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Date Created: 09/22/15

A Sernipararnetric Stochastic Mixed Model for Increment Averaged Data with Application to Carbon Sequestration in Agricultural Soils F Jay Breidt Colorado State University Joint work with Nan Jung Hsu National Tsing Hua University and Stephen Ogle Colorado State University The work reported here was developed under STAR Research Assistance Agreements CR 829095 awarded by the US Environmental Protection Agency EPA to Colorado State University This presentation has not been formally reviewed by EPA EPA does not endorse any products or commercial services mentioned in this report Space Time Aquatic Resources Modeling amp Analysis Program o STARMAP EPA funded STAR grant study probability surveys of aquatic resources CSU focuses on modeling side partners with sister program at OSU design side a Um agricultural soils y aquatic resources spin off from other work on nonparametric survey regression estimation Greenhouse Effect a Solar energy transmitted to earth as Visible and ultraviolet radiation lAtrnosphere Surface Reflected 25 5 Absorbed 25 45 0 Radiation absorbed by surface gets re radiated as infrared 0 Greenhouse Gases GHGS pass Visible and UV but trap infrared include water vapor C02 methane nitrous oxide others Kyoto Protocol a December 1997 meeting in Kyoto Japan a Resulted in Kyoto Protocol signed by over 170 nations including US 0 Binding commitment for US to reduce emissions of six GHGs reduce 7 from 1990 levels during 200872012 0 US never rati ed Kyoto no commitment by developing nations potentially large economic impact Chicago Climate Exchange 0 www chicagoclimatex com 0 Participants set voluntary limits on CHC emissions legally binding commitments for reductions Commitments Baseline 2003 2004 2005 2006 avg199872001 1 2 3 4 Why Submit to Binding Commitments o Creates a cap and trade market can make own reductions can buy credits from others that have extra reductions to sell CCCCC my Why Participate 0 Participants include Ford Motorola DuPont City of Chicago Waste Management 0 Reduce long term costs of carbon reduction 0 Get nancial bene ts eg reduced energy costs a Enhance environmental leadership reputation 0 Build trading skills help create market rules Examples of GHG Mitigation and Offset Projects o Renewal energy systems wind power solar power 0 Energy ef cient process innovations a Recovery use of land ll and agricultural methane a Carbon sequestration tree biomass agricultural soils no till agriculture Agriculture 001 Tillage a Traditional Tillage after harvest eld contains crop residues tillage turns over the soil to bury residues often repeated several times prior to planting a Conservation Tillage Reduced T ill limited tillage substantial crop residues on surface No T ill NT doesn t use tillage all crop residues left on surface Economically feasible due to new technology Advantages of No T ill Management 0 NT results in lower production costs fewer management steps cheaper lower horsepower tractors 0 NT leaves crop residue on soil surface reduces soil loss due to wind and water erosion reduces ow of sediments nutrients and pesticides into surface waters improves soil fertility enhances soil organic matter sequesters carbon Carbon Sequestration Via No T ill 0 Original soil carbon content has been reduced 50 since invention of the steel plow aeration due to tillage leads to faster decomposition decomposition releases C02 to atmosphere 0 No till signi cantly slows decomposition widespread adoption could restore original carbon within 40 years could cut projected growth in US C02 emissions by 20 Show Me the Money a Carbon as a cash crop a Government could subsidize switch to no till o Corporations faced with caps on CHC emissions could buy carbon credits some Canadian companies have already paid lowa farmers for carbon credits 0 Estimates of value range from 4i40 per acre Key Question for Carbon Credits o How much carbon is sequestered in switching to no till 0 Several studies have compared no till with traditional tillage on paired elds a Measure carbon difference after one or more years since management change Y no till carbon traditional tillage carbon Available North American Studies o 63 studies compare tillage types on paired elds Soil Core Samples 0 Use probe to select one or more cores 0 Separate cores into increments may be xed increments eg 0715 cm 15730 cm may be determined by soil pro le eg plow layer A horizon7 B horizon7 C horizon a Mix matching increments in a bucket a Bag a subsample and send to the lab Aside on Core Data 0 Like a time series with depth replacing time speci cally a ow de ned on interval income expenditure precipitation not a stock instantaneous interest rate temp o Other cores ice cores increment represents one year of snowfall vertical ozone pro les like time series tend to be many regularly spaced relatively narrow increments a Not true for soil cores Increment Averaged Data a Difference in metric tons C ha 1 VS depth average carbon content difference depth in centimeters Ad Hoe Methods Midpoint Assignment a True model is i d Y2 Whiz aw 6 d gakdz j dz j i di3 1gklttgt dt 6 gakgmiffjk Ezj where tgjk E dijj1dz jgt 0 But we regress on ngZj y gkt jk measurement error problem 0 Least squares estimators are biased and inconsistent DO We Need to Worry o Simulate from the tted model 1 d Yzj Md 1ltoao 061lt1 15 dt Ezj 2 23 1 Simulation Results 0 Simulate from true model then estimate with midpoint assignment use actual increments from data set resample if necessary sample size reps a0 017 041 332 218 10000 EMU 017 Moll 421 10000 1 020 017 021 418 0 Over 25 relative bias in slope estimate Ad Hoc Methods Adjustment of Increments a Study 1 measures Y11 carbon stock change over increment 0715 cm Y12 carbon stock change over increment 15730 cm a Study 2 measures Y21 carbon stock change over increment 0750 cm a Adjustment of Y Values Yf Y11 Y12 represents 0730 cm Y 50321 represents 0730 cm Ad Hoc Methods for Increments o Midpoint assignment leads to bias inconsistency o Adjustment leads to loss of information likely bias 0 One nal method simply drop studies with non matching increments obvious loss of information 0 Need to recognize the increment nature of the data Key Data Features o Increment averaging irregular wide dif cult to specify parametric model a Within core dependence increments within same core may be correlated o Other effects time since change to no till climate regime soil type Semiparametric Stochastic Mixed Model Longitudinal 0 Zhang Lin Raz Sowers 1998 JASA Yzj Xz Tj 9152 3 Zszbi Uilttzjgt Ezj B p unknown regression coefficients X M sz known covariates 90 twice differentiable function of time bi independent q vectors of random effects Ultt independent stochastic processes ezj independent errors a Does not handle increment averages Sernipararnetric Stochastic Mixed Model Increments o Increment average in 2th core jth increment 1 d him 905 dtZgbiUij ij T Yz j X j3d j1 zj dz j i B p unknown regression coefficients X M sz known covariates 90 twice differentiable function of depth bi independent q vectors of random effects Uzj increment averaged stochastic process ezj independent errors Assumptions for Semiparametrie Stochastic Mixed Model 0 De ne T UZZltUi17UZ2aaUmZgt o Assume UZ are independent norma10 Pi where E GOV Uta Uz 5 bi are independent norma10 Dq5 where D is a positive de nite matrix cm are independent norma10 02 bi UZ and cm are mutually independent Integrated Stochastic Process Speci cation 0 Increment averaged form 1 4 U t dt Ui Z z 7 dz j dz J l H where UN are independent mean zero Gaussian stochastic processes a Choices for instantaneous process Wiener process integrated Wiener process non homogeneous Ornstein Uhlenbeck process Nt xnnogeneousrnsn anH enbeCkiProcess 0jovar uuxaSmiumurefbrinstantaneous procesg varltU lttgtgt W eXp o 6115 COIrltU lt8gt7 UM eXpJLO ls tl olt3cnuiances nfincrennent averaged process COVltUzja Um 50 dzj dij 1gtltdz k dzk 1gt 6fhjlt 00 eakgr 1 CK 09k 09 eariJ 7 1 31 O X a a Integrated Nonparametric Function Speci cation 0 Parameter glttgt is in nite dimensional 0 Assume gt is natural cubic spline knots at distinct right hand endpoints cubic polynomial between knots g g 9 continuous at knots hence everywhere linear before rst knot after last knot Splitting Up the Integral o Distinct right hand endpoints t t1 t2 t T o Increment total d H 905 dt t t tk d2 tkkc1i19lttgt d1 tkkjfglttgt dt tkf 1 90 d1 O0110OG for some non negative integer k and some positive integer p where G I t 90 cit12 91 dt ii 90 CWT Value Second Derivative Representation 0 De ne gi 9a and Vi g ti fort12r W1 w 0 0 F01quot hi hurl ti we have 2 2 2 t 151 t1 hiti G1tdtti 7 if i 0 9H 1 2m 91 2h192 6 2V2 and 3 t h h Gi1 tll19lttgt dt 2 92 9H1 231 VH1 fort127 1 Ah Linear oSoG39isalinear function ofgglg7aT and7ltfy27 7fyT 1gtT g I G A A AA li2l7lli2lRiQT where A1 A2 R and Q are known matrices depending only on o In fact G is linear function of 9 alone 97 Finite Dimensional Model a Integrals 0f in nite dimensional parameter now reduced to 7 unknown constants 7d n Ida Z w 273 1 jil 0 Xz l Gl Zz bil Uil EZ CT 271139 X BNigZibiUiei Likelihood o Log likelihood given covariance parameters Q5 5 and 02 1 ltBgYgt 210gV ltY X3 NggtTV 1ltY X3 Nggt where V diagV1 V2 Vm Vi ZiDq5ZZT n 021 Penalized Likelihood o Log likelihood given covariance parameters Q5 5 and 02 1 ltBgYgt 310g WI 1 2ltY X3 NggtTV 1Y X3 Nggt o Penalized likelihood 2 A lt gYgt 2lrlg lttgtl d1 where A gt 0 controls smoothness A gt 0 implies interpolating A gt 00 implies global regression Penalized Likelihood Continued o Penalized likelihood can be rewritten A 2 A 5amp9 Y 267quot l9 lttgtl d1 6amp9 Y QQTKg where K QR 1QT is known 0 Now differentiate with respect to B and g Maximum Penalized Likelihood Estimators 0 Assume A as well as Q5 5 and 02 are given a MPLEs solve the following linear equations XTV 1X XTV lN 3 XTV lY NTV 1X NTV 1N AKl g NTV lY a Unique solution Uniqueness of MPLEs o By Theorem 41 of Green and Silverrnan 1994 unique solution if X N1 N t has full column rank 0 Result If 601 61m Z ColX where m is vector of increment midpoints then XTV 1X XTV lN 3 NTV 1X NTV 1N AK 9 has a unique solution B Q MPLE XTV lY NTV1Y Yet Another Representation Linear Mixed Model a From Green 1987 SR it turns out that g lt17 tgt62gtltl Bar 2gtlt17 Where B is a known matrix derived from K 0 Linear mixed model representation Y XNTl lNBaZbUe X3NBaZbUe where a is normal0 TI and 739 lA T rieky not the data generating mechanism 0 BLUE of and BLUP of g from mixed model MPLEs Estimation of Covariance and Smoothing Parameters 0 Use REML Restricted Maximum Likelihood standard method for linear mixed models De ne V TNBNBT V o REML criterion for T 5T 02 T is mob 5T 0 2 o Y 10g log 1 2ltY XBgtTVgtk 1ltY 0 Given REML estimates MPLEs are immediate Back to the Carbon Sequestration Data 0 Now have all the tools needed for inference in the carbon data set 0 Goals iden yjrnporunn1 xede ects idernd yin1pcntaln3sourceslt3fxeuiationanricorrek jorl estnnate depth 1nctnn1 estirnate expected carbon sequestered due to no t l IPCC 030 XOTO 9a dt where X 00 covariates at 20 years after management change The Plow Layer a Top 15720 cm of agricultural soil N Plow Layer carbon difference in metric tons ha 0 20 40 60 80 depth in centimeters 100 Model Speci cation o Fixed effects wet dry climate aquicnon aquic soils years since management change parametric or nonparametric yeardepth interactions 0 Random effect soil core 0 Stochastic process non homogeneous OU Some Results 0 Estimation results from REML and MPLE Model 1 Model 2 Parameter Estimate Standard Error Estimate Standard Error aquic 01383 00587 01371 00592 wet 01725 00593 01720 00596 years 00157 00047 00031 00032 yeardepth 00003 00001 qb 00078 00077 00094 00093 02 00073 00091 00064 00102 50 08260 02368 07911 02463 51 00601 00205 00643 00219 04 02607 00447 02693 00490 739 1 00326 00046 00189 00026 73995 A921 000005 0000008 Residual Diagnostics 0 Accounting for OU dependence structure residuai residuai O O O 20 40 60 80 10 20 30 40 50 60 increment midpoint increment Width 0 0 E 2 9 8 n u u u 72 0 2 4 3 2 ii 0 i 2 3 residuai Ouantiies of Standard Normai Intergovernmental Panel on Climate Change lntegrals 0 Estimated IPCC integrals Carbon Change in Aquic Soils l i 0 WetChmale 0 DryClimate mt t Ch 410 a5 0 5 10 15 I l 0 5 10 15 20 25 30 years Smce management cnange Carbon Change in NonAquic Soils mt t Ch 40 a5 0 5 10 15 Summary and Future Research o Increment averaging cannot be ignored in soil core data a Semiparametric stochastic mixed model exibly models increment averages handles xed and random effects ts in standard linear mixed model framework 0 Further work more modeling and diagnostics for carbon data extension to generalized linear mixed models Section 12 3 Colorado Niche Beef Producers amp Products Amanda Ziehl Department of Agricultural amp Resource Economics Colorado State University if Section Summary Look at specific activities and enterprises of niche beef producers within Colorado Determine popular product attributes and distribution methods used by Colorado producers if Survey Overview Survey conducted In Fall 2003 using gt Colorado producer directories gt information from Extension other producers amp current customers Includes 22 Colorado producers Includes name of company contact information product claims amp distribution methods Other producers may not be included in this list as new Colorado niche beef producers emerge into the marketplace over time 3 Product Attributes Most producers sell natural beef with no antibiotics and hormones Several producers sell grassfed andor range finished beef that is healthy Several producers sell beef that is humanely raised and environmentally safe Other attributes include organic cornfinished mountainraised local traceable predator friendly safe highquality and dryaged V Distribution Methods Most producers sell freezer beef Producers most often take orders by phone although some take Internet orders Several producers ship their beef through the mail to their customers Some producers participate in farmer s markets Some producers sell to restaurants and grocery stores or have their own retail store if Idea For Your Company Electronic newsletter gt Some producers surveyed use electronic newsletters to contact their customers gt Keep customers informed on new products give out recipes and cooking tips gt Inform customers about other important information such as mad cow disease gt Creative way to encourage repeat purchases 7 References and Resources Colorado Eat Wild Web site httpwwweatwildcomproductscoloradohtml Colorado Organic Producers Association Web site httpwwworganiccoloradoorg Colorado Food and Agricultural Directory Web site httpwwwagstatecousmkt CategorySearchResultasp Colorado Proud Program website httpwwwagstatecousmktCOProudmeateggs html Contact Information Product Attributes Product Distribution u E or w o E r g d m a i u quot m u gt w 3 5 u g I u or w g 39D a 5 a or 1 2 a w 5 id 5 or s I or E E or r or m s E pg 2 gt u a or a u 0 u 5 r 7 n w t 1 w E E u o w 22wE3 Euoaammiwn EEFEEE isssi9 sagaiia r m r E a in in Vi ii 15 CompanyContact 530Eggggg E8 85 EE u u Brand Name Person Address PhoneFax Wehsrte Emall z 5 5 c U E 3 F g I 00 I E 1 g I o E E E E 0 0 0 0 CoiemanNatuai 5i40RaceCt iii P 3032979393 Wromieman miananmie treason WW Denver COBUZi rain29741426 natumicnm mannatuaimm o o o o o o o o o o FOBaxi r Cnieman MeiCaiemaanme d co Hmwmw paim50i0 Randies inc ir eariiiiininet 0003 Cninradr Naman ii323000tane Biesiiame Smtii FEOHiE CO Pym5274350 deadheeicam deadheeicdn o o o o o o o o o o o o o o SENZN GaErRandi F7i9052 Zi33 NaturdFrmds WM mmw Frioaszaiaa WWW o o o 7 GuideianeCSA nte SBASNCtdeZB Wrouguideiane intnguide DEWH39WLuveiard COBUSSBF 9704610272 a stmetarmcam o o o o o HinesRancii Ciripand FOanim iiit Beet iudyHines Carson COBUBZS MB39BWN o o o o o o Rnhert P000201 irdiamiaFann anoes anttnun CO Pin6004640 O O O O eiharn 0006 Wintames heetiames JamesRandi Jam DWOCOW Porn2475652 mm mm 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 CiWeE 360 CR i00 iaiiisimmiidi Jairnmn amiidanioe Hmms comz P 9705003470 Mama 0 O O O O O O O iaiinm 46001106001de WW0 DEW Bir oggenrco Pan5495311 0 o Nauri Pippa 00652 KWFarms Winger Namasa cowm 7105000429 imi9tnnenet O O O O O O O 1188 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY NOTES AND CORRESPONDENCE VOLUME 15 Feasibility of Retrieving Cloud Condensation Nucleus Properties from Doppler Cloud Radar Microwave Radiometer and Lidar G FEINGOLD S YANG Department of Atmospheric Science Colorado State University Fort Collins Colorado R M HARDESTY NOAAEnvironmental Technology Laboratory Boulder Colorado W R COTTON Department of Atmospheric Science Colorado State University Fort Collins Colorado 9 July 1997 and 31 October 1997 ABSTRACT This paper explores the possibilities of using K band Doppler radar microwave radiometer and lidar as a means of retrieving cloud condensation nucleus CCN properties in the stratocumuluscapped marine boundary layer The retrieval is based on the intimate relationship between the cloud drop number concentration the vertical air motion at cloud base and the CCN activation spectrum parameters The CCN properties that are sought are the C and k parameters in the N CS relationship although activation spectra based on the lognormal distribution of particles is also straightforward Cloud droplet concentration at cloud base is retrieved from a Doppler cloud radar combined with a microwave radiometer following a previously published technique Cloud base is determined from a lidar or ceilometer Vertical velocity just above cloud base is determined from the vertically pointing Doppler cloud radar By combining the simultaneous retrievals of drop number and vertical velocity and assuming theoretical relationships between these parameters and the subcloud aerosol parameters the Cparameter can be derived under the assumption of a xed k If a calibrated backscatter lidar measurement is available retrieval ofboth C and kparameters is possible The retrieval is demonstrated for a dataset acquired during the 39 Tran ition 39 39 A ea 1 39 39 39 39 39 en itivitv to assumptions used in the retrieval is investigated It is suggested that this technique may afford the acquisition of longterm datasets for climate monitoring purposes Further investigation with focused experiments designed Cooperative Institute for Research in the Atmosphere NOAAEnvironmental Technology Laboratory Boulder Colorado to address the issue more rigorously is require 1 Introduction Nearly four decades ago Twomey 1959 explored the relationship between cloud condensation nucleus CCN parameters the production of supersaturation due to adiabatic expansion in a vertically moving parcel and the number of droplets activated from the CCN phase The resulting relationship called an activation spectrum is of the form Nd csk 1 V Corresponding author address Graham Feingold NOAACIRA 325 Broadway Boulder CO 80303 Email gfeingoldetlnoaagov 1998 American Meteorological Society where N is the number of droplets activated S is the maximum supersaturation produced near cloud base and C and k are parameters related to a powerlaw size distribution of aerosol particles During the course of the intervening decades 1 has become a standard in CCN measurements The determination of C and kpro vides a critical link between aerosol and droplet micro physics and important information for studying the aero sol indirect effects whereby aerosol particles affect in coming solar radiation by allowing droplets to condense upon them Thus these parameters are central to the prediction of the impact of CCN on clouds and future climate states Although we present here a method that uses 1 as its representation of the activation spectrum other rep OCTOBER 1998 resentations such as those based on a lognormal rep resentation of CCN are equally feasible provided they do not have more than two parameters Thus a lognorm al distribution of CCN can be used if the breadth param eter is assumed constant The advantage of the lognor mal function is that there is copious evidence of its ability to match the aerosol population with delity and that it allows for the curvature frequently observed in the logilog graphical representation of activation spec tra eg von der Emde and Wacker 1993 However to demonstrate the fundamental concepts of this work we prefer to use 1 at this point In section 4 we explore use of the lognormal function further CCN measurements at various geographical locations generally distinguish between continental spectra hav ing high C W1000 cm and high k 1 and maritime spectra with low C W100 cm and low k W05 However C and k vary greatly depending on a number of factors such as the source of the atmospheric aerosol its age whether or not it has been processed by clouds the altitude of the measurement etc Values of C and k are derived from measurements in thermal gradient diffusion chambers or isothermal haze chambers These measurements are performed in situ either at the earth s surface eg Twomey and Wojcie chowski 1969 or on board aircraft ying in the vicinity of cloud base eg Hudson 1989 Because of the im portance of these parameters in the aerosolicloudicli mate puzzle longterm monitoring of CCN parameters at different geographical locations is essential Unfor tunately the in situ measurement of CCN is limited in that it provides only a point measurement or line mea surement in the case of aircraft with limited spatial resolution Surface measurements suffer from the ad ditional limitation that they may not be representative of the properties at cloud base On the other hand a remote measurement if successful could supply long term datasets with ne temporal resolution and valuable information for climate monitoring studies Feingold and Grund 1994 explored one such avenue by using multiwavelength lidar measurements to retrieve CCN parameters That study indicated the strong sensitivity of the retrieval to measurement errors as well as as sumptions about the chemical composition of the par ticles In this study we explore a different technique By utilizing a combination of Doppler Kaband radar microwave radiometer and lidar a method is developed whereby both C and k can be retrieved The technique builds on the previously published technique of Frisch et al 1995 where Kaband radar and microwave ra diom eter were used to retrieve the number concentration of cloud droplets in stratocumulus The retrieval as sumes that drop number is invariant with heightia good assumption in nondrizzling stratocumulus with negligible cloudtop shear In this work that same re trieval of N d is then combined with simultaneous Dopp ler vertical motion w and using known relationships between w supersaturation production number of drop NOTES AND CORRESPONDENCE 1189 lets activated and the C and kparameters an inversion ofthe value of C is performed for xed k If in addition to the aforementioned measurements a calibratedback scatter lidar measurement is available it is shown how both C and k can be retrieved using a least squares minimization technique The method is outlined in section 2 Thereafter it is applied to data acquired during the Atlantic Stratocu mulus Transition Experiment ASTEX39 Albrecht et al 1995 A discussion of the results is presented in section 4 followed by a summary 2 Method The wellknown equations for droplet activation and growth in a parcel rising at a constant velocity see eg Squires 195839 Twomey 195939 Pruppacher and Klett 1978 are used The equation in supersaturation S is given by 51539 dn T P i T P 2 dt M w 1M dt where w represents the updraft velocity and r1 is the liquid water mixing ratio Here 1111 and 1112 are functions of temperature T and pressure P their exact for mulations can be found in the aforementioned refer ences The growth equation for a droplet of radius rd is giv en y drd a Gm T PS m T 3 Here Grd T P determines the rate of condensational growth of a droplet and yrd T represents the surface tension and solute correction terms to the saturation eld around the droplet Using 3 drKdl can be written as d 7 47139 Grd T PS 7 yrd Trdnrd drd Pa 4 Twomey 1959 solved 2 by assuming that droplets grow from an initial radius of zero ignoring kinetic effects ie the dependence of G on rd and neglecting y He then approximated I rdnrd drd through I SI dt This together with 3 yielded a maximum supersaturation of the form AT Pw3Z 1k2 CkB 32 k 2 Here AT P is given by Twomey 1977 and B is the complete beta function Equation 1 is then used to determine Nd Feingold and Heyms eld 1992 offered an alternative more accurate formulation based on em pirical relations from detailed parcel model calculations simulating the growth of droplets formed on completely soluble ammonium sulfate particles The essence of the parameterization is not substantially different from the 5 1190 JOURNAL OF ATMOSPHERIC Twomey formulation and therefore will not be repeated ere Based on 1 and 5 we may write Nd f1C 1 w T P 6 where f1 is a known function The rst step in the retrieval of C and k involves measurement of Nd and w at known T and P Then using 1 and 5 or an alternative for mulation represented by the parameter C can be re trieved for an assumed value of k If an additional mea surement is available for example a calibrated lidar back scatter measurement the method can be extended as fol lows The lidar backscatter is de ned as 130 1 gm m Amine dn 7 where L is wavelength Qr m L is the backscatter ef ciency at particle radius r complex index of refrac tion m and L and nr is the size distribution of aerosol particles In accordance with 1 we assume that nr can be described by a Junge 1952 powerlaw distri bution dN 73 dlnr C r 8a or quot7 C1775quot 8b although 3 is not xed as in the aforementioned work It is then straightforward to show see appendix that the two free parameters in 8 C and 3 are related to C and k as follows C 15kCD 3 15k 9 andD is a function of particle chemistry see appendix A further interesting relation is that between the Ang strom coef cient a which describes the backscatter de pendence on wavelength and k a37215k72 10 Substituting 8b into 7 and using 9 7 may be written in the general form amp f2C Thus a measurement of 3 enables determination of kprovided C is known Also if a measurement of the Angstrom coef cient a is available 10 can be used to determine k Unfortunately measurements of a are usually performed at the surface so that their utility in this retrieval may be limited unless the boundary layer is well mixed and the settling velocity of particles is small enough so that vertical gradients in the size dis tribution are minimal A retrieval algorithm for C and khas been formulated f3k 11 AND OCEANIC TECHNOLOGY VOLUMEIS based on the above discussion It consists of using 61 and 11 together with a least squares minimization tech nique to determine the free parameters The technique assumes that the CCN distribution changes on time scales longer than the sample period for which C and k are sought The vertical velocity w on the other hand changes rapidly and generates local activation of new droplets Finding C and k from the observations is equivalent to a problem that has two unknowns C and k and a number of observations that link C and k through nonlinear equations 6 and 11 This problem can be solved using the least squares method Speci cally optimal values of C and k can be obtained by minimizing the squared difference between the observed drop concentration and backscatter and their theoretical values calculated for a given C and k This difference is termed a cost function de ned as JC k 2 W1N N202 W203 3021 12 where 139 indicates the ith sample39 Na and 3a represent the observed drop concentration and lidar backscatter respectively39 N g and 3 represent the theoretical values of drop concentration and backscatter calculated ac cording to 6 and 11 from the observed w T and P39 and w1 and w2 are the weightings for drop concentration and backscatter observations and are assigned to the reciprocals of the variances of their observational errors Here JC k is minimized using the quasiNewton con jugategradient algorithm Liu and Nocedal 1989 It is worth noting that C and k should be properly scaled in order to obtain a convergent solution In this paper C and k are scaled as C CCs k kks 13 where C and k5 are the scales of C and k respectively Here C5 500 cm 3 and k5 05 Numerical tests have shown thatJ converges to its minimum as long as C E 50 5000 and k E 01 50 3 Results The algorithm described in section 2 is applied to a dataset acquired by the National Oceanic and Atmo spheric Adm inistration s NOAA Environmental Tech nology Laboratory on the island of Porto Santo in the Madeiras during ASTEX The primary instruments were the NOAA Kaband 86mm Doppler radar Krop i et al 1995 1059pm lidar Pearson 1993 and micro wave radiometer Hogg et al 1983 all collocated on the island Radar data were recorded at 3s temporal resolution and 375m vertical resolution39 radiometer data yielding cloud integrated liquidwater path were recorded at 1min intervals The lidar recorded range Equation 6 will be represented in the form given by Feingold and Heyms eld 1992 OCTOBER 1998 measurements of power and Doppler velocity at a tem poral resolution of 1 s The lidar system was still in its infancy and unfortunately was not calibrated Therefore we have had to apply a calibration factor to these data in order to yield reasonable retrievals Sensitivity to this calibration factor will be investigated in section 4 A ceilometer run by Colorado State University was also available and provided data on cloud base at 1min res olution A lookup table of the normalized lidar backscatter 11 as a function of k was built a priori assuming a Junge distribution de ned over the range 001 mil pm and a relative humidity of 95 Aerosol particles were assumed to be composed of pure ammonium sul fate and at their equilibrium sizes Sensitivity to these assumptions will be explored in section 4 Data are conditionally sorted into updrafts and down drafts with only the updrafts considered since it is they that are responsible for activation of new droplets No other sorting of the data has been performed although in principle the technique should only be applied to active growth regions of cloud and not to decaying cloud The value of w at one radar range gate 375 m above cloud base determined by lidar or ceilometer is used Data are also thresholded so that the retrieval is not attempted when the maximum radar re ectivity fac tor is larger than 715 dBZ to avoid contamination by drizzle The presence of drizzle drops would bias the w measurement and mean that the measured w is some indeterminate mix of vertical air motion and drop ter minal velocity The retrieval is also not attempted when updrafts are less than 5 cm s l The data are ingested into the retrieval algorithm 12 for a short period of time for which simultaneous ver tically pointing radar and lidar returns were available These data spanned the period 103571135 UTC on 22 June 1992 during which there were no indications of a shift in air mass Although copious amounts of data were collected by these instruments during ASTEX the lidar was still in its prototype design phase and we are un comfortable with applying the lidar data beyond this period Fortuitously the UK C130 instrumented air craft ew over the measurement site albeit over 4 h later 1558 UTC Nevertheless the CCN measurements from that ight provide some in situ measurement against which to evaluate this remote sensing technique Table 1 indicates the retrieved C and k parameters for the sample period Other values reported relate to the sensitivity studies discussed in section 4 The retrieved C and k parameters for the baseline simulation are C 175 k 155 These values are quite reasonable for typical marine boundary layers eg Hudson 1980 The log reports of the Meteorological Research Flight A215 Johnson et al 1992 indicated that on this day there existed a continental airmass with several very dis tinct haze layers and very sharp concentration tran sitions from 100 to 2000 cm 3 measured by the PCASP NOTES AND CORRESPONDENCE 1191 TABLE 1 Retrieved C and k parameters for base case and sensitivity tests The base case assumes an aerosol composition of ammonium sulfate rmm 001 and rmax 1 Mm Relative humidity at the height of the backscatter measurement is assumed to be 95 Experiment C k Details Base case 175 155 See text S0 677 184 039 143 in IV retrieval s1 175 180 rm 2 m s2 164 100 a 10 dB S3 172 206 8 i 10 dB S4 218 140 T 5 C S5 175 178 In situ 789 130 Aircraft measurement 45 h later above cloud 2 The data from this day are not ideal for intercomparison rst because they were not collocated and simultaneous and second because of the degree of variability in the vertical The comparison of the remote sensing data with the in situ data is therefore of limited value but is nevertheless included in Table 1 It is seen that the observed C is signi cantly 45 times larger than the base case retrieval39 however the observedvalue ofk 130 is fairly close to the retrieved value of 155 The only way that the retrieval technique can match this high C would be if the retrieved N were signi cantly higher As noted earlier the Frisch et al 1995 tech nique is the source of the N measurement By increasing the assumed breadth parameter of the drop size distribution used in that retrieval from a geo metric standard deviation of 120 or relative dispersion of 018 as used here to one of 143 relative dispersion of 037 one achieves larger Nd and C 677 k 184 indicated by S0 in Table 1 Although no in situ mea surements of drop spectra are available for this event we do not believe the breadth of the drop size distri bution was this large in the nonprecipitating clouds an alyzed here An alternative explanation for the discrep ancy might lie with measurement uncertainty associated with the CCN spectrometer 4 Discussion 1 Sensitivity to assumptions used in the retrieval A number of experiments have been performed to establish the sensitivity of results to assumptions used in the retrieval These appear in Table 1 and are labeled S17S5 Here S1 assumes that the upper cutoff of the aerosol distribution is an order of magnitude larger in mass or a factor of about 22 larger in radius This does not affect the retrieved value of Cbut increases kto 18 compared to the base case value of 155 The reason is that the presence of larger particles tends to produce enhanced backscatter but under the constraint of equal 2 The PCASP or passive cavity aerosol spectrometer probe mea sures particles in the size range 0173 Mm 1192 JOURNAL OF ATMOSPHERIC 3 the algorithm converges on a larger k or smaller number of large particles see 8 and 9 Experiments S2 and S3 were performed bearing in mind the uncer tainties in the lidar backscatter calibration An increase in 3 of 10 dB a factor of 10 results in a drop in C from 175 to 164 and a more signi cant decrease in k from 155 to 100 This increase in 3 has to be provided by a larger fraction of large particles or smaller k A decrease in 3 of 710 dB has little effect on C but k is increased to 206 Again this decrease in 3 has to be accounted for by a much smaller number of large particles or larger k Note that these are rather extreme values of uncertainty typical 3 errors are about 3 dB or a factor of 2 Experiment S4 assumes that the cloudbase temper ature is 5 C higher than used for the base case Here there is a strong increase in C 218 vs 175 and a drop in k to 14 This emanates from the sensitivity of the function f1 to T and the fact that supersaturations are lower at higher T all other elds being equal Ac cording to 1 a lower S must result in a higher C if N is constant This results in a higher C for a xed number of N d assuming constant k39 clearly the situation is a little more complex when the assumption of xed k is relaxed Finally S5 indicates the sensitivity to the base case assumption that particles are in an environ ment of 95 relative humidity by increasing this value to 98 Here Cremains unaffectedby this change while k increases from 155 to 180 The explanation for this again relates to the fact that a higher RH of 98 tends to produce enhanced backscatter but if 3 is held con stant a reduction in the relative number of large par ticles or larger k must follow The result for S5 is very similar to that of S1 where rmax is larger which is consistent with the fact that a higher RH results in en hanced deliquescence growth of particles In general the k parameter is more sensitive than C The reason is that with the exception of S4 the exper iments all affect 37 which is a strong control on k see 11 In the case of S4 the change in Tmainly affects C through the close relation between Nd S and C Ex periments S2 and S3 form the lower and upper bounds on k and indicate that lidar backscatter must be well calibrated if this technique is to be applied If an acoustic sounder is collocated with the other instruments mea surement of cloud base T is straightforward and any errors incurred should be small Errors in P have almost no effect on the retrieval On the other hand assump tions about RH rm and particle chemistry are more dif cult to constrain An average value of RH of 98 may be more reasonable than 95 considering that only updraft regions are considered This assumption will also depend on the lidar range gate used as a source of the 3 data In these retrievals a single range gate was used because cloud base was fairly constant39 however if cloud base varies and crosses from one range gate to another it is quite conceivable that there will be some level of variation in RH associated with the vertical AND OCEANIC TECHNOLOGY VOLUMElS resolution of the lidar The rmax assumption is a dif cult one to constrain although meteorological conditions may be of some help For example if the atmosphere is stably strati ed there should be no large particle source from the surface and given that residence time decreases with increasing size the subcloud aerosol should be devoid of giant particles On the other hand in a wellmixed marine boundary layer breaking waves could be a source ofgiant CCN some fraction of which make their way up to cloud base Finally some information on particle chemistry would be valuable It is noted that the general form of the relationship between wet and dry aerosol radius is given by 14 eg Fitzgerald 1975 The coef cient 8 varies with chemical composition and RH whereas 7 is a function of RH only For example sodium chloride has a value of 8 that is about 35 larger than that for ammonium sulfate Fitzgerald 1975 so that incorrect assumptions about chemistry would be equivalent to a 35 enhance ment in particle size 7 7 rm 57W 1 Dependence on cloud droplet number retrieval The CCN activation spectrum retrieval technique de pends on a good measure of drop number Nd In this work that parameter has been derived following Frisch et al 1995 A few words in this regard are in order The method retrieves pro les of cloud water and median drop size under the assumption of a constantinheight number concentration and spectral breadth To the extent that these conditions are met in real clouds the scheme will perform well Feingold et al 1995 However in many cloud scenarios these conditions may not be met and alternative techniques for retrieving cloud base Nd should be considered One such possibility is the com bination of radar and lidar It has been shown Eberhard et al 1997 that by taking the ratio of radar and lidar backscatter and assuming a functional form of the drop spectrum two parameters of the spectrum number and size can be retrieved Because lidars suffer from atten uation as they penetrate clouds this retrieval is only of use in the region of cloud base but since this is the region of interest in our case it could provide a useful alternative to the Frisch et al 1995 scheme 0 Relative role afC and w in determining Nd Twomey 1977 discusses how for a particular T and P 1 and 5 yield a dependence of N on C and w that varies with k Nd M C1kk 2w1 5kk 2 15 For k 05 N X C 3 w 3 indicating that drop number is primarily determined by C and much less so by w With increasing k w takes on an increasingly OCTOBER 1998 important role in determining Nd so that for k 15 N X C057w05quot and for k 2 N X C 5 w 75 The k values listed in Table 1 are all greater than unity and thus w is an important factor It is interesting too that in recent years the advent of a new generation of in situ instruments eg Hudson 1989 that measure down to low S on the order of 001 indicate larger k than some of the earlier instruments that could not measure at very low S Because the low Smeasurements measure the larger CCN particles this could be indicative of a variable slope in the activation spectrum plotted on a logilog plot as has been measured by Hudson and Fris bie 1991 When converting this spectrum to a size distribution for an assumed particle chemistry this translates to a curvature in the size distribution when plotted on logilog ordinates and deviation from 8 A variable k across the size spectrum could also be indic ative of a sizedependent chemical composition of par ticles 51 Limitations of the C k representation As pointed out by Twomey 1977 the Junge power law distribution 8 and the closely associated C and k parameters 9 is limited in that it requires a minimum radius for a closed solution to the particle concentration In addition various other physical parameters do not exist for speci c values of 3 Perhaps more importantly measurements of particle size spectra indicate that the Junge distribution is only approximate and that a log normal distribution is more appropriate In fact many workers eg Jaenicke 1988 have suggested a bi or even trimodal lognormal distribution Clearly for the purposes of climate monitoring it makes little sense to use parameterizations that would require the retrieval of up to nine parameters for the case of a trimodal log normal from remote measurements Nevertheless it is feasible to perform the same retrieval presented here under the assumption of a single lognorm al distribution of CCN with xed breadth nr L ex iln2rr 2 11120 16 27712 11107 p g where N is the total CCN concentration rg is the median size of the CCN particles and 039 is the geometric standard deviation is a constant Equation 6 is then written as Nd f4N w T P 17 a 7g where f4 is a function given by Ghan et al 1993 Von der Emde and Wacker 1993 have also used the log normal form for the CCN spectrum and shown that in stead of the characteristic straight line on a logilog dis play predicted by 1 the activation spectrum has a concave downward curvature The proposed algorithm is easily extended to the lognormal function using 7 with nr represented by a lognorm al function Equation 11 then has an analog of the following form NOTES AND CORRESPONDENCE 1193 1 37 mg 18 By analogy to the proposed method 6 and 11 retrieval of Na and rg is performed with the aid of 17 and 18 The lognormal form is thus a viable alternative to the C k representation and at this point it appears a matter of preference which form is more desirable Clearly both are limited in that they simplify the form of CCN distribution The large body of data on C and k at different locations in the world might indicate a preference at this stage for the C k representation On the other hand some general circulation models eg Ghan et al 1993 include prognostic equations based on a lognormal distribution e Multiplesensor issues The proposed technique requires a combination of three remote sensors each having a different eld of view For example the beam divergences for the Ka band radar microwave radiometer and lidar are re spectively 05 25 and 0004quot Note that the radar radiometer pair have beam divergences with the least disparity and therefore sampling volumes are not grossly different within the boundary layer To the extent that this is a problem retrievals of Nd will be affected One method of alleviating it is to temporally average the data and this is currently under investigation as part of a eld experiment that includes in situ measurements of N to verify the technique The disparity in volumes sampled by the instruments is especially important when a combination of the measurements is used to derive a physical property of the particles in partially overlap ping sample volumes However here the difference in sample volumes between the radarradiometer pair and the lidar is not a serious problem because the former is used to retrieve incloud drop spectra while the latter is used to retrieve subcloud aerosol properties In other words different physical properties are retrieved by the radarradiometer and lidar each in their respective sam pling volume 5 Summary A technique for retrieving CCN parameters using a combination of a Doppler Kaband cloud radar a mi crowave radiometer and a calibrated lidar has been pre sented The retrieval relies on the technique of Frisch et al 1995 to retrieve cloud droplet number concen tration Using theoretical relationships between drop number radarmeasured vertical velocity supersatura tion production and the CCN parameters together with a calibrated backscatter measurement the C and k pa rameters in the N CS relationship proposed by Twomey 1959 are derived A least squares retrieval has been applied here to a dataset acquired during AS TEX Unfortunately lidar backscatter measurements 1194 JOURNAL OF ATMOSPHERIC were not calibrated and so a calibration has been as sumed39 sensitivity studies have beenperformed by vary ing this calibration over a broad range Results show that retrieved values of C and k are quite reasonable although there is no validation for these retrievals other than an in situ aircraft measure ment that was taken 4 h later Analysis indicates sen sitivity of the k parameter to assumptions about the backscatter calibration upper truncation of the distri bution and cloudbase RH For this technique to be useful at some level a wellcalibrated lidar system will be important In addition because the cloud droplet number retrieval relies on radar and radiometer mea surements these instruments will also have to be well calibrated It is stressed that the various assumptions made in this retrieval are simpli cations of the real world39 the assumption of the Junge powerlaw size distribution is perhaps the most severe Nevertheless we are of the opinion that the advantages of being able to provide copious amounts of data with a level of detail that is appropriate for analysis of rstorder aerosol effects as well as for use in general circulation models are im portant enough to warrant pursuing measurements of this kind Focused experiments that collect data specif ically with this problem in mind will be required in cluding in situ veri cation with thermal gradient dif fusion chambers If a greater understanding of the re mote sensing retrieval is achieved by intercomparison with in situ measurements the true utility of this tech nique will be clearer Thus it is not suggested that this technique replace the detailed in situ measurement but rather augment it Given the importance of CCN mea surements in the aerosolicloudiclimate problem we feel this is a worthy pursuit The alternativeinamely a reliance on surface measurements of CCN or the oc casional airborne campaigniseems inadequate Further testing of this technique could be performed at the Atmospheric Radiation Measurement Cloud and Radiation Testbed site in Oklahoma where all instru ments required by this retrieval are located In situ mea surements of activation spectra during intensive obser vation periods will establish the groundtrut acti vation spectra Two lidar systems the NASA micropulse lidar Spinhirne et al 1993 and Raman systems Gold smith et al 1996 once calibrated will provide impor tant information on both backscatter and relative hu midity in the case of the Raman lidar that will be essential for successful implementation of this algo rithm Acknowledgments The NOAAEnvironmental Tech nology Laboratory s ETL lidar and radar groups are thanked for their efforts expended in acquiring this da taset In particular thanks are due to A S Frisch J Snider A Weickmann and L Olivier of ETL Dr S Cox of Colorado State University is thanked for pro viding the ceilometer data from Porto Santo Support AND OCEANIC TECHNOLOGY VOLUMElS for this work has been provided by NOAA GF as well as an NSF Grant ATM9529321 entitled Simulations of cloudradiative responses to variations in CCN GF and WRC Partial support came from NOAA under Contract NA37RJ02021TEM 14 WRC APPENDIX Derivation of Eq 9 The approximate Kohler relation is b s 1 5 7 3 A1 739 l where 331075 4339 a and Pl A2 T M Here 139 is the van t Hoff factor approximately equal to 2 m is the aerosol mass and M5 is the molecular weight of the solute When dSdr 0 we obtain the relation between S and particle radius 4a3 12 S 27 Substituting A3 into 1 we obtain after some rear ranging A3 N CD Wr WZ A4 where 3113M D A5 274313971715 and p5 is the aerosol density Now since N J39 nr dr J39 C1743qu A6 we have N gr 3 A7 3 Comparing A4 and A7 we obtain 9 C 15kCDkZ39 3 15k A8 REFERENCES Albrecht B A C S Bretherton D Johnson W H Schubert and A S Frisch 1995 The Atlantic Stratocumulus Transition Ex perimentiASTEX Bull Amer Meteor Soc 76 8897904 Eberhard W L S Y Matrosov A S Frisch and J M Intrieri 1997 Microphysical retrievals from simultaneous radar and optical or microwave measurements Proc WMO Workrhop on Measure mentr of Cloud Properties for Forecasts of Weather Air Quality and Climate Mexico City Mexico World Meteor Org 2487 254 Feingold G and A J Heyms eld 1992 Parameterizations of con densational growth of droplets for use in general circulation models J Atmor Sci 49 232572342 OCTOBER 1998 and C J Grund 1994 On the feasibility ofusing multiWave length liudj 4 39 39 J Atmos Oceanic Technol 11 154371558 A S Frisch B Stevens and W R Cotton 1995 Radarra diometer retrievals of cloud liquidwater and drizzle Analysis using data from a 3D LES simulation of marine stratocumulus clouds Proc Fourth Atmospheric Radiation Measurement Sci ence Team Meeting Charleston SC US Dept ofEnergy 1417 144 Fitzgerald J W 1975 Approximation formulas for the equilibrium size an aerosol particle as a function of its size and composition and ambient relative humidity J Appl Meteor 14 104471049 Frisch A S C W Fairall and J B Snider 1995 On the mea surement of stratus cloud and drizzle parameters With a K band Doppler radar and a microwave radiometer J Atmos Sci 52 278872799 Ghan S J C C Chuan and J E Penner 1993 A parameterization of cloud droplet nucleation Part1 Single aerosol type Atmos Res 30 1977221 Goldsmith J E M F H Blair and S E Bisson 1996 Implemen tation of a tumkey Raman lidar for pro ling atmosphericWater vapor and aerosols at the US southern Great Plains climate study site Advances in Atmospheric Remote Sensing with Lidar A Ansmann et al Eds Springer 3377340 Hogg D C and Coauthors 1983 A steerable dualchannel micro Wave radiometer for measurement of Water vapor and liquid in the troposphere J Climate App Meteor 22 7897806 Hudson J G 1980 Relationship between fog condensation nuclei and fog microstructure J Atmos Sci 37 185471867 1989 An instantaneous CCN spectrometer J Atmos Oceanic Technol 6 105571065 NOTES AND CORRESPONDENCE 1195 and P R Frisbie 1991 Cloud condensation nuclei near marine stratus J Geophys Res 96 20 795720 808 Jaenicke R 1988 Aerosol physics and chemistry Zahlenwerte und Funktionen aus Naturwissenscha en und Technik G Fischer Ed SpringerVerlag 3917457 Johnson D W G M Martin J Taylor and M Gibbs 1992 ASTEX Flight Summary for UK C1 0 0 Tech Rep 50 pp Avail ab e from Meteorological Research Flight Facility Farnsborough Hants GU14 6TD United Kingdom Junge C 1952 Die Konstitution des atmospharischen AerosolsAnn Meteor 5 175 Krop i R A and Coauthors 1995 Cloud physics studies With 8 mm Wavelength radar Atmos Res 35 2997 13 Liu D C an J Nocedal 1989 On the limited memory BFGS method for large scale optimization Mat Prog 45 5057528 Pearson G N 1993 A highpulserepetitionfrequency CO2 Doppler lidar for atmospheric monitoring Rev Sci Instrum 64 11557 1157 Pruppacher H R and J D Klett 1978 Microphysics ofClouds and Precipitation D Reide 714 Spinhirne J D 1993 Micropulse lidar IEEE Trans Geosci Remote Sens 31 48755 Squires P 1958 The microstructure and colloidal stability ofwarm clouds Part II The causes in the variations in microstructure Tellus 10 2627271 TWomey S 1959 The nuclei of natural cloud formation Part II The supersaturation in natural clouds and the variation of cloud droplet concentration Geo s Pura Appl 43 2437249 1977 Atmospheric Aerosols Elsevier 302 p and T A Wojciechowski 1969 Observations ofthe geograph ical variation of cloud nuclei J Atmos Sci 26 6847688 von der Em e K an U Wacker 1993 Comments on the relation 39 etween aerosol spectra equilibrium drop size spectra and CCN spectra Beitr Phys Atmos 66 1577162 Part 3 Atmospheric Thermodynamics The Gas Laws Eq of state pV mRT mass gas constant for 1 kg of a gas 0 m v pszT 0r pazRT where azlp For dIy air pd paRdT where R Z R K universal gas constant 83143 Id 6g d M kmol d 35 z m M d i 2 weighted over molecular wt 2 ml M 1 M1 2 molecular weight Rd 2 287J deg 1 kg 1 For water vapor e pVRVTV R R 461Jde 71k 1 v MW g g Rd MW De ne 8 RV Md 0622 Dalton s Law pZpipde mdmv Y V V pd pv 36 ezp39VRvTandpd 2R T v v 7 6 6 pzpdpvz RdT RVT d Rd RV Rd L E RdTl pl 8 e Define TV T 1 1 5 Then Virtual temp T that dry air must have in order to have the same density as moist air at the same pressure 37 Adding moisture to air has the effect of raising TV Moist air is less dense than dry air Hydrostatic Equation Hydrostatic pressure is a result of the weight of an air column Consider an air slab haVing unit crosssectional area 38 Weight of slab mg gde p is a forcearea Therefore 5p represents a net force acting in the vertical direction and since p decreases with height 8p acts upward To be in equilibrium 5pgp5z OI dp E 2 pg hydrostatic equation 39 Alternate View Vertical equation of motion dw l d p g Viscosity dt 0 dz acceleration P grad gravitational in vertical forceacceleration acceleration Ignoring Viscosity assume E 0 Then 1 d P 2 g p dz or dp dz Pg 40 To find p at any height 2 p00 oo I dp 2 gpdz 112 2 0r 192 Igp dz Z Geopotential I is the work that must be done against gravitational field to raise a mass of 1 kg from sea level to a given height 41 d g dz Change in potential energy from the hydrostatic relation dp g pdz gt gdz Odp 01 d Odp The geopotential at height 2 is MD 2 z I d Jgdz 00 0 We define geopotential height zz zijgdz g0 gOO 42 where go is globallyaveraged g at earth s surface g0 98ms 2 Z is what is plotted on our weather maps Express Z in terms of Tp Eq Of state Then Integrating from 61 adp gdz P deTv d Rdzdm p p1 gtp2 2 P2 2 1 Rdj Tvd Inp P1 43 Dividing by go we have R P2 22 21 dj Tvdlnp 0P1 Replace Tv by average fthrough a layer Zz Z1 ln 172171 go or Z Z1 RdTV In p1p2 2 0 Called thickeness eq 44 The geopotential height of the 500 mb surface is ZSOO Rd TV m p2alevel 500 0 Z500 will be low if psea level is low Conversely for a given psea level Z500 is low if mean TV between surface and 500 mb is low 45 First Law of Thermodynamics Internal Energy Sum of kinetic energy of molecules Increases in internal energy in the form of molecular motions is manifested as increases in temperature Consider a unit mass of gas which absorbs a certain quantity of heat energy 6 in joules ie by radiation or thermal conduction As a result the gas may do a certain amount of external work W The excess of energy supplied to the gas over the external work done by the gasis q szuzu2 u1 In differential form dq dw 2 Ch First Law of Thermodynamics incremental differential differential heat elemental internal work energy Consider a gas contained in a cylinder of fixed crosssectional area with a moveable frictionless piston Vocd since area is constant 46 The work done by the gas in expanding is equal to the force exerted on the piston pA multiplied by the distance dx through which the piston moves Thus dV r IH dw pAdx pdV called pV work For a unit mass of gas in the free atmosphere we have dw pda OI dw d6 2 ch pdot First law 47 Speci c Heats Suppose that a small amount of heat dq is given to a unit mas of gas and its temperature increases from Tto T 5T The ratio dqdT is called the specific heat If the volume is held xed then CV 2 dT 0 const If 0cconst 0 dq du 9 48 but for an ideal gas then or specific heat at const p CV 2 dT opeonst u uT only CV 2 dz dq CVdT pda C 2 p pc0nst dq CVdT pda CVdT dp06 adp since pa 2 RT dq CVdT dRT adp 49 OI dq 2 CV RdT adp if p const 016 dqCVRdTgt CVRC dT p p or C p 2 CV R Thus dqdeT adp Enthalpy Define h u pa as enthalpy dh dudpa Remember du CVdT From 1St Law dq CVdT pda du dpa adp dh dq dh adp Rem dq cpdT adp Thus or with h0 T0 Previously we defined geopotential d gdZ adp dq dh adp dh dCpT Called Montgomery stream function If no heat is added or taken away from a parcel in a hydrostatic atmosphere then dq0gth const The Montgomery stream function is conserved along isentropic surfaces Potential Temperature It is often convenient to define a variable that is conservative under adiabatic motion Consider an adiabatic process dq 0 deT adp Using the eq of state pa 2 RT Then d C C dT RT pz 0 zid nT d np p R P integrating from p001 000 mb where we let T 9 to p i d nTz id np T 9 P00 C Yp nTQ 2 Zn 71700 9 Tp00 pRCF Poisson s Eq The potential temperature is the temp that the parcel of air would have if it were compressed adiabatically from its initial level pl to sea level pressure p00 1000171 The Adiabatic Lapse Rate Consider an infinitely small parcel of air that is Thermally isolated from its enVironment such that heat is not added or taken away from the parcel adiabatic The parcel immediately adjusts to the hydrostatic pressure at any level Its motion is small so that its KE remains small For a parcel experiencing adiabatic transformations dq 0 deT adp Take derivative with respect to Z dT 0 CF p 0 dz dz Since pressure adjusts to hydrostatic p at any level the hydrostatic equation Gives us 0119 gp dz 0139 dT dT C g 0gt gC p d dz p Define yd gCp 98 Ckm Parcel Stability The buoyancy of a parcel relative to its environment is defined as B D T D 0 0 V y g To 00 g Vertical accelerations are then dw 039 TV D g g 611 00 To d0 Suppose the env1ronInental lapse rate is d gt 0 Z x l v M 39 F 56 ul i 3 12 Environmental lapse a rate 3 32 O 0 039 0 0 dw For adiabatic rnotion 8constant thus A 0 lt 0 and lt 0 I 0 0 The parcel will return to 0 Conversely if it is brought down to 6quot 6 6 B M gt O and parcel will return to O 60 60 d6 Thus if d gt O the environment is said to be stable Z d6 Now suppose d lt 0 Z Environmental lapse rate 0 an 9 a 57 6A 60 dw Parcel lifted to A Wlll have buoyancy gt 0 or E gt 0 It Wlll 0 continue to accelerate upwards 63 60 Parcel depressed to B Wlll experience buoyancy 6 lt 0 dw 7 lt 0 and parcel will accelerate downwards The lapse rate is unstable I d6 If d 0 then the atmosphere is neutral Z Stability in terms of TandFd gCp Fe zd T Z M 2 Stable Unstable 59 Summary Stable Neutral Unstable F lt Pd F 2 rd F gt Pd gt 0 0 lt 0 dz dz dz 60 Water Vapor Moisture Parameters Mixing ratio mv mass 0 f water vapor md mass of dry air expressed in gkg rv Tropics marine air at surface 20 gkg rv hot summer day in Colorado 6 gkg rv cold dry Winter day in Colorado 01 gkg 61 Speci c Humidity mV mass of watervapor V md mV mass ofmoistair Saturationva or ressure M 5 LatMum e s e 8 9 a I I I cs39Gi i 30 40 1433 3 quot 62 Saturatlon vapor pressure over roe em lt em Saturation Mixing Ratios ramp eSRVT R d esj S Md pd peRdT RV pes Generally es D p or 63 Relative Humidity RH RH100gtltr V Dew Point Temperature air becomes saturated when it is cooled isobarically at const P RH IOOX FSWITS p rs at T p Lifting Condensation Level LCL Level at which a parcel of moist air can be lifted adiabatically before it becomes saturated 64 V BRCH 39 LIFf KINX SHET TOTL CHPE CINS LFC 8208221200 72469 DEN B 0 844 9988 9888 3999 9398 9888 900 45 40 35 30 25 20 15 10 5 0 5 10 15 20 25 3O 35 40 TMPC DHPC 65 During lifting 9 and rv are constant but rs decreases until rs rv at the surface Wetbulb temperature TE Temperature to which a parcel of air is cooled by evaporation of water into it at constant p until the air is saturated with respect to water It is measured with a wellventilated moistened wick covered with a thermometer In general The evaporation of the wetted bulb the wick adds a certain amount of water vapor rvy to the air thus making VsTwVVTrL but VSEVVT Hence rSTW gt 1379 Tw gt Td 66 If once a parcel of air becomes saturated we retain all the condensate the process can be considered reversible since the latent heat liberated during ascent will be consumed by evaporation of water during descent Such a process is said to be saturated adiabatic If all the condensate is dropped out of the parcel during ascent however the process is irreversible since the water is not available during descent for evaporation The process is called pseudoadiabatic 67 SaturatedAdiabatic and Pseudoadiabatic Processes When an air parcel is liften dry adiabatically until it becomes saturatured Condensation on droplets is initiated upon further ascent Latent heat is released and the parcel cools at a lesser rate becomes warmer than a dry lifted parcel than the dry parcel Consider the First law for a system composed of dry air water vapor rv and liquid water Q CF rVCVp QCZdT lrv iv adp dq d 68 Derivation of pseudoadiabatic eguation Latent heat due to condensation or evaporation is dq Ldrv Thus if a certain amount of water is condensed er drv Since rV 1x10 3rz 5 gtlt10 3 CV C P LQN 1 RV ltlt1 Cp Rd C Cp1j CW 3 dT P P 0 1r V ad Ldr 3 P v 69 Thus deT adp Ldrv Assume parcel is exactly saturated drv drs then deT adp Ldrs Let adp gdZ hydrostatic eq Differentiate with respect to z C Ldrs p dz g dz or 70 The pseudoadiabatic lapse rate is dT L drs d dz Cp dT or F n S dz 1 L dr C s p dT Since drs gt0then FSltFd Assumptions 0 No supersaturation o No heat storage on drops 0 Water dropped out of parcel 71 Equivalent Potential Temperature Q We now derive a variable which is conservative for pseudo adiabatic motions Return to Let deT adp dq RT 06 ignoring morsture Remember Thus dT ds entrophy p T p T RH dlnT dznp RCp 6 4 29 n6 2 HT RCp np const 72 or de nQ de nT Rd np Thus for a pseudoadiabatic process Cd n z z P T T czan zidrs d us 40 i CPT CPT CPT Can be shown to be small OI 6e6exp Lrs CPT 6g is the potential temperature of a parcel that has been lifted until rs gt 0 On thermodynamic diagram lift pseudoadiabatically until pseudoadiabat parallels 9 39S then compress dry adiabatially to 1000 mb gt yields 9e 6g is conserved during pseudoadiabatic ascent and descent 73 Conditional Instability Consider the above sounding F Suppose that the air with early morning surface temperature T1 is heated to T2 and lifted dry adiabatically to 0 At 0 the parcel is at the same temperature as the environment If it is lifted to A LCL it will become saturated and thereafter cool at 12 when lifted If lifted to B it will again equal the environmental lapse rate and if lifted further TV gt 0 and the parcel will accelerate upwards 74 Summag The atmosphere is said to be conditionally unstable whenever 75 Convective or potential Instability Consider an inversion layer AB which is at the top of a moist layer Suppose the entire layer AB is lifted The LCL for A is A The LCL for B is B much higher than B After the bottom of the layer near A becomes saturated it will cool pseudoadiabatically whereas B continues to cool dry adiabatially Eventually the entire layer is unstable 59 Condrtlon e lt 0 62 76 y l I nb Poigh i ial I was tact on men mm m m DMAAC 47a nowran rm man an sch pman 9 W CHART CU 77 SIGNIFICANT LEVELS Significant level data are convenient indicators of atmospheric stability This information is very important to analyzing a sounding s vertical structure with respect to lapse rates and moisture content Lifting Condensation Level LCL This is the height at which a parcel of air becomes saturated when it is lifted dry adiabatically The LCL is typically used to identify the base of clouds from lifting due to terrain and frontal systems The LCL for a surface parcel is always found at or below the CCL Convective Condensation Level CCL This is the height to which a parcel of air if heated sufficiently from below will rise adiabatically until condensation starts This is typically used to identify the base of cumuliform clouds which are normally produced from surface heating and thermal convection By using the day s maximum surface temperature along with the corresponding dewpoint and necessary upperlevel temperature adjustments the maximum potential instability of the day s atmosphere can be determined Level of Free Convection LFC This is the height at which a parcel of air lifted dryadiabatically until saturated the LCL then lifted moistadiabatically thereafter would first become warmer than the surrounding air sounding temperature profile at the LFC The parcel will then continue to rise freely above the LFC until it becomes colder than the temperature profile surrounding air at the EL see below The LFC is a critical component to other variables which are dependent on how the LFC is determined Lifted Parcel Level LPL Identifies the level from which parcel lifting processes begin Equilibrium Level EL This is the height where the temperature of a buoyantly rising parcel again becomes equal to and then cooler than the temperature of the environment The EL is commonly used to estimate the tops of convective clouds and thunderstorms ELs are usually defined as two different types the EL resulting from convection initiated from the CCL chEL and the EL resulting from convection initiated from the LFC lfc EL WetBulb Zero WBZ height This is the height where the wetbulb profile transitions from a positive to a negative temperature The wetbulb profile can be displayed when viewing the sounding diagrams skewt or emagram WBZ data is commonly used as one of many factors in estimating hail size and severe weather potential The CCL and cclEL levels are derived from a mean mixing ratio regardless of the selected LPL 78 79 SOUNDING INDICES Sounding indices are convenient numerical indicators of atmospheric stability 7 Standard severe weather threshold values are Iisteo below All temperatures are Celsius C unless othenvise indicated 7 Showalter Index SI The Si is dependent upon 850 mb data and is most reliable when the moist layer extends above the 850 mb level Hart amp Korotky 91 The SI is determined by following the moistadiabat from the 850 mb based LCL to 500 mb and then subtracting the found temperature from the 500 mb sounding temperature SI Thunderstorm Potential 3 4 7 Weak 4 to 4 Moderate lt 4 Strong Lifted Index Ll The Ll is a modified SI and eliminates the 850 mb dependency Johns amp Doswell 92 The Ll is determined from the same process as that used for the SI but the low level LCL is found by using the mean moisture content in the lower 3000 feet Ll Thunderstorm Potential gt 3 Weak 3 to 5 Moderate lt 5 Strong K Index The K Index is a function of 850 700 and 500 mb temperature and moisture information Hart amp Korotky 91 Index T850 T500 ngso Td7oo TDepression700 K Index Thunderstorm Potential lt 25 Weak 25 to 35 Moderate gt 35 Strong See Sturtevant 1995 for additional discussion of these indices and related meteorological data including interpretation and application 80 Thompson Index TI The TI is primarily used to determine thunderstorm potential in the Rocky Mountains Sturtevant 95 TI K Index LI Tl Thunderstorm Potential lt 30 Weak 30 to 35 Moderate gt 35 Strong Jefferson Index JI The JI has been tested and used in both maritime and arid areas and is therefore very important that threshold values be adjusted for the local area of interest Nonfrontal thunderstorms can be expected for index values of 27 or 28 and above Jefferson 6366 JI 16 WBPTBso T500 5 TDepression700 8 iwhere WBPT is wetbulb potential temperature KO Index This index was developed by the Deutsches Wetterdienst German Weather Bureau to estimate thunderstorm potential in Europe It is more sensitive to moisture than other more traditional stability indices and is best used in cooler moist climates AWSFM90001 799500 99700 799350 991000 K0 Index 7 r j 2 2 rwhere Se is equivalent potential temperature KO Index Thunderstorm Potential gt 6 Weak 2 to 6 Moderate lt 2 Strong Boyden Index This index was developed by the British Meteorological Office to forecast the probability of frontal thunderstorms in the UK Boyden 63 Available documentation indicates that it forecasts correctly approximately 60 to 65 percent of the time If the index is 394 then thunderstorms are most likely Boyden Index Tm 1000700mb Thickness 200 10 where the thickness is measured in meters 81 8 index This index was deveIOped by the German Military Geophysical Office as documented by 2WWFM8800 l and is primarily used to indicate thunderstorm potential from April through September 8 index TT T TD700 K 7T and TD are 700mb temps where K is defined as 0 when VT 25 2 when VT gt22 and lt25 6 when VT 22 where VT is the Vertical Totals index T350 T500 S Index Thunderstorm Potential lt 40 39 None 40 to 46 Possible gt 46 Likely Total Totals TT index This index is commonly used as a severe weather indicator and is based on temperature and moisture data AWST R 200 TT T850 Tdsso 2 Tsoo TT Index Thunderstorm Potential lt 45 Weak 45 to 55 Moderate gt 55 Strong SWEAT Index SWEAT was specifically created to help predict severe thunderstorms and tornadic activity AWSTR 200 Caution SWEAT should not be used to predict ordinary thunderstorms SWEAT 12 ngso 20 TT49 2 F850 Fsoo 125 82 where ngso is set to zero if value is negative TT49 is set to zero if value is negative where TT Total Totals F850 is speed of 850 mb wind in knots F500 is speed of 500 mb wind in knots S is the Sine of 500 mb 850 mb wind directions and 7125 S2 is set to zero if any of the following are not met 850 mb wind must be in the 130 through 250 range 500 mb wind must be in the 210 through 310 range 500 mb 850 mb wind direction must be positive 500 mb and 850 mb wind speeds are at least 15 knots 82 SWEAT Thunderstorm PotentiaL lt 300 Weak 39 300 to 399 Moderate 400 to 599 Strong gt 600 High CAP CAP strength also called the Lid Index is determined by finding the maximum temperature difference between the environmental and the lifted parcel profiles within the layer bounded by the lifted parcel level and the LFC The lifted profile is defined by the dry adiabat below the LCL and the moist adiabat above the LCL Fog Stability Index FSl FSI was developed by USAF meteorologists for use in Germany but can be applied to similar climates It was introduced by the USAF publication 2WWfTN79008 and is designed to indicate the potential for radiation fog FSI 4 Ts 2T850 ms F550 where Ts Surface Temp C TDs Surface Dewpoint C FFsso 850mb Wind Speed kts FSI Likelihood of Radiation Fog gt55 Low 31 to 55 Moderate lt31 High Fog Point This value indicates the temperature at which radiation fog will form AWSFM90001 It is determined by following the saturation mixing ratio line from the dew point curve at the LCL pressure level to the surface temperature Fog Threat This value indicates the potential for radiation fog as described in AWSFM 90001 7 Fog Threat WBPT850 Fog Point where WBPTgso 850mb wet bulb potential temperature Fog Threat Likelihood of Radiation Fog gt 3 Low 0 to 3 Moderate lt 0 High 83 Bayesian Selection of Geostatistical Regression Models Devin S Johnson PhD Department of Mathematics and Statistics and Institute of Arctic Biology University of Alaska Fairbanks WNARIMS June 21 24 2005 L Fairbanks AK Sponsor The work reported here was developed under the STAR Research Assistance Agreement CR829095 awarded by the U S Environmental Protection Agency EPA to Colorado State University This presentation has not been formally reviewed by EPA The views expressed here are solely those of presenter and the STARMAP the Program he represents EPA does not endorse any products or commercial services mentioned in this presentation This research is funded by US EPA Science To Achieve Results STARProgram 1 TM Grant UNIVERSITY or ALASKA FAIRBANKS L 2 program Bayesian Selection of Geostatistical Regression Models p1 Introduction 1 0 Environmental data is often collected within a spatial domain obtained through GIS layers 0 Failing to take spatial correlation into account can affect regression model selection results 9 Previous methods to account for spatial correlation in model selection 0 Ver Hoef et al 2001 Spatial stepwise selection 9 Thompson 2001 Bayes factor approx a Hoeting et al 2005 Spatial AICC I Bayesian Selection of Geostatistical Regression Models p2 Bene ts of RJMCMC approach 1 Q Allows inclusion of expert knowledge in selection of regression covariates 9 Spatial stepwise and AICC methods treat all covariates equally 9 Selection over large model spaces Q In both the Bayes factor and AICC methods parameters in each model must be separately estimated 0 Straightforward extension to spatial GLMMs Bayesian Selection of Geostatistical Regression Models p3 Geostatistical regression models 1 YS 30 X18 l Xp8 p 58 where Q s e D C R2 is a spatial location a Zs is the response variable of interest 9 X7siS an explanatory variable 239 1 p 9 65 8 e D is a Gaussian random field with covariance function that decreases with distance I Bayesian Selection of Geostatistical Regression Models p4 Covariance function 001K509 58 02pm ab Var6s 02 where 9 h s 5 Q 02 is the sill 0 lt 02 lt oo 0 gb are the spatial correlation parameters 9 ph gb is a nonnegative correlation function eg pm cb exp lth hgt12 I Bayesian Selection of Geostatistical Regression Models p5 Spatial GLMMs quot Data Model Yltsgtzltsgt Lid Pltg 1zltsgtgt where EiYsZsi g1zltsgt Parameter model 2 ltzlts1gtzlt5ngtgt Mam 2 where 2 is defined by a geostatistical covariance I Bayesian Selection of Geostatistical Regression Models p6 Bayesian model selection 9 Model incorporated as another parameter M with sample space M mg mK 9 For each mk we have 19k k 02 gb Z 1 Inference for the model can be made through the posterior model probability PMP 13071le 0lt DOWkWkP19kmkPmkd19k PYmkPmk Bayesian Selection of Geostatistical Regression Models p7 Model prior distribution 1 A classic model prior is derived by treating inclusion of the p coefficients as a series of independent Bernoulli trials with probability 79 The result is the following prior 19 I Pmk Hwyk 1 7Tj1 1quot j1 where ij is the indicator that my y O I Bayesian Selection of Geostatistical Regression Models p8 Markov Chain Monte Carlo 1 9 Appropriate for large model spaces 9 Objective Draw a sample 193142Mlt1gt 19MWU from P19kmkY 9 Construct a Markov chain with stationary distribution PW mkY CL Approximate PmkY by the proportion of M that equal mk a Posterior Coefficient Probability PCP Plt j 7A OIY Zk k o FUNHY Bayesian Selection of Geostatistical Regression Models p9 Reverse Jump MCMC 1 For current state a 19k 771k 1 Propose move of type 239 to mk from distribution J7a 2 Draw 19k from G7a 77 3 Accept new state 513 with probability mm 1 Px YJ x G x PltxiYgtJiltxgtGiltxgt Bayesian Selection of Geostatistical Regression Models p10 Dif culty with RJMCMC quot 9 Low acceptance rate Even ifthe appropriate model is chosen bad parameter proposals will hinder mixing 9 Conjecture Proposals distributions Cx close to Pt9kmk Y will produce the best results Acceptance probability for Pt9kmk Y mm1 Pmk YJmk 7 PmkYJmk Bayesian Selection of Geostatistical Regression Models p 11 Partial analytic RJMCMC 1 Q Godsill 2001 for AR order selection 9 Use parameter proposal 9 Propose k N P kmk 02 gb Z Y Q 891 0 gbkx Zkl 03927 gb 0 Acceptance probability Pmk 027 7 Pmk027 7Z7YJmk No need to actually simulate m values I Bayesian Selection of Geostatistical Regression Models p12 Acceptance ratio for GLMM 1 91 Suppose PWklmk NPltW 7 Vic 1 Since Z X kJr Zlmk 02 cb NnltXkuk XkaXg 2 Z7027 027 1 OC exp Z1Z gtlt Bayesian Selection of Geostatistical Regression Models p13 Sampler for spatial regression A 97915910 1 Update 02 from P02 Update gb from Pgb Update Z from PZ IfGLMM is used Update mk using partial analytic RJMCMC Update k from P k goto step 1 Bayesian Selection of Geostatistical Regression Models p14 Fish abundance in MAHA region a In 1994 1995 n 119 stream sites were sampled by EPA in MidAtlantic highlands region of US MAHA a Abundance of pollution intolerant fish important indicators of stream health 9 Environmental Covariates Strahler order Elevation Watershed area Dissolved O2 conc fine sediments Road density watershed disturbed Habitat quality index fish cover Bayesian Selection of Geostatistical Reg 1 ression Models p15 Fish sampling locations o Q 2 0quot O Q 00 000 A 51963 0 1 0 O O 0 0 o i quot O 3 Wi d 39 1 1 7 319 G n J 3933 539 w 39 0 O Circles are proportional to abundance of pollution intolerant fish I Bayesian Selection of Geostatistical Regression Models p16 Parameters and Priors 9 Model 79 075 for Strahler Order Elevation and Watershed Area and 79 05 for others a uniform prior was also used a m N Np010002XXk 1 Np update a 61 log 02 N N0 10 Q ltb77 11 62 10g77N071 Bayesian Selection of Geostatistical Regression Models p17 Proposal details For current state Z 02 qb k mk Q Normal proposals used for spatial parameters Q LangevinHastings proposal for Z Draw 2 N Nn Z 98 logPZ m Q Random walk proposal for model jumps Propose to add or drop a randomly selected covariate JmkJmk 1 Q Gibbs update for k I Bayesian Selection of Geostatistical Regression Models p18 Model chain summary PMP Covariate PCP 012 006 005 005 004 Strahler order 088 a o Elevation 029 0 Area 043 Road density 038 o o Disturbance 079 a o Habitat quality 074 a o Dissolved 02 015 Fish cover 010 Fine sed 013 419 out of 512 models visited Bayesian Selection of Geostatistical Regression Models p19 I Model Averaged Coef cients 00 04 08 12 10 00 10 20 Strahler Order 00 04 08 12 2 1 0 1 Watershed Disturbed 00 04 08 12 00 04 08 12 10 00 10 20 Watershed Area 05 05 15 Habitat Quality Index Bayesian Selection of Geostatistical Regression Models p20 Comments Future work 1 0 Partial analytic RJMCMC provides a straightforward method of model update in an MCMC sampler 9 Simple addition to a standard Gibbs sampler 9 Future Transformed Gaussian models possible 9 Future Covariate based model proposals 9 Straightforward extension to generalized linear spatial models a Hierarchical centering allows partial analytic approach a Future Robust hierarchical centering Bayesian Selection of Geostatistical Regression Models p21 Responding to Webwork Jeff Achter Phoenix 2004 jachtercolostateedu Webwork works best when feedback loops are closed and tight 7p11 calculuscolumbia jan 2002 l Calculus required for many students I 12 sections of calculus 1 calculus 2 l Limited teaching resources 0 staffing O classrooms 7p21 calculuscolumbia jan 2002 l Calculus required for many students I 12 sections of calculus 1 calculus 2 l Limited teaching resources 0 staffing O classrooms 7p21 calculuscolumbia jan 2002 l Calculus required for many students I 12 sections of calculus 1 calculus 2 l Limited teaching resources 0 staffing O classrooms 7p21 calculuscolumbia jan 2002 l Calculus required for many students I 12 sections of calculus 1 calculus 2 l Limited teaching resources 0 staffing O classrooms meager resources for graders 7p21 calculuscolumbia jan 2002 l Calculus required for many students I 12 sections of calculus 1 calculus 2 l Limited teaching resources 0 staffing O classrooms no space for recitations 7p21 jdacoumbia jan 2002 l Calculus required for much of my teaching I N1 3 of calc students in my sections l Limited teaching resources 6 staffing 6 administrative overhead 7p31 jdacoumbia jan 2002 l Calculus required for much of my teaching I N1 3 of calc students in my sections l Limited teaching resources 6 staffing 6 administrative overhead 7p31 jdacoumbia jan 2002 l Calculus required for much of my teaching I N1 3 of calc students in my sections l Limited teaching resources 6 staffing 6 administrative overhead 7p31 jdacoumbia jan 2002 l Calculus required for much of my teaching I N1 3 of calc students in my sections l Limited teaching resources 6 staffing 6 administrative overhead 7p31 jdacoumbia jan 2002 l Calculus required for much of my teaching I N1 3 of calc students in my sections l Limited teaching resources 6 staffing 6 administrative overhead maintaining ow of hundreds of pages per week tore into class time 7p31 Initial configuration I sysadmin downloaded installed I graduate TA wrote questions I jda taught 7p41 Initial configuration I sysadmin downloaded installed I graduate TA wrote questions I jda taught troubleshot 7p41 Initial configuration I sysadmin downloaded installed I graduate TA wrote questions I jda taught troubleshot That s the social con guration Technical Dell foar processor server 700 Mhz PHI 4G RAM RAID array 7p41 Successes l great feedback to students I students kept working problems I TA found employment 7p 51 Challenges l problems with problems I talking back to webwork 7p61 FeedbackD New problems have buggy answers 7p7 1 FeedbackD New problems have buggy answers 0 students waste time get frustrated 7p71 FeedbackD New problems have buggy answers 0 students waste time get frustrated O subsequently students assume computer s wrong 7p71 FeedbackD New problems have buggy answers 0 students waste time get frustrated O subsequently students assume computer s wrong 0 generates lots of email 7p71 FeedbackD New problems have buggy answers 0 students waste time get frustrated O subsequently students assume computer s wrong 0 generates lots of email 0 better debugging 7p71 FeedbackD New problems have buggy answers 0 students waste time get frustrated O subsequently students assume computer s wrong 0 generates lots of email 0 better debugging 0 have students write directly to TA closing the feedback loop solved the problem 7p71 Feedback ll Very limited support for free answers 7p81 FeedbackH Very limited support for free answers Closes off potentially useful line of communication 0 instructor can t ask openended questions In this application what does the derivative measure Explain 0 students can t explain themselves Is this a syntax error or am I actually wrong 0 students can t comment on the course I m feeling very shaky now 7p81 FeedbackH Very limited support for free answers Closes off potentially useful line of communication 0 instructor can t ask openended questions In this application what does the derivative measure Explain 0 students can t explain themselves Is this a syntax error or am I actually wrong 0 students can t comment on the course I m feeling very shaky now 7p81 FeedbackH Very limited support for free answers Closes off potentially useful line of communication 0 instructor can t ask openended questions In this application what does the derivative measure Explain 0 students can t explain themselves Is this a syntax error or am I actually wrong 0 students can t comment on the course I m feeling very shaky now 7p81 FeedbackH Very limited support for free answers Closes off potentially useful line of communication 0 instructor can t ask openended questions In this application what does the derivative measure Explain 0 students can t explain themselves Is this a syntax error or am I actually wrong 0 students can t comment on the course I m feeling very shaky now 7p81 FeedbackH Very limited support for free answers Closes off potentially useful line of communication 0 instructor can t ask openended questions In this application what does the derivative measure Explain 0 students can t explain themselves Is this a syntax error or am I actually wrong 0 students can t comment on the course I m feeling very shaky now 7p81 What s wrong with email Facilities exist for emailing responses but this doesn t address all concerns I inconvenient for instructor I hard to process answers efficiently I not integrated with grades database I no snapshot of class 7p91 What s wrong with email Facilities exist for emailing responses but this doesn t address all concerns I inconvenient for instructor I hard to process answers efficiently I not integrated with grades database I no snapshot of class 7p91 What s wrong with email Facilities exist for emailing responses but this doesn t address all concerns I inconvenient for instructor I hard to process answers efficiently I not integrated with grades database I no snapshot of class 7p91 What s wrong with email Facilities exist for emailing responses but this doesn t address all concerns I inconvenient for instructor I hard to process answers efficiently I not integrated with grades database I no snapshot of class 7p91 What s wrong with email Facilities exist for emailing responses but this doesn t address all concerns I inconvenient for instructor I hard to process answers efficiently I not integrated with grades database I no snapshot of class 7p91 What s wrong with paper Intrinsically nothing but I sarne weaknesses as email inefficient separate from grades database no snapshot l Vitiates administrative benefit of electronic classroom 7p101 What s wrong with paper Intrinsically nothing but I same weaknesses as email inefficient separate from grades database no snapshot l Vitiates administrative benefit of electronic classroom 7p101 What s wrong with paper Intrinsically nothing but I sarne weaknesses as email inefficient separate from grades database no snapshot l Vitiates administrative benefit of electronic classroom 7p101 Proposed solution Add support to WeBWorK for freeforrn answers I students can enter short text responses I WeBWorK stores but doesn t grade I TA instructor can easily 0 write back to student 0 Change student score I student can easily View all text answers and instructor responses 7p111 Proposed solution Add support to WeBWorK for freeforrn answers I students can enter short text responses I WeBWorK stores but doesn t grade I TA instructor can easily 0 write back to student 0 Change student score I student can easily View all text answers and instructor responses 7p111 Proposed solution Add support to WeBWorK for freeforrn answers I students can enter short text responses I WeBWorK stores but doesn t grade I TA instructor can easily 0 write back to student 0 Change student score I student can easily View all text answers and instructor responses 7p111 Proposed solution Add support to WeBWorK for freeforrn answers I students can enter short text responses I WeBWorK stores but doesn t grade I TA instructor can easily 0 write back to student 0 Change student score I student can easily View all text answers and instructor responses 7p111 Proposed solution Add support to WeBWorK for freeforrn answers I students can enter short text responses I WeBWorK stores but doesn t grade I TA instructor can easily 0 write back to student 0 Change student score I student can easily View all text answers and instructor responses 7p111 Implementation WeBWorK already stores the text of student answers 80 harness this with modifications I introduce fields P roblemRe spon s e p rp l prp2 where instructor can respond to students answers and utility routines l Database add putProblemResponse etc I Interface add buttons to login pl profLoginplTALoginpl l Answer evaluator dummystrcmp always gives credit 7p121 Implementation WeBWorK already stores the text of student answers 80 harness this with modi cations I introduce fields P roblemRe spon s e p rp l prp2 where instructor can respond to students answers and utility routines l Database add putProblemResponse etc I Interface add buttons to login pl profLoginplTALoginpl l Answer evaluator dummystrcmp always gives credit 7p121 Implementation WeBWorK already stores the text of student answers 80 harness this with modi cations I introduce fields P roblemRe spon s e p rp l prp2 where instructor can respond to students answers and utility routines l Database add putProblemResponse etc I Interface add buttons to login pl profLoginplTALoginpl l Answer evaluator dummystrcmp always gives credit 7p121 Implementation WeBWorK already stores the text of student answers 80 harness this with modi cations I introduce fields P roblemRe spon s e p rp l prp2 where instructor can respond to students answers and utility routines l Database add putProblemResponse etc I Interface add buttons to login pl profLoginplTALoginpl l Answer evaluator dummystrcmp always gives credit 7p121 Implementation WeBWorK already stores the text of student answers 80 harness this with modi cations I introduce fields P roblemRe spon s e p rp l prp2 where instructor can respond to students answers and utility routines l Database add putProblemResponse etc I Interface add buttons to login pl profLoginplTALoginpl l Answer evaluator dummystrcmp always gives credit 7p121 Implementation ll and extensions I respond pl Lets instructor View all responses to given problem If desired instructor can write back to students and update score I readresponses pl Lets student View all instructor responses to her his answers I new kinds of homework questions 7p131 Implementation ll and extensions I respond pl Lets instructor View all responses to given problem If desired instructor can write back to students and update score I readresponses pl Lets student View all instructor responses to her his answers I new kinds of homework questions 7p131 Implementation ll and extensions I respond pl Lets instructor View all responses to given problem If desired instructor can write back to students and update score I readresponses pl Lets student View all instructor responses to her his answers I new kinds of homework questions 7p131 Implementation ll and extensions I respond pl Lets instructor View all responses to given problem If desired instructor can write back to students and update score I readresponses pl Lets student View all instructor responses to her his answers I new kinds of homework questions 7p131 Implementation ll and extensions I respond pl Lets instructor View all responses to given problem If desired instructor can write back to students and update score I readresponses pl Lets student View all instructor responses to her his answers I new kinds of homework questions 7p131 Outcome I allowed openended questions I more student feedback about course I more instructor feedback about student performance 7p141 Outcome I allowed openended questions I more student feedback about course I more instructor feedback about student performance 7p141 Outcome I allowed openended questions I more student feedback about course I more instructor feedback about student performance 7p141 Outcome I allowed openended questions l more student feedback about course I more instructor feedback about student performance I invoked unsuspected feature in database 7p141 One more request Students and faculty have different access to university ID information 7p151 One more request Students and faculty have different access to university ID information Student authentication is difficult 0 students have troubles figuring out their original password 0 lots of account turnover at beginning of semester 7p151 One more request Students and faculty have different access to university ID information Student authentication is difficult 0 students have troubles figuring out their original password 0 lots of account turnover at beginning of semester Perhaps WeBWorK can be changed to use university s Kerberos server 7p151 Postscript Final Configuration I all NLOOO calculus 1 2 students use WeBWorK l sysadmin administers system I instructors instruct I one TA maintains WeBWorK for calculus 1 and another for calculus 2 l same technical configuration 7p161 Postscript Final Configuration I all 1000 calculus 1 2 students use WeBWorK l sysadmin administers system I instructors instruct I one TA maintains WeBWorK for calculus 1 and another for calculus 2 l same technical configuration jda thanks you for your attention 7p161

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