160: EGM2008(NPavlis&al_EGU2008.ppt - 18.5 MB) - Presentation given at the 2008 European Geosciences Union General Assembly held in Vienna, Austria, April13-18, 2008.•Background Papers from Earlier Symposiums on the New EGM。

综合egm2008、剩余地势模型及地面重力数据的大地水准面精化

综合EGM2008、剩余地势模型及地面重力数据的大地水准面精化中文摘要（作者：康靖玉指导教师：吴琼）GPS测量具有较高的效率，能够在地面观测不通视的情况下测得WGS84坐标基准下的经纬度以及大地高，并且不依赖于气候条件。

GPS测量结合（似）大地水准面模型可以获得测量点的正（常）高，利用GPS测量大地高并将其转为高程的精度由两个因素决定：第一是根据GPS测量获得的大地高的精度，第二是由大地水准面模型导出的大地水准面差距（高程异常）精度。

随着GPS测量技术的发展以及测量方式的改进，通过使用GPS接收机联测IGS跟踪站或者使用CORS系统等方式均能快速获得高精度的大地高，因而如何获取高精度的大地水准面模型成为GPS高程测量中需要解决的关键问题。

本文根据EGM2008全球重力场模型的原理、特点及其计算方式，介绍了恢复重力场模型高频信号的剩余地势模型法、模型的基准面不一致改正方法，以及确定大地水准面差距的Stokes理论和实测重力数据的处理方法，并根据“移去-恢复”法计算了大地水准面差距。

以兴城娘娘庙地区以及吉林省地区为例，首先根据实测的GPS数据计算出了基准面不一致参数，对水准面模型进行改正；之后基于剩余地势模型恢复了EGM2008全球大地水准面模型的高频信号；基于实测重力数据和“移去-恢复”法建立了重力大地水准面模型；并根据实测GPS/水准数据，对两种计算方法进行精度评估，提出了综合剩余地势模型与地面测得的重力数据精化大地水准面的计算方法。

在使用综合方法的过程中进行残差拟合时，分别用了三次样条差值法、反距离加权、二次曲面拟合法三种方法，对每种方法的计算效果进行对比和分析，并比较其差异。

最后根据研究区的GPS数据和水准数据分别对剩余地势模型法、“移去-恢复”计算法以及综合法的精化结果进行精度评定，并根据不同地区的位置、地势、范围大小等特点，分析各种改正的效果，确定最佳改正方案。

结果表明，在兴城娘娘庙研究区内，单独使用剩余地势模型改正后的EGM2008模型比原始模型精度提高约1cm，而利用重力数据计算获得的大地水准面差距的精度比原始的EGM2008模型提高了2.5cm左右，其精度以及可靠性均高于RTM模型。

MODIS Land Surface Temperature as

MODIS Land Surface Temperature as an index of surface air temperature for operational snowpack estimationEylon Shamir a ,⁎,Konstantine P.Georgakakos a ,ba Hydrologic Research Center,12555High Bluff Drive,Suite 255,San Diego,CA 92130,USA bScripps Institution of Oceanography,UCSD,La Jolla,CA 92037,USAa b s t r a c ta r t i c l e i n f o Article history:Received 26November 2013Received in revised form 26April 2014Accepted 1June 2014Available online 24June 2014Keywords:MODIS LSTLand Surface Temperature Surface air temperature Snow pack Snow 17Regional operational modeling systems that support forecasters for the real-time warning of ﬂash ﬂood events often suffer from lack of adequate real-time surface air temperature data to force their accumulation and ablation snow model.The Land Surface Temperature (LST)product from MODIS,which provides four instantaneous readings per day,was tested for its feasibility to be used in real-time to derive spatially distributed surface air temperature (T a )forcing for the operational snow model.The study was conducted in the Southeast region of Turkey using an atypically dense network of hourly T a ,daily snow depth,snow water equivalent (SWE),and rainfall datasets for the period:October 2002–September 2010.A comparison between the T a and the corre-sponding LST grid-cell data indicated close associations that are different in nature for periods with and without snow on the ground.The LST-derived T a was compared with that obtained from on-site gauge-based interpola-tion procedures and climatological time series.The LST-derived T a was found inferior only to the T a derived from the interpolation of the dense gauge network (31-gauges).Snow-pack simulations using estimated T a time series were compared to simulations that were forced by the observed T a at each site of 18sites.The LST-derived T a performed well in simulating snow mass and maximum SWE magnitude,while it did not represent well the timing of the annual peak of SWE and the duration of spring melt.Our study concluded that the MODIS/LST product can be a valuable additional source of real time forcing data for regional operational snow models,especially in remote mountainous areas with sparse telemetric data.©2014Elsevier Inc.All rights reserved.1.IntroductionSnow accumulation and ablation models that track the snowpack seasonal evolution of the energy and mass balance have been applied routinely worldwide to evaluate the snowpack and melt characteristics.In operational setups which require real-time measurements to repre-sent watershed scale snowpack characteristics,surface air temperature is often a key observed variable that serves as an index for a range of energy ﬂuxes in the atmosphere –snow pack interface and the internal snowpack (e.g.Anderson,1973and 2006).A major uncertainty source in the output of these snow models stems from the commonly insuf ﬁ-cient density of the in-situ meteorological observation network that is required to derive reliable and accurate estimates of energy ﬂuxes and their spatial variability.In mountainous regions with complex terrain and few gauges this uncertainty is expected to be larger (e.g.Bales et al.,2006).In addition,in complex terrain the use of a constant to describe the lapse rate and associate temperature in different elevations adds uncertainty,because in these regions the lapserate varies considerably,being dependent on the synoptic conditions (e.g.Lundquist &Cayan,2007).Snow model sensitivity to surface air temperature (T a )is well demonstrated in Fig.1.The upper panel of the ﬁgure compares simula-tions of snow water equivalent (SWE)for a selected meteorological station in Southeastern Turkey using observed and interpolated surface air temperatures from a relatively (and atypically)dense observation network (lower panel).The details about the snow model,air surface temperature dataset and the interpolation procedure will be further discussed in the following sections.Clearly,these two surface air temperature time series yield a substantial difference between the simulations of the snow water equivalent (SWE)during the winter accumulation and spring ablation time.The sensitivity of the snow model to the surface air temperature even during periods with small temperature differences is attributed to the nonlinear behavior of the snow model and the cumulative effect of the error with the progression of the snow season (e.g.Shamir &Georgakakos,2006).This uncertainty associated with the interpolation of surface air temperature is added to the uncertainty associated with the representa-tion of the energy ﬂuxes as functions of surface air temperature to yield the model simulation uncertainty.Remote Sensing of Environment 152(2014)83–98⁎Corresponding author.E-mail address:Eshamir@ (E.Shamir)./10.1016/j.rse.2014.06.0010034-4257/©2014Elsevier Inc.All rightsreserved.Contents lists available at ScienceDirectRemote Sensing of Environmentj o u r n a l h om e p a g e :w w w.e l s e v i e r.c o m /l o c a t e /r s eIn this study we explore the applicability of the Moderate Resolution Imaging Spectroradiometer (MODIS)Land Surface Temperature (LST)product as a source of information to develop real-time surface air temperature forcing for the snow model.Our motivation to use the MODIS LST product as a proxy for surface air temperature is driven by its potential to resolve the ﬁner scale features of variability which are commonly not inferable from in-situ gauge networks in complex terrain.In the following we present a feasibility study conducted in South-east Turkey.To our knowledge,this is one of the handful studies of LST MODIS in snow environment conducted in latitudes lower than 40°N.Following literature review,introduction of the study area,data,and procedures (Section 2)we discuss and evaluate the association between the LST and observed surface air temperature.In Section 3the availability of valid LST reports for the study region is evaluated,and a comparison between surface air temperature derived from the LST product and other commonly available time series is presented.Section 4compares the effect of the interpolated T a on the simulation of snow pack.Study conclusions are provided in Section 5.1.1.Literature reviewEnergy exchange ﬂuxes at the land-surface are largely in ﬂuenced by T a ,LST and their difference.Unlike T a that is commonly measured at a 2m height,LST has not been routinely measured in meteorological stations.In the U.S.for example,less than 2%of the snow measuring automatic stations measure radiant LST (Raleigh,Landry,Hayashi,Quinton,&Lundquist,2013).The complicated interlinks between LST and T a can be demonstrated by examining the energy exchange equation between the land surface and the immediate atmosphere:1−αðÞR s ↓−R L ↑þR L ↓−G ¼H þλEð1Þwhere R S ↓is the incoming shortwave radiation,R L ↓and R L ↑are the incoming and outgoing longwave radiation,respectively,αis the surface albedo,G is the ground heat ﬂux,and H and λE are the sensible and latent heat ﬂuxes,respectively.The units of the energy ﬂuxes are in power per area (e.g.W m −2).All the energy ﬂux terms in Eq.(1)are dependent on LST,T a ,or their difference.The longwave radiation ﬂux association with temperature is explained by the Stephan –Boltzmann's equation,which ascribes R L ↓as a function of T a and the atmospheric emissivity,and R L ↑as a function of LST and the surface emissivity.The magnitude and direction of the turbulent latent and sensible heat ﬂuxes,as seen in the equations below,are controlled by the difference between T a and LST.H ¼C H U T a −LST ðÞð2ÞλE ¼C E U e a −e s ðÞð3Þwhere C H and C E are the bulk transfer coef ﬁcients for heat and moisture,respectively (kJ/m 3°C),U is the wind speed above the surface (m/s),and e s and e a are the vapor pressure (Pa)of the snow surface and the air surface,respectively.Notice that the vapor pressure is monotonically and positively associated with temperature.The ground ﬂux (G )is often estimated as a function of the LST,and the surface albedo is often associated with LST (Jin &Dickinson,2010),which determines the absorbed shortwave radiation at the land surface.As a general rule the land surface warms up in the morning to yield a positive LST −T a difference,whereas during the night LST cools faster and the difference is negative.In dry soil,calm air,and clear sky condi-tions the LST follows the radiation cycle with a short time lag (~1h).In similar conditions for wet soil,the LST is considerably reduced because of latent heat releases from evaporation.Vegetation canopy cover050100150200250300350400450S n o w W a t e r E q u i v a l e n t (m m )-30-20-10010206-Hour IntervalT a i r (o C )Fig.1.SWE simulation using observed (solid black)and interpolated (red)air temperatures for Gauge #17920(2005–2006).The interpolation was based on a network of 31gauges.Lower panel shows the observed (solid black)and interpolated (red)surface air temperatures.84 E.Shamir,K.P.Georgakakos /Remote Sensing of Environment 152(2014)83–98depends on type and the fractional cover changes the albedo and modulates the fraction of shortwave radiation absorbed at the surface. The dominant impact of vegetation however,is associated with its evaporative cooling.Plant canopy actively exchanges absorbed solar radiation through evaporation and thus maintains daytime canopy temperature that is close to the ambient T a(e.g.Nemani,Pierce, Running,&Band,1993).Remotely sensed LST measurements often represent a mixture of soil and vegetation canopy temperatures.The observed portion of soil and vegetation varies with the viewing angle and the fraction and type of vegetation.LST measurements in these mixed areas are also inﬂuenced by the lower atmosphere and the temperature difference between the vegetation canopy and the soil background(Weng,Lu,& Schubring,2004).In snow,because of its low thermal conductivity and insulating characteristics,the surface layer quickly reaches temperature equilibri-um with the atmosphere even for large differences between LST and T a(e.g.,Liston,1995).Snow pack LST cannot exceed melting temperature(0.0°C)except for cases of vegetation litter(e.g.,Hardy,Davis,Jordan, Ni,&Woodcock,1998)or dust(e.g.,Painter et al.,2007).During calm nights radiative cooling can reduce the LST below the T a(5–10°C) and cold surface temperature often causes near surface inverse stable conditions that reduces turbulent mixing.Warmer snow surface temperature,as might seem counter intuitive,in general,decreases the snowpack's melting rate.This is because sensible and latent heat ﬂuxes decrease as a consequence of smaller temperature and vapor pressure gradients(see Eqs.2and3),and increase of outgoing longwave radiation(e.g.Sade et al.,2011).During periods of complete areal snow cover it is reasonable to assume that horizontalﬂuxes between the grid cells are less dominant than the verticalﬂuxes.However,in patchy snow pack during melt season the intergrid cell differences in vertical energyﬂuxes between the snow covered and uncovered areas create considerable horizontal ﬂuxes that introduce large uncertainty to the energy balance equation (Liston,1995).The relationship between LST and surface air temperature was stud-ied by many,both from theoretical and empirical perspectives.A few large scale global studies concluded that this relationship is dependent on local variables such as land use/cover,soil moisture,snow cover, frozen ground,regional microclimate conditions,terrain and local landscape features(drexler,Zhao,&Running,2011;Jin& Dickinson,2010;Prihodko&Goward,1997;Sade,Rimmer,Litaor, Shamir,&Furman,2011).1.2.MODIS LSTDetails of the MODIS LST algorithm,its day–night split-window regression,and its theoretical basis can be found in Wan and Dozier (1996),Wan and Li(1997),Wan(2003)and Wan(2008).The MODIS LST product can be made operationally available in near real-time, within hours of the satellite overpass,from the MODIS Rapid Response System(Pinheiro et al.,2007).Land Surface Temperature(LST)is an instantaneous measurement of the skin temperature that is mapped from the radiometric(kinetic) temperature and derived from the thermal IR(TIR)radiation emitted from the land surface.Because TIR measures radiance at the top of the atmosphere,during cloudy conditions the measured signal is from the cloud-top and therefore the MODIS LST is retrieved only in clear-sky conditions.Sensible heatﬂuxes emitted from the earth surfaces are detected by the MODIS satellite and are used to estimate instantaneous LST.The LST generalized split-window algorithm depends on the MODIS brightness temperature bands(11and12μm channels31 and32)and surface emissivity estimates for these two channels.The emissivity in the thermal infrared is estimated using regression coefﬁcients that depend on land cover classiﬁcation(MOD12Q1)and daily binary snow cover(MOD10_L2).During snow and ice conditions the emissivity is set to0.993and0.99for bands31and32,respectively (Wan,2008).The algorithm also requires dynamic estimates of the atmospheric transmission,which is derived directly from the MOD07_L2product of vertical atmospheric temperature and vapor proﬁles and the MODIS cloud masking product(MOD35_L2).In addition to night and day LST estimates,the product also includes information on the emissivity,view-angle,cloud-cover,time of acquisi-tion,and quality-control assessment for each grid cell.Under clear sky conditions the accuracy of the MODIS/LST product was reported to be within1°C for a temperature range of−10to50°C.It was also report-ed to perform better in desert regions(Wan et al.,2004).The performance of MODIS LST in snow and ice covered areas and under clear sky conditions was reported to be accurate within +/−1°C(Wan,Zhang,Zhang,&Li,2002)for a temperature range of −15–0°C(Hall et al.,2008).The accuracy of MODIS LST in snow and ice conditions and temperature below−15°C is still unknown.For these conditions it is reasonable to assume+/−1°C accuracy as report-ed by Wan et al.(2002)(Hall et al.,2008).Although the accuracy is assumed reasonable,Westermann,Langer,and Boike(2012)claim that during clear sky conditions the snow surface temperature is usually colder than during cloudy conditions,and therefore the MODIS LST in snow and ice has a systematic cold bias.MODIS LST was used to detect the thermal regime of permafrost. Several studies conducted in the Arctic found MODIS LST to correspond well with permafrost and the detection of thawing(e.g.,Langer, Westermannb,Heikenfelda,Dorna,&Boikea,2013for the Lena River Deltain in Siberia;Hachem et al.,2012for Northern Quebec,Canada). Hall,Nghiem,Schaaf,DiGirolamo,and Neumann(2009)used the MODIS LST product to detect melting areas in Greenland's ice sheet. Compared with shallow ground based measurements,the LST was found to have better association with surface air temperature and reported to perform better during periods with snow cover(Hachem, Duguay,&Allard,2012).Another study from the Arctic that compared among various LST products derived from polar orbiter satellites con-cluded that the MODIS LST product had the best association with surface air temperature(Urban,Eberle,Hüttich,Schmullius,&Herold,2013). Tight association between monthly MODIS LST and T a was also reported at the Lambert Glacier basin in East Antarctica(Wang,Wang,&Zhao, 2013).All the above described validation studies for LST in snow,ice and permafrost environment were conducted in high latitude relatively homogeneous snow and ice surfaces.However,only a handful of studies evaluated the performance of MODIS LST in mid-latitude mountainous terrain with seasonal snow pack.One example is aﬁeld study from the Italian Alps that reported correlation between the skin temperatures of snow and observed surface air temperature to be greater than0.85 and0.89for day and night,respectively(Colombi,De Michele,Pep,& Rampini,2007).This strong association enables derivation of detailed maps for surface air temperature in snow dominated mountainous terrain that can potentially improve the estimation of the spatial variability of melt.2.Data2.1.Study regionThe study was conducted in the Southeast region of Turkey (black thick rectangular outline in Fig.2).The study domain (500km×330km)contains the headwaters of the Euphrates and Tigris Rivers and other east-ﬂowing rivers draining into the Caspian Sea.The domain expands across two continental climate zones:south-eastern Anatolia and Eastern Anatolia.In the western part of the domain is the Anti-Taurus mountain range with average peak elevation at 3000m;while the eastern part near the border with Armenia is the Armenian Highlands mountain range that peaks at Mount Ararat85E.Shamir,K.P.Georgakakos/Remote Sensing of Environment152(2014)83–98(5137m).Turkey's largest lake,Lake Van,is situated in the mountains at an elevation of 1546m.The Southern slopes of the Anti-Taurus Mountains constitute a region of rolling hills and a broad plateau surface that extends into Syria.The region experiences severe winters with frequent heavy snow-fall events and warm summers.The dominant land cover is grasslandwith patches of closed and open shrubland and extended areas of cultivated agricultural land.The MODIS LST product was extensively validated for these dominant land cover of grassland and cultivated agriculture by Wan et al.(2002)and Wan (2008),respectively.Because of therelativelyFig.2.A map of the study area.123456789101112MonthsV a l i d L S T R e p o r t s (F r a c t i o n )31-gauges Oct 2002-Sept 2010Fig.3.Aqua and Terra monthly summary of the valid day and night LST reports (Oct 2002–Sep 2010)for grid-cells that are associated with the 31gauges of surface air temperature.MONTHP E R C E N TAvg. Reports Binary Snow Product: Sept 2002 - Oct 2010Fig.4.Classi ﬁcation of the Snow Cover Area (SCA)reports from Aqua into valid reports (red),clouds (green)and no data (black).In addition,the valid reports that indicated snow cover are in blue.86 E.Shamir,K.P.Georgakakos /Remote Sensing of Environment 152(2014)83–98low vegetation canopy,the seasonal snow pack in such landscape is relatively homogeneous.Therefore,it is expected that during periods of snow cover the LST signal will represent well the snow surface temperature as reported in the validation studies mentioned in Section 1.2.2.In situ observationsMeteorological time series of observations for 1October 2002–30September 2010were received from the Turkish State Meteorological Service (TSMS).For the study domain we identi ﬁed 31stations with hourly surface air temperature and daily snow survey information,such as snow water equivalent and snow depth (white triangles in Fig.2).Eighteen of those locations also have daily precipitation records.The range of station elevations is 370–2300m,with a mean average el-evation of 1300m.Most of the stations are located in areas that were identi ﬁed as either cultivated crop or grassland.This dataset represents a fairly dense observation network with an average distance between nearest neighbors equal to 37km (range 11to 80km).Such network density is considered highly dense by operational network standards,especially for high mountains and rugged terrain regions,as in the study domain.2.3.MODIS LST data for the study areaThe LST product from the Aqua and Terra spacecraft MODIS sensors was retrieved from NASA's Earth Observing System Data and Informa-tion System for 1October 2002–30September 2010.We used version 5of the daily Aqua and Terra Land Surface Temperature/Emissivity L3which is available globally at 1km in a sinusoidal projection (MOD11A1and MYD11A1)and use the generalized split-window algorithm (Weng et al.,2004).The study area is covered by a single tile that includes 1200rows and columns (h21v05).Most overpasses for Terra [Aqua]occurred around 11:00and 23:00[13:00and 01:00]local time.2.3.1.LST product availabilityFor the study region and the duration of analysis the LST dataset has 505missing daily products (out of 6574),most having occurred prior to 2005.The availability of valid LST reports,which are cases with actual temperature reports,for the study duration and for the grid-cells that match the 31surface air temperature gauges was summarized by month and is presented in Fig.3.It is seen that valid LST products are available less than 40%of the time during November –April.The availability of valid LST products rises to about 60–80%during June–parison of MODIS LST and 31surface air temperature gauges.Black dots indicate gauges of snow measurement and gray are gauges with measurement of no snow.Red lines are regression estimates and con ﬁdence bounds (5%and 95%)for the snow and no snow separately.87E.Shamir,K.P.Georgakakos /Remote Sensing of Environment 152(2014)83–98September for Aqua and Terra,respectively.We note that only 2%and 0.5%of the missing LST products from both Terra and Aqua lasted longer than 15and 26consecutive days,respectively.Fig.4shows the status of the MODIS snow cover product for the same period and domain available from the Aqua 500m daily snow cover product (MYD10A1)(Hall,Riggs,&Salomonson,1995).As mentioned above,the snow cover product is being used as input to the MODIS LST algorithm.In Fig.4we consider the entire dataset as valid and invalid reports.Valid reports are considered as grid-cells that report snow cover,bare ground,or water;and invalid reports are grid-cells that considered as cloud masking and missing data that are ﬂagged by the quality assessment index.In addition,as a point of reference,we present the percent of snow cover reports (out of the total),which is also part of the valid report category.During November –April,the period with reported snow cover (blue line),the cloud cover was reported in 50–70%of the time (red line).On the other hand,cloud cover decreased considerably during the summer months,and valid daily reports of the product (green line)became abundant.Notice however,that for this period (summer)the number of snow cover reports is marginal (blue line).The black line,which ranges between 8and 12%,summarizes the periods in which the SCA product quality assessment index indicated a problem with-15-10-50510152025Days: 1 Aug 2005 - 31 July 2006T e m p e r a t u r e D i f f e r e n c e (L S T -T a )o CFig.6.An example of annual plot (August 2005–July 2006)of LST,T a ,and snow water equivalent from a gauge that is located in the southeast part of the study domain.The upper panels show the day and night Terra (left)and Aqua (right)LSTs (red and black circles,respectively),day and night gauge air surface temperatures (red and black stars,respectively)and snow water equivalent (cm)(black line).The differences between the LST and the T a are shown for Terra and Aqua (left and right,respectively)and for day and night red and black markers,respectively.88 E.Shamir,K.P.Georgakakos /Remote Sensing of Environment 152(2014)83–98the SCA product.The prevalence of cloud cover in the study region,which obscures the MODIS/SCA product during the snow accumulation season,was previously documented by Tekeli,Akyürek,Şorman,Şensoy,and Şorman (2005and 2006)and Tekeli,Şensoy,Şorman,Akyürek,and Şorman (2006).The analyses presented in Figs.3and 4indicate that the LST product is highly infrequent during the winter,when snow is likely to accumulate,and becomes available with higher frequency during spring –summer (melting period).2.3.2.LST correspondence with surface air temperatureThe correspondence between the LST and gauge surface air temper-ature for the study region is shown in Fig.5.The black [gray]dots in this ﬁgure indicate cases in which snow [no snow]was reported at the gauge locations.The red lines show the estimated linear regressions and the 5and 95percentile prediction con ﬁdence intervals,calculated separately for gauges with and without snow.The data in the ﬁgure was screened through the quality control procedure that is described in Section 3.3.Although there is an overall monotonic association between the gauges and the LST values,there is signi ﬁcant scatter in the association.It is also seen that the relationships during snow and no snow periods are different as seen by the slopes of the regressions.In general during periods of snow on the ground small changes in LST correspond with a larger range of surface air temperature values.During these periodsthe LST values are below or equal to 0°C,while the surface air temper-ature can possibly rise above freezing temperature.The intra-annual correspondence between LST from Terra and Aqua,gauge surface air temperature,and SWE is further demonstrated in Fig.6,for a high elevation meteorological station (~2300m)in a grass-land dominated land cover at the southeast corner of the study domain.During periods of no snow the LST day –night amplitude brackets the surface air temperature with LST being warmer [colder]than surface air temperature during the day [night](upper panels).During periods with snow measured at the gauge (black solid line),the day –night amplitudes of the LST and surface air temperature were comparable.The LST temperature remained consistently below zero while the surface air temperature occasionally increased above zero.Although snow was measured at the gauge as late as mid-March,the LST increased above zero about a month earlier.The lower panels,which depict the temperature differences between the LST and T a ,show that the daily LST was warmer than T a during no snow periods.This difference ranges 0–20°C with highest differences observed in the summer (June –September).During periods with snow on the ground the daily LST was occasionally colder than the T a .The night LST was consistently colder than T a throughout the year and about an additional 5°C colder than the T a during periods with snow cover.The daily LST values from both sensors were comparable for this location,and although one might expect a warmer Aqua LST because its overpass is closer to the daily maximum short wave irradiance,no-40-200204000 Local time-40-200204006 Local Time200400600-200204060 October 2002 -September 2004 (Days)12 Local Time200400600-40-200204018 Local Time-40-200204000 Local time-200204006 Local Time200400600-20204060October 2002 -September 2004 (Days)200400600-200204018 Local Time-40-200204000 Local time-40-200204006 Local Time-200204060October 2002 -September 2004 (Days) -200204018 Local Time -40-2002000 Local time-40-20204006 Local Time-20204060October 2002 -September 2004 (Days)-40-200204018 Local TimeT e m p e r a t u r e (o C )T e m p e r a t u r e (o C )T e m p e r a t u r e (o C )T e m p e r a t u r e (o C )Fig.7.Two years (1October 2002–30September 2004)of LST and surface air temperature at 00,06,12,and,18hour local time for four selected stations that represent different elevations.89E.Shamir,K.P.Georgakakos /Remote Sensing of Environment 152(2014)83–98distinct differences are apparent.The night time LST however,shows a persistent colder temperature (about 2°C)for the Aqua sensor.Spring LST values that exceeded 0°C although snow was measured at the gauge indicate considerable uncertainty in the association between LST and T a .This uncertainty is likely attributed to the patchi-ness of snow cover that is prevalent during spring times and the classi ﬁcation of the LST product as snow-free grid-cells while patches of snow existed in high elevations.During spring it is expected that in-situ snow reports will often misrepresent the patchiness of snow cover at their corresponding grid-cells.In addition,it is expected that during spring time when daily surface air temperature can rise to relatively warm temperatures the snow surface temperature can't rise above freezing.The generalized split-window algorithm for version 5ingests binary snow cover estimates and therefore grid cells that are fractionally covered with snow are assigned to either completely covered or not covered.This in turn increases the uncertainty in the LST estimates and likely to under (over)estimate in cases of low (high)snow fraction-al cover (Rittger,Painter,&Dozier,2012).The recent addition of the fractional snow cover to the MODIS operational products can potentially be incorporated to improve LST estimates in fractionally snow covered grid cells (Dozier,Painter,Rittger,&Frew,2008).Another depiction of the relationship between LST and surface air temperature is provided in Fig.7.Two years of LST and T a daily time series for 0,6,12,and 18hour local time are plotted from four gauges in a grassland environment that were selected in order to represent different elevations.The interpolation of the MODIS LST from the obser-vation times to these hours and for the periods for which LST valid reports are unavailable is described in Section 3.1.The general patterns among these selected gauges are similar except that in the higher-elevation gauges (lower panels)during the winters there are more missing LST values,and the reported LST values for these periods are based on climatological values for their estimates and thus appear as straight lines.The plot further demonstrates theStart t ing g tim m e Oc c tob b er 1 (6-h h our r tim m e int t erv v als)Fig.8.Sensitivity of simulated SWE to biased air temperature (+/−3°C)on each of the three processes:temperature threshold to distinguish between rain and snow (left),snow melt(middle)and snowpack energy balance during non-melt periods (right).-40-2002040MOD 11am-40-2002040MOD 22pm-40-2002040Surface Air Temperature (o C)MYD 13pm-40-2002040MYD 02amM O D I S L S T (o C )Fig.9.Scatter plots of observed hourly surface air temperature from 31gauges and the corresponding LST values for the day and night overpasses from Terra and Aqua satellites.The red dots indicate cases in which the difference between T a and LST exceeds 15°C.90 E.Shamir,K.P.Georgakakos /Remote Sensing of Environment 152(2014)83–98。

基于EGM2008模型确定的昆明地区似大地水准面精化

基于EGM2008模型确定的昆明地区似大地水准面精化作者：胡翀来源：《价值工程》2014年第02期摘要：本文主要介绍了EGM2008模型与利用GPS水准点与EGM2008模型相结合进行大地水准面精化的方法。

结合昆明市呈贡区部分GPS水准点数据在EGM2008模型的基础上利用GPS水准点与EGM2008模型结合计算出高程异常然后采用多项式拟合的方法拟合，最后对拟合后的结果进行了一个检验和精度评估。

Abstract： This paper mainly introduces the model EGM2008 and how to use the GPS leveling point and the model EGM2008 to refine the geoid. With the partial leveing point data of Chenggong District of Kunming and the model EGM2008， we can calculate the height anomaly which is to be used for polynomial fitting， and finally conduct text and accuracy evaluation for the result.关键词：大地水准面精化方法；EGM2008模型；GPS/水准；多项式拟合Key words： refining of geoid；the model EGM2008；GPS/level；the polynomial fitting function中图分类号：P223 文献标识码：A 文章编号：1006-4311（2014）02-0182-020 引言EGM2008模型是由美国国家地理空间情报局（NGA：US National Geospatial-IntelligenceAgency）在奥地利维也纳科学联盟大会上推出的最新一代全球重力场模型[1]。

Temperature dependent Electron Land'e g-Factor and Interband Matrix Element in GaAs

a r X i v :c o n d -m a t /0608534v 4 [c o n d -m a t .m t r l -s c i ] 15 N o v 2007Temperature dependent Electron Land´e g-Factor and Interband Matrix Element inGaAsJ.H¨u bner,∗S.D¨o hrmann,D.H¨a gele,†and M.OestreichInstitute for Solid State Physics,Gottfried Wilhelm Leibniz University of Hannover,Appelstr.2,30167Hannover(Dated:February 4,2008)Very high precision measurements of the electron Land´e g-factor in GaAs are presented using spin-quantum beat spectroscopy at low excitation densities and temperatures ranging from 2.6to 300K.In colligation with available data for the temperature dependent eﬀective mass an unexpected temperature dependence of the interband matrix element within k ·p -theory is revealed.A strong decrease of the interband matrix element with increasing temperature consistently resolves a long lasting discrepancy between experiment and theory and substantially improves the modelling of both parameters.The ﬁndings are supported by estimating the impact of phonon induced ﬂuctuations of the interatomic spacing on the electronic band structure.PACS numbers:78.55.Cr,78.20.Ci,71.18.+y,78.47.+pThe semiempirical k ·p -theory is a universal tool to calculate the band structure in semiconductors and semi-conductor heterostructures and is regularly employed in such diﬀerent ﬁelds as the physics of semiconductor laser design,the quantum Hall eﬀect,and spintronics.The part of the theory describing magnetic ﬁeld related phe-nomena has been extensively improved since its intro-duction by Kane [1],Luttinger and Kohn [2]in the mid ﬁfties.Nowadays,5-and more band k ·p -models are state of the art and many low temperature experiments have conﬁrmed the incredible accuracy of k ·p -calculations [3,4,5,6,7].All these experiments support the validity of k ·p -theory whereas a single but central experiment,which measures the temperature dependence of the elec-tron Land´e g-factor in GaAs,shows a strong discrepancy between experiment and k ·p -theory [8].The temperature dependent measurement of the elec-tron Land´e g-factor g ∗has been carried out more than 10years ago but the astonishing discrepancy between theory and experiment has still remained an unresolved problem.The most commonly believed explanation for the discrepancy have been uncertainties in the experi-mental data.However,in the ﬁrst part of this paper it is shown that extremely high precision,temperature de-pendent measurements of the electron Land´e g-factor un-ambiguously endorse this discrepancy.A detailed study which takes into account the experimental error bars of the temperature dependent energy band-gaps conﬁrms that the experimental uncertainties can not be responsi-ble for the deviations of the electron Land´e g-factor val-ues between theory and experiment.This work will show that a temperature dependent interband matrix element yields a consistent explanation for the temperature de-pendence of the electron Land´e g-factor and the eﬀective mass within k ·p -theory.The k ·p -theory is a perturbation theory calculating the electronic band-structure by expansion around a single point in the Brillouin zone.In direct semiconductors like GaAs,the well-investigated high symmetry Γ-point is thenatural expansion point.Input parameters are only the measured band-gaps at k =0and the interband matrix elements (P 2).The change of the band-gap energies with the lattice temperature are very well known for GaAs,making it an ideal material system to test and improve the accuracy of k ·p -theory.The only remaining parameter which does not pos-sess a direct experimental access is the interband ma-trix element P 2,whose temperature dependence has been assumed to be marginal:The interband matrix ele-ment is inverse proportional to the interatomic distance a since P 2depends on the spatial overlap of valence-and conduction band states |s and |p ,respectively:P 2∼| s | p |p |2∼12 ducting magnet and excited with circular polarized lightpulses from an80MHz picosecond laser.The sampletemperature is varied from2.6K to room temperature,whereas the excess-energy of the exciting light is about6meV above the direct band gap for temperatures upto80K.At higher temperatures,the contribution of theexcess energy is negligible compared to the thermal en-ergies present in the sample lattice[12].The excitedcarrier densities amount to about6·1015cm−3.The pho-toluminescence from the sample is collected in backwarddirection.Energy-and time-resolution is performed bya spectrometer followed by a synchroscan streak cam-era.The electron Land´e g-factor is deduced from theoscillating time evolution of the cross circular polarizedcomponent of the photoluminescence via the relationg∗=ωL /(µB B),withωL being the Larmor precessionfrequency of the conduction electron spins,µB Bohr’smagneton,and B the magneticﬁeld.Great care has been taken in the time calibration ofthe detection system as well as in the correct determina-tion of the magneticﬁeld present in the superconductingmagnet.Superconducting magnets can often show unap-parent remanenceﬁelds and incorrectﬁeld calibrationswhich easily inﬂuence the experimental data.There-fore the magneticﬁeld is calibrated with a precise Hallsensor for all appliedﬁelds.The often cited value ofg∗=−0.44±5%[13]in GaAs applies for donor boundelectrons,whereas for the free conduction band electronshigher g-factor values are reported[14,15].From thedata presented here a very high accuracy extrapolatedelectron Land´e g-factor of g∗=−0.484±0.7%at T=0K,B=0T,and P exc.=0mW is determined.Figure1shows the electron Land´e g-factor versus sam-ple temperature.Each value is extrapolated to zero mag-neticﬁeld and zero excitation power from measurementsat diﬀerentﬁelds and excitation powers at constant tem-perature to eliminate any residual eﬀects of those entitieson the g-factor.The measurements from2.6to62K arecarried out with alternating excitation ofσ+-andσ−-light and a small tilt of the sample against the magneticﬁeld.This technique enables us to monitor and subtractthe inﬂuence of the eﬀective nuclearﬁeld on the electronLand´e g-factor measurement(see Ref.[16]for details).The experimental results are compared with estab-lished k·p-theory.The central5-level k·p-result forg∗and m∗was presented by Hermann et al.[6]:g∗3…1E(Γc6−Γv8)+∆0«−P′2E(Γc7−Γc6)−1m∗=1+P2E(Γc6−Γv8)+1 3…1E(Γc8−Γc6)«+C(2)Here,g0=2.0023is the free electron Land´e g-factor,m0electrong-factortemperature [K]FIG.1:High precision measurement of the temperature de-pendence of the electron Land´e g-factor in bulk GaAs(ﬁlled circles).The red solid line is aﬁt of the experimental data by Eq.1with a strong temperature dependent interband ma-trix element.The green dashed line is the calculated g∗with P2(T)depending only on the anharmonic lattice expansion.the free electron mass,and E(Γv8,Γc6,Γc7,Γc8)and∆0are the energies of the band extrema at the center of the Brillouin zone and the split-oﬀenergy of theΓv7valence band,respectively.The contributions from higher bands are summarized in the constants C′=−0.02and C=−2 [6].The temperature dependence of the band-gap ener-gies E(T)are well known by experiment and described by the semi-phenomenological model introduced by Vi˜n a et al.[17]E(T)=E B−αB1+2k B T ek B T e.(4)The integration starts at the minimum of the conduc-tion band,D3D(E)=1/2π2 2m∗/ 2 3/2√TABLE I:Parameters usedE BαBΘeV meV K Ref.of P2.Now,P2(T)is estimated by taking into account the statistics of phonon induced lattice elongations.The eﬀective average lattice distance is stretched by the vari-ance of the mean squared strain(MSS)due to the adia-batic following of the electrons:a eﬀ=a(T)(1+σMSS). Here the MSS is given byσMSS= ∂x u and u′2 =1Mω2u20.Hence,the MSS can be written as2u′2 =1v2s Z(ω) 2 dω(5)where n(ω)is the Bose-Einstein distribution n(ω)= (eβω−1)−1,β= /k B T,and M=M1+M2is the mass of the two atomic basis in the fcc lattice.The 3D Debye density of states in the unit volume V in the continuum approximation for one phonon polarization is: Z(ω)=V14π2v5s Ming length.Acc.to Eq.3P2changes starting from≈40K.[25]Landolt-B¨o rnstein,Group III Condensed Matter,vol.41,(Springer,2001).[26]−ζ4+ζ2P∞l=1(−lζ+3)e lζl3−e lζ15。

基于EGM2008模型的小区域似大地水准面模型构建与优化

基于EGM2008模型的小区域似大地水准面模型构建与优化摘要：本文阐述利用EGM2008模型构建优化区域似大地水准面，利用EGM2008的2190高阶模型，结合区域已知水准成果，进行模型的拟合优化，针对模型误差较大的区域，进行实地几何水准测量，改善误差较大区域的水准成果。

利用EGM2008模型构建的正常高成果精度为±13.7cm，利用水准改进的2次模型优化，正常高成果精度为±4.8cm，满足一般工程测量的应用。

关键字：EGM2008；似大地水准面；模型优化；高程异常Abstract: this paper using EGM2008 model building optimization area like the geoid, the use of EGM2008 2190 high order model, in combination with regional known level results, the fitting model optimization, in view of the model error larger area, field geometry level measurement, improve the level of the large area error results. EGM2008 model is built using the normal GaoChengGuo accuracy of + /-13.7 cm, use level improvement 2 times model optimization, normal GaoChengGuo accuracy of + /-4.8 cm, meet general engineering measurement applications.Keyword: EGM2008; Like the geoid; Model optimization; Abnormal height 引言似大地水准面模型在全国多地已经开展，国内应用较多的是利用武汉大学李建成院士的移去恢复法，利用WDM94等重力模型构建的省域似大地水准面模型，利用该法可有效建立高精度区域似大地水准面。

基于EGM2008和SRTM模型确定的新疆地区似大地水准面及其精度评估

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( 2) * * T H H = ∋ [ 1 - 2 ( 1 + f + m - 2f sin2B ) + 3( ) 2 ]- 1 ! a a 0 ( 3) 其中 , C 为地面点的地球位基数 , T 为地面点处扰动位 , B 为该点大地纬度 , f 为椭球扁率 , a 为椭球长半轴长度 , m 可以由椭球参数求得 ( m = ∀ 2 a3 ( 1 - f ) /GM , ∀ 为自转角速度 ) , !0 为地面点沿椭球法向所对应的椭球表面点处正常重力 , 可由椭球参数求出 [ 2] 。确定了高程异常 , 即确定了该处似大地水准面的位置。为求得某区域的似大地水准面 , 需要事先知道该区域地表的空间位置 ( 用以计算重力场模型位 ) 。然而 , 存在的问题如下 : GPS 站点的精度很高 , 但其分布过于稀疏 ; 数字高程模型可以达到很高的空间分辨率 , 但其位置精度较差。文献 [ 3] 论证了利用数字高程模型结合重力场模型确定似大地水准面的可行性 , 指出数字高程模型的位置误差对高程异常计算结果的误差影响很小。因此 , 可以根据数字高程模型求得地表格网点的位置 , 利用重力场模型 EGM 2008 求解具有一定分辨率的区域似大地水准面。
[ 2] [ 3]
[ 4] 分 ) 见表 1 。根据 15∃ % 15∃ 似大地水准面的高程异常网格点分布值 , 利用三次样条插值得到 15 个 GPS /水准检验点处的高程异常 , 此为计算值。检验点的大地高与正常高的差值作为实测值 , 计算值与实测值的比较见表 1 。 15 个检验点处高程异常值的残差统计分析见表 2 。由表 2 可知 , 基于 EGM 2008 重力场模型和 SRTM 高程模型求得新疆地区似大地水准面 , 与 15 个 GPS /水准检验点的平均差异为 0 11m, 精度为 0 23m。 [ 5]

M eissel-S to kes 核函数应用于区域大地水准面分析

M eissel-S to kes 核函数应用于区域大地水准面分析荣敏;周巍;任红飞【期刊名称】《测绘工程》【年(卷),期】2015(000)009【摘要】In order to improve the accurate of the regional determination ,based on the potential coefficient of gravity field model EGM2008 ,the features ,the truncation error coefficient and the truncation error of the Meissel-Stokes kernel function are analyzed .In the test region ,the accuracy of the geoid determination is estimated using the Meissel-Stokes kernel function by the remove-restore technique .The results show that the convergences of the Meissel-Stokes kernel function and its truncation error coefficient are accelerated , and its truncation error is small and stable .If the integral radius is too small to reach the required range , the Meissel-Stokes kernel function used in the geoid determination is better than the standard Stokes .%为提高区域大地水准面计算精度，基于EGM2008地球重力场位系数模型分析Meissel-Stokes核函数、截断误差系数以及截断误差。

Dynamical Characterization of Peak-Current-Mode-Controlled Buck Converter With Output-Current Feedfo

444IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 22, NO. 2, MARCH 2007Dynamical Characterization of Peak-CurrentMode-Controlled Buck Converter With Output-Current FeedforwardMatti Karppanen, Mikko Hankaniemi, Student Member, IEEE, Teuvo Suntio, Member, IEEE, and Mika SippolaAbstract—The paper investigates the effect of unity-gain outputcurrent-feedforward in a peak-current-mode-controlled (PCMC) buck converter. A consistent theoretical basis is provided showing that the unity-gain feedforward can improve signiﬁcantly the load invariance and transient performance of a PCMC buck converter. The nonidealities associated to the scheme would, however, deteriorate the obtainable level of invariance. The nonidealities can be maintained at acceptable level, and therefore, the scheme would provide a viable method to reduce signiﬁcantly the load interactions as well as improve the load-transient response. The theoretical predictions are supported with comprehensive experimental evidence both at frequency and time domain as well as comparisons between three different buck converters. Index Terms—Buck converter, load–current feedforward, output impedance, peak-current-mode control (PCMC).Fig. 1. Interconnected regulated system.I. INTRODUCTION achieved only, if the open-loop internal output impedance is designed to be zero [9], [13]. It was demonstrated in [15] that even the zero open-loop output impedance does not necessarily ensure load invariance, because the load may interact the converter dynamics via the internal input impedance at the presence of the source impedance. The use of output-current feedforward has been demonstrated to improve the output-voltage transient performance for the load-current changes in a hysteretic current-mode-controlled (HCMC) buck converter in [7]. According to the applied theory, the zero output impedance would be achieved by using unity-feedforward gain. The peak-current-mode-controlled (PCMC) buck converter is treated in [8]. The effect of output-current feedforward on the output impedance of the converter is comprehensively analyzed. Close to unity-feedforward gain is stated to give the minimum output impedance. The general conditions for achieving zero output impedance have been derived in [9]. It was stated that the zero output impedance can be implemented in any converter regardless of topology but the validations were only carried out by using a buck converter. A voltage-mode-controlled (VMC) buck converter has been treated in [10] but the theoretical basis for the design approach is not explicitly deﬁned and therefore, the validation of the method is difﬁcult. The experimental load transients shown in [12] imply that the zero output impedance may not be achievable in a boost converter by applying output-current feedforward, i.e., a better transient behavior may be achieved by optimizing the voltage-loop-controller design. A theoretically consistent treatment of the effect of outputcurrent feedforward in a regulated converter is presented in [13].INTERCONNECTED regulated power supply systems—known also as distributed power supply (DPS) systems (Fig. 1)—are extensively used to supply different electronic loads [1], [2]. The nonlinear nature of the associated regulated converters would make the interconnected systems prone to stability and performance problems [3]. Basically it is a question of the interactions caused by the different impedances [i.e., the output impedance of the source system ( , Fig. 1) and the input impedance of the load system ( , Fig. 1)] associated to the speciﬁed interface within the system [4], [5], [14]. A natural desire would be to get rid of those interactions. It is well known that the load impedance (i.e., , Fig. 1) may affect adversely the voltage-loop gain of the converter (i.e., the supply converter in Fig. 1) [4]–[6], [14]. It is claimed explicitly in [4], and implicitly in [5] and [6] that the load invariance may be achieved by designing the voltage-loop controller in such a way that makes the closed-loop internal output impedance small. According to sound scientiﬁc theory, the load interactions are reﬂected into the converter dynamics via the open-loop internal output impedance [14], [15]. Therefore, it may be obvious that the perfect load invariance at arbitrary load may beManuscript received November 3, 2005; revised June 10, 2006. Recommended for publication by Associate Editor F. L. Luo. M. Karppanen, M. Hankaniemi, and T. Suntio are with the Department of Electrical Engineering, Institute of Power Electronics, Tampere University of Technology, Tampere FI-33101, Finland (e-mail: teuvo.suntio@tut.ﬁ). M. Sippola is with Efore Oyj, Espoo FIN-02211, Finland. Digital Object Identiﬁer 10.1109/TPEL.2006.8899210885-8993/$25.00 © 2007 IEEEKARPPANEN et al.: DYNAMICAL CHARACTERIZATION OF PCMC BUCK CONVERTER445functions constituting the set in (1) can be deduced by means and the output of the variables in the input vector , where denotes the general control variable, vector respectively. The set in (1) can be equally represented by using a linear two-port model shown in Fig. 2(a), which deﬁnes also explicitly the used input and output variables in (1)A. Load Interactions The effect of load impedance [i.e., in Fig. 2(b)] on the dynamics of a converter can be approximated by computing and from Fig. 2(b), when the nonideal load is connected, giving (2). Replacing and in (1) by means of (2) gives the load-affected dynamical representation of a converter as shown in (3) as a function of the nominal model and the load. The same load-interaction formalism (3) may be also derived using the extra element theorem introduced in [18] by means of the linear two-port modelFig. 2. Linear two-port models (a) nominal model with ideal source (^ ) and u j load (^ ) and (b) nominal model with nonideal source (u , Z ) and load (^ , i ^ Z ).It deﬁnes explicitly the required conditions for theoretical zero output impedance at open loop based on the well-known transfer functions of the associated converter. According to it, the zerooutput-impedance conditions are impossible to implement in a converter exhibiting nonminimum phase behavior due to the right-hand-plane zero. The dynamical effect of output-current-feedforward with a unity gain in a peak-current-mode-controlled (PCMC–OCF) buck converter is treated in this paper. The consistent theoretical basis has been provided earlier in [13]. The theoretical predictions are proved by means of experiments both in frequency and time domain. The effect of nonidealities is addressed in detail. Comparisons between VMC, PCMC, and PCMC-OCF buck converters are provided by using the same power stage with different control systems. The results show that the unity-gain output-current-feedforward scheme in a PCMC buck converter is a viable method to improve the load insensitivity. In addition, the PCM control in a buck converter would reduce also the source interactions (i.e., input or source invariance) due to high input-to-output attenuation at open loop [16] and as a consequence also the load interactions via the source impedance. II. SOURCE-LOAD-INTERACTION MECHANISM The dynamics of a switched-mode converter is usually represented by means of a set of transfer functions at open loop as deﬁned in (1). The load of the converter is not commonly known, when the converter is designed, produced, and delivered. Therefore, it would be most convenient to give such a set of transfer functions, which only represents the internal dynamics of the converter without the effect of source or load impedances. Such a set is known as an unterminated set [17], which can be usually measured using an ideal constant-voltage source and an ideal constant-current load [Fig. 2(a)]. The set will be known in this paper as a nominal set due to its speciﬁc nature representing the pure internal dynamics of the associated converter. The transfer(2)(3) The control-to-output transfer function is one of the key elements in the voltage-loop gain [15]. Therefore, changes in it would reﬂect directly changes also in the voltage-loop gain or in the dynamics of the converter. According to (3), the load-affected control-to-output transfer ) would stay intact (i.e., function (i.e., ), if the internal output impedance 0. Similarly, the load-affected open-loop input admittance (i.e., ) would stay intact (i.e., ), if the open-loop input-to-output transfer function 0. B. Source Interactions The effect of source impedance [i.e., in Fig. 2(b)] on the dynamics of a converter can be approximated by computing and from Fig. 2(b), when the nonideal source is conand in (1) by means of nected, giving (4). Replacing (4) gives the source-affected dynamical representation of a converter as shown in (5) as the function of the nominal model and and in (5) are deﬁned in (6), respectively. source. is known as an ideal input admittance and as an open-loop short-circuit admittance [15]. Both of the special admittances are load invariant [15]. The ideal input admittance is speciﬁc for a given topology but the conduction and control modes do not affect it, i.e., its value can be computed according446IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 22, NO. 2, MARCH 2007to (6) by using the voltage-mode transfer functions [15]. The short-circuit admittance is dependent on the control mode, and therefore, the corresponding transfer functions in (6) have to be the nominal transfer functions of the associated topology and control mode [15](4)(5)(6) According to (5) and (6), the source-affected control-to-output transfer function (i.e., ) and the internal output impedance(i.e., ) at open loop would stay intact, 0. if the open-loop input-to-output transfer function As a summary we may state that a converter having zero output impedance and zero input-to-output transfer function at open loop would provide both load and supply invariance. In practice, perfect load and source invariance may not be achievable due to parasitic circuit elements, circuit nonlinearities, and control delays, etc. III. ZERO OUTPUT IMPEDANCE (Fig. 3) The required output-current-feedforward gain to achieve theoretically zero open-loop output impedance can be derived from the control-block diagram shown in Fig. 3(a), is the equivalent output-current sensing resistor, and where is the gain factor between the control voltage ( , Fig. 3) and the control signal ( , Fig. 2) (i.e., ). According to Fig. 3(a), we can compute the set of transfer functions deﬁning the output dynamics of the converter to be (7). The corresponding set for the input dynamics (8) can be computed from Fig. 3(b), respectively [8] but the methodology to come up to the conclusions is erroneous as criticized in [11]. can be deThe zero-output-impedance condition for 0, which gives (9). The zerorived from (7) letting impedance conditions presented in [9] may be the same as (9) but the deﬁnitions of the variables used in the corresponding equation are not deterministic and the statement that the derived feedforward gain can be implemented for any converter does not hold: The nonminimum-phase converters (e.g., boost and buck-boost) have right-hand-plane (RHP) zero in the control-tooutput transfer function, and it cannot be implemented without causing instability. Therefore, theoretically correct feedforward gain (9) may be implemented only for the minimum-phase converters such as a buck converter (9) (8) According to (7) and (8), the load-current feedforward would affect only the output impedance and the output-to-input . All the other transfer functions would transfer function stay virtually intact. Similar conclusions are also presented in It may be obvious that the unity-feedforward gain (i.e., 1) would be desired due to easiness of implementation. The lack of resonant behavior and rather high low-frequency open-loop output impedance make the PCM-controlled buck converter an optimal medium for achieving low open-loop output impedance by using unity-feedforward scheme [7],Fig. 3. Control-block diagrams for solving the dynamics of a converter with load-current feedforward (a) output dynamics and (b) input dynamics.(7)KARPPANEN et al.: DYNAMICAL CHARACTERIZATION OF PCMC BUCK CONVERTER447functions with those speciﬁed in (10) and (11). The resulting output impedance becomes as shown in (13) According to (13), we can make the following conclusions. 1) If , then the unity-feedforward output impedance would resemble the output impedance of a VMC buck converter without resonant behavior, which is known to be small except at the vicinity of the resonant frequency [20]. , then the magnitude of the output impedance 2) If would approach the magnitude of the PCMC buck converter along the increase in the match error. , then the phase of the output impedance 3) If would start at 180 at low frequencies, when 0 as explained also in [8]. It may be obvious that the minimum low-frequency output impedance would take place, when 0. , then the phase behavior of the output 4) If impedance is similar to the open-loop output impedance of the PCMC buck converter. , (10)] is If the input-to-output transfer function [ small as with the optimal compensation stated above, then the source interactions would be minimal [16]. If the compensation deviates from the optimal value (i.e., overcompensation), then the load interactions may be reﬂected into the input side according to (3) and (5). The key factor in this sense , would be the output-to-input transfer function which may be given as (14). The overcompensation means decreases compared to the that the duty-ratio gain optimal compensation, and as a consequence, the poles and zeros of (14) would approach each other. This means that approaches unity also at low the magnitude of frequencies boosting the load reﬂections into the input side ]. In a PCMC buck [(3), would decrease along the converter, the magnitude of as may be concluded according to (15), and decrease in therefore, the load reﬂections would be reducedFig. 4. Experimental buck converter.[13]. The unity-feedforward scheme in a VM-controlled buck converter does not give desired results due to resonant behavior of the output impedance: The feedforward scheme should be effective at the frequencies close to the resonant frequency of the converter as is implicitly stated in [10]. We do not consider further the methods needed for the other converters to implement the zero or close to zero output impedance but the basis for the methods or their existence may be concluded from (9) by means of appropriate transfer functions. IV. UNITY FEEDFORWARD SCHEME The buck converter under consideration is shown in Fig. 4 with the relevant power-stage parameters. The nominal transfer functions corresponding to (1) for the PCMC buck converter can be given as shown in (10) and (11) [13], [19], where the duty-ratio gain , , is the slope of the artiﬁcial compensation (i.e., for optimal compensation [19], [20] giving 0), and are de1 , where ﬁned in (12). In the case of PCM control, is the equivalent inductor-current-sensing resistor(10) (14) (15) (11) The authentic unity-feedforward PWM modulator and control system corresponding to Fig. 4 are shown in Fig. 5. The control-block diagram in Fig. 5(a) clariﬁes the overall control system implementation, where the error-amp (EA) feedback el20 k , ements are the same as shown in Fig. 5(c) and respectively. The popular PCM-PWM modulator (UC38C42) is utilized to form the time base, protection and latch functions as(12) The open-loop output impedance resulting from the application of the unity-feedforward scheme can be computed from (7) letting 1, and replacing the corresponding transfer448IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 22, NO. 2, MARCH 2007Fig. 5. Practical control-process implementation. (a) Simpliﬁed block diagram illustrating the implementation of the overall control system. (b) PWM modulator and artiﬁcial compensation. (c) Voltage-loop controller and load-current feedforward (R = 75 m).shown in Fig. 5(b). A constant-current source is used to create a , Fig. 5(a)] as shown in linear compensation ramp [i.e., Fig. 5(b), respectively. The type-2 compensator and the physical realization of the load-current feedforward are shown in Fig. 5(c). V. EXPERIMENTAL MEASUREMENTS The VMC, PCMC, and PCMC–OCF buck converters used for the experimental frequency and time-domain measurements are built by using the same power stage shown in Fig. 4. The control system is changed accordingly to the type of converter. The voltage-loop controllers have been designed to give the same nominal control bandwidths (11 kHz) and close to same nominal phase margins (VMC: 48 , PCMC: 55 ) at the input voltage of 50 V [Fig. 6(a)]. A. Internal Nominal Dynamics The voltage-loop gains of the converters are shown in Fig. 6(a). Fig. 6(a) shows clearly that the load-current feedforward does not change the nominal loop gain as predicted in (7). The measured nominal open-loop output impedances are shown in Fig. 6(b), respectively. The effect of the load-current feedforward is obvious. All the converters were subjected to a constant-current-type load change from 0.2 to 2.5 A (250 mA s). Even if the loop-gain-related dynamical parameters are virtually identical, the output-voltage responses are quite different as shown in Fig. 6(c). The differences can be explained by means of the differences in the open-loop output impedances [Fig. 6(b)].Fig. 6. Nominal output dynamics of VMC, PCMC, and PCMC-OCF buck converters at the input voltage of 50 V. The current sensing resistors of PCMC–OCF converter were close to identical, i.e., 75 m. (a) Voltage-loop gains. (b) Openloop output impedances. (c) Output-voltage responses to a load change from 0.2 to 2.5 A (250 mA=s).KARPPANEN et al.: DYNAMICAL CHARACTERIZATION OF PCMC BUCK CONVERTER449Fig. 7. Varying of nominal open-loop output impedance along the match error in the current-sensing resistors.Fig. 8. Output-voltage response of PCMC–OCF converter to a constant-currenttype load change from 0.2 to 2.5 A (250 mA=s) at the input voltage of 50 V, 1 (upper subplot) and R =R 1.1 (bottom subplot). when R =R==B. Effect of Mismatch in Sensing Resistors The current-sensing-resistor ratio (13) was varied and the open-loop output impedance of the PCMC–OCF converter was measured. The results are shown in Fig. 7. The pos1, and the negative sign that itive sign means that 1, respectively. The behavior of the internal open-loop output impedance follows the rules implied in (13). It was demonstrated in [13] that the analytical predictions based on (13) would give accurate results but we have omitted the predictions in this paper, because the purpose of the paper is not to treat the small-signal modeling and its accuracy. It is noteworthy that the model in (13) predicts that the minimum output impedance would take place, 1. Fig. 7 shows that the output impedance when 0.99 is larger than the output impedance at at 1.02 implying that the prediction is correct. The PCMC–OCF converter was subjected to the same constant-current-type load change as in Fig. 6(c), when the resistor ratio is 1.1 or 10%. The corresponding output-voltage response is shown in Fig. 8 (bottom subplot) compared to the response at the ideal resistor ratio (upper subplot). The phase behavior of the open-loop output impedance (i.e., low-frequency phase close to 180 ) results in a slight overshoot in the response as discussed also in [8]. C. Load Sensitivity It was shown in Section II that the direct load interactions would be reﬂected into the converter dynamics via the openloop output impedance as depicted in (16). It is known that the impedance ratio [14] would predict the load-imposed is the closed-loop output instability of the converter, where impedance of the converter. As a consequence, the instability boundary in respect to the load impedance can be deﬁned ex, which means that , and plicitly as 180 (16)The measured closed-loop output impedances of the PCMC–OCF and PCMC converters are shown in Fig. 9, when (Fig. 9: 1), and (Fig. 9: 2). If the phase of the output impedance is 90 or higher then the converter tends to be sensitive to the capacitive or resonant-type load. If the phase is less than 90 , then the sensitivity may be addressed to the switched-mode converters with EMI ﬁlter as a load. According to Fig. 9, both of the converters are sensitive to converter loads at high frequencies. The match error in the sensing resistors would clearly increase the capacitive-load sensitivity of the PCMC–OCF converter and extend it up to rather high frequencies, which is more severe effect than the slight overshoot observed in the time domain transient behavior in Fig. 8. It was stated earlier that the difference in the output-current between PCMC–OCF (14) and PCMC susceptibility (15) converter would make the PCMC–OCF converter more susceptible to the load interaction reﬂected back due to changes in the open-loop input admittance. The measured output-current susceptibilities are shown in Fig. 10. When the input-to-output is same for both of the converters, transfer function it may be obvious that the reﬂected load sensitivity would be much higher in the PCMC–OCF converter than in the PCMC converter at high frequencies as discussed earlier in Section III, i.e., (3). VI. CONCLUSION The dynamical effect of unity-load-current feedforward in a PCMC buck converter was investigated. A sound theoretical formulation was deﬁned and applied to obtain an analytical description of the internal dynamics of such a converter. The dynamical characterization proved the previous ﬁndings but introduced also new earlier unobserved features. It was stated that a PCMC–OCF buck converter may posses both high invariance to supply and load side interactions, if an optimal slope compensation and a match in the inductor- and load-current sensing resistors exist. The match error in the sensing resistors would increase the open-loop output impedance and make the converter450IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 22, NO. 2, MARCH 2007Fig. 9. Closed-loop output impedances of PCMC–OCF and PCMC buck converters.[6] D. Lee, B. Choi, J. Sun, and B. H. Choi, “Interpretation and prediction of loop gain characteristics for switching power converters loaded with general load subsystem,” in Proc. IEEE PESC’05, 2005, pp. 1024–1029. [7] R. Redl and N. O. Sokal, “Near-optimum dynamic regulation of DC-DC converters using feed-forward of output current and input voltage with current-mode control,” IEEE Trans. Power Electron., vol. PE-1, no. 3, pp. 181–191, Jul. 1986. [8] G. K. Schoneman and D. M. Mitchell, “Output impedance considerations for switching regulators with current-injected control,” IEEE Trans. Power Electron., vol. PE-4, no. 1, pp. 25–35, Jan. 1989. [9] L. Varga and N. Losic, “Synthesis of zero-impedance converter,” IEEE Trans. Power Electron., vol. 7, no. 1, pp. 152–170, Jan. 1992. [10] S. Kanemaru, T. Hamada, T. Nabeshima, T. Sato, and T. Nakano, “Analysis and optimum design of a buck-type DC-to-DC converter employing load current feedforward,” in Proc. IEEE PESC’98, 1998, pp. 309–314. [11] B. Johansson, “Analysis of DC-DC converters with current-mode control and resistive load when using load current measurements for control,” in Proc. IEEE PESC’02, 2002, pp. 165–172. [12] S. Hiti and D. Borojevic, “Robust nonlinear control for boost converter,” in Proc. IEEE PESC’93, 1993, pp. 191–196. [13] M. Hankaniemi, T. Suntio, and S. Ruotsalainen, “Load invariant buck converter—analysis and implementation,” in Proc. EPE’05, 2005, pp. 1–10. [14] M. Hankaniemi, T. Suntio, and M. Sippola, “Load-impedance based interactions in regulated converters,” in Proc. IEEE INTELEC’05, 2005, pp. 569–573. [15] ——, “Characterization of regulated converters to ensure stability and performance in distributed power supply systems,” in Proc. IEEE INTELEC’05, 2005, pp. 533–538. [16] T. Suntio, K. Kostov, T. Tepsa, and J. Kyyrä, “Using input invariance as a method to facilitate system design in DPS applications,” J. Circuits, Syst., Comput., vol. 13, no. 4, pp. 707–723, 2004. [17] T. Suntio and I. Gadoura, “Dynamic analysis of switched-mode converters using two-port modeling technique,” in Proc. PCIM’02 (Europe), 2002, pp. 387–392. [18] R. D. Middlebrook, “Null double injection and extra element theorem,” IEEE Trans. Education, vol. ED-32, no. 3, pp. 167–180, Aug. 1989. [19] T. Suntio, “Uniﬁed derivation and analysis of duty-ratio constraints for peak-current-mode control in continuous and discontinuous modes,” in Proc. IEEE IECON’02, 2002, pp. 1398–1403. [20] R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics, 2nd ed. Norwell, MA: Kluwer, 2001.Fig. 10. Open-loop output-current susceptibilities (T and PCMC buck converters.) of PCMC–OCFmore sensitive to load interactions. It may be possible to maintain the match error sufﬁciently small, and therefore, the unity output-current-feedforward would provide a method to signiﬁcantly reduce the load and supply side interactions. REFERENCES[1] L. Brush, “Distributed power architecture demand characteristics,” in Proc. IEEE APEC’04, 2004, pp. 342–345. [2] M. P. Sayani and J. Wanes, “Analyzing and determining optimum on-board power architectures for 48 V-input systems,” in Proc. IEEE APEC’03, 2003, pp. 786–789. [3] B. Choi, B. H. Cho, and S.-S. Hong, “Dynamics and control of DC-to-DC converters driving other converters downstream,” IEEE Trans. Circuits Syst. I, vol. 46, no. 10, pp. 1240–1248, Oct. 1999. [4] B. Choi, B. J. Kim, B. H. Cho, and C. M. Wildrick, “Designing control loop for DC-to-DC converters loaded with unknown AC dynamics,” IEEE Trans. Ind. Electron., vol. 49, no. 4, pp. 925–932, Aug. 2002. [5] P. Li and B. Lehman, “Performance prediction of DC-DC converters with impedances as loads,” IEEE Trans. Power Electron., vol. 19, no. 1, pp. 201–209, Jan. 2004.Matti Karppanen was born in Kajaani, Finland, in 1982. He received the M.Sc. degree in electrical engineering from the Tampere University of Technology, Tampere, Finland, in 2006 where he is currently pursuing the Ph.D. degree. Since 2006, he has been with the Institute of Power Electronics, Tampere University of Technology, as a Research Scientist. His current research interests include power electronics, switched mode power supplies, and distributed power supply systems.Mikko Hankaniemi (S’06) was born in Adelaide, Australia, in 1979. He received the M.Sc. degree in electrical engineering from Tampere University of Technology, Tampere, Finland, in 2005 where he is currently pursuing the Ph.D. degree. Since 2005, he has worked at the Institute of Power Electronics, Tampere University of Technology, as a Research Scientist. His current research interests include power electronics, dynamic modeling, control design, and identiﬁcation of switched mode power supplies. Mr. Hankaniemi is a member of the IEEE Industrial Electronics and IEEE Power Electronics Societies.KARPPANEN et al.: DYNAMICAL CHARACTERIZATION OF PCMC BUCK CONVERTER451Teuvo Suntio (M’98) received the M.Sc. and D.Sc. degrees in electrical engineering from the Helsinki University of Technology, Espoo, Finland, in 1981 and 1992, respectively. From 1977 to 1991, he worked at Fiskars Power Systems as a Design Engineer and R&D Manager. From 1991 to 1992, he worked at Ascom Energy Systems Oy as an R&D Manager. From 1992 to 1994, he was an independent Consultant Engineer, and from 1994 to 1998 he worked at Efore Oyj as a Consultant and Project Manager. Since 1998, he has been a Professor specializing in power converter technologies ﬁrst at the Electronics Laboratory, University of Oulu, and since August 2004 at the Institute of Power Electronics, Tampere University of Technology. He holds several international patents. His current research interests include dynamic modeling, control design, and optimal EMI design of switched-mode power converters. Dr. Suntio is a member of the IEEE Industrial Electronics, IEEE Power Electronics, and IEEE Circuits and Systems Societies.Mika Sippola was born in Helsinki, Finland, in 1971. He received the M.Sc, Lis.Sc., and Dr.Sc. degrees in electrical engineering from the Helsinki University of Technology, Espoo, Finland, in 1997, 2001, and 2003, respectively. He has been a Design Engineer for an UPS manufacturer, a Researcher at the Applied Electronics Laboratory, Helsinki University of Technology, an entrepreneur in design consultancy, and a Senior Research Engineer at Schaffner EMV AG Global R&D. Since 2004, he has been with Efore Oyj, Espoo, Finland, where he as Executive Vice President of Technology is responsible for research and technology development. He has number of publications and patents. Dr. Sippola received the IEEE Power Electronics Society “Best Paper Award” in 2003.。

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