NOTES AND CORRESPONDENCE. Uncertainties of Derived Dewpoint Temperature and Relative Humidity

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1 MAY 4 NOTES AND CORRESPONDENCE 81 NOTES AND CORRESPONDENCE Uncertinties of Derived Dewpoint Temperture nd Reltive Humidity X. LIN AND K. G. HUBBARD High Plins Regionl Climte Center, School of Nturl Resource Sciences, University of Nebrsk Lincoln, Lincoln, Nebrsk 5 July 3 nd 8 December 3 ABSTRACT This pper presents n evlution of derived dewpoint temperture nd derived reltive humidity, in which the dewpoint temperture is clculted using mesured mbient ir temperture nd mesured reltive humidity vribles nd the derived reltive humidity is clculted from mesured dewpoint temperture. The derived dewpoint temperture nd reltive humidity re clculted using lgorithms provided by the World Meteorologicl Orgniztion. The method of uncertinty nlysis, provided by the Ntionl Institute of Stndrds nd Technology, is pplied to clculte the uncertinties of n indirect mesurement of derived dewpoint temperture nd derived reltive humidity. The results from the uncertinty nlyses of derived nd observed vribles suggest tht the use of derived dewpoint temperture nd derived reltive humidity involves risk becuse the uncertinties of modern dewpoint temperture nd reltive humidity sensors cn crete severl degrees Celsius of error in the derived dewpoint temperture nd severl percent in the derived reltive humidity. 1. Introduction A single ir humidity sensor or system usully is cpble of directly mesuring single humidity-relted vrible such s reltive humidity by use of solid-stte humidity sensor or dewpoint temperture from chilled mirror hygrometer. Use of these technologies is common with wether-sttion networks or controlled environmentl systems. However, in mny griculturl nd engineering pplictions, the dewpoint temperture could be the required vrible when the reltive humidity ws mesured, or vice vers. In such cses, we usully perform some clcultions to derive the desired humidity vribles from existing dt nd vilble lgorithms. The question for us is, How ccurte re these derived humidity vribles nd how cn we quntify their uncertinties? The concept of the uncertinty of mesurements ws newly defined in set of guidelines by the Ntionl Institute of Stndrds nd Technology (NIST 1994) tht is technicl summry of longer Interntionl Orgniztion for Stndrdiztion guide (ISO 1993), which ws updted nd corrected in The uncertinty is prmeter, ssocited with the result of mesurement, tht chrcterizes the dispersion of the vlues tht resonbly could be ttributed to the mesurnd. This con- Corresponding uthor ddress: Dr. Kenneth G. Hubbrd, 44 L. W. Chse Hll, School of Nturl Resource Sciences, University of Nebrsk Lincoln, Lincoln, NE E-mil: khubbrd1@unl.edu cept hs been interntionlly ccepted s the bsis for the determintion of the mesurement uncertinty. Appliction of these guidelines hs extended beyond clibrtion nd reserch lbortories nd into the industril domin of mnufctured products. Our intent is to obtin the uncertinties of derived dewpoint temperture nd derived reltive humidity when derived using the clcultion lgorithms provided by the World Meteorologicl Orgniztion (WMO 1996).. Method More thn dozen simple or complicted empiricl equtions exist for clculting the sturtion vpor pressure from mbient temperture nd mbient ir pressure or clculting the ctul vpor pressure from the dewpoint temperture bsed on the Goff Grtch (1946) formultion. The equtions tht re commonly used in tmospheric sciences nd other relted disciplines re from Wexler (1976), Buck (11), nd WMO (1996). In this study, the WMO equtions were tken for the uncertinty nlysis s follows (WMO 1996): 17.6T ews f (P)6.11 exp nd (1) 43.1 T e w RH, () e ws where e ws is sturtion vpor pressure (hp) with respect to wter t the ir temperture T ( C), f(p) [ Americn Meteorologicl Society

2 8 JOURNAL OF APPLIED METEOROLOGY VOLUME 43 FIG. 1. Uncertinty of derived dewpoint temperture under (top) the ccurcy combintion of. C for known mbient temperture nd % for known reltive humidity nd (bottom) the ccurcy combintion of.3 C for known mbient temperture nd 5% for known reltive humidity. ( )P.74P 1 ] is the enhncement fctor t the ir pressure P (hp), RH is reltive humidity (%), nd e w is ctul vpor pressure (hp). The sturtion vpor pressure e ws is function of ir temperture nd site tmospheric pressure, wheres the ctul vpor pressure e w is function of dewpoint temperture T d nd site tmospheric pressure if the sturtion vpor pressure is with respect to the wter or is function of frostpoint temperture if the sturtion vpor pressure is with respect to ice. Thus, the WMO Eqs. (1) nd () re often used to clculte the derived T d from known vlues of RH nd T nd to clculte FIG.. Uncertinty of derived reltive humidity under (top) the ccurcy combintion of. C for known mbient temperture nd.3 C for known dewpoint temperture nd (bottom) the ccurcy combintion of.3 C for known mbient temperture nd.5 C for known dewpoint temperture. the derived RH from known vlues of the T d nd T. The clcultions for both RH nd T d obviously require the mesurements of site tmospheric pressure. In our uncertinty nlysis, the site or sttion tmospheric pressure is considered to be constnt ( hp). In uncertinty nlysis, the individul stndrd uncertinty u i is defined s the uncertinty of the result of mesurement expressed s its stndrd devition (NIST 1994). For the mnufcturer s stted ccurcy

3 MAY 4 NOTES AND CORRESPONDENCE 83 specifictions of ir temperture, reltive humidity, nd dewpoint temperture sensors, it is resonble to use rectngulr distributions in uncertinty nlysis (NIST 1994). Thus, one must ssume rectngulr distribution for ll sensors tht mesure T, RH, nd T d bsed on the mnufcturer specifictions with lower limit nd n upper limit (e.g.,. C for T or T d nd % for RH). Therefore, the stndrd uncertinty (NIST 1994) of T d nd RH is u(x i), (3) 3 where ( )/ for both dewpoint temperture nd reltive humidity, nd x i represents T, T d, nd RH. The combined stndrd uncertinty u c cn be derived from the originl Eqs. (1) nd () using the lw of propgtion of uncertinty, commonly clled the root sumof-squres (RSS) method for specific output quntity (T d or RH). Multiplying coverge fctor k (NIST 1994) by the combined stndrd uncertinty, one cn obtin the expnded uncertinty U. In ccord, norml coverge distribution will be used for expressing uncertinty in this study nd is defined by setting k such tht the uncertinty intervl for U hs 95% confidence level. We ssume tht, in ech system, the mesurement of dewpoint temperture or the mesurement of reltive humidity is sttisticlly independent of ir temperture mesured in observtions. Therefore, the expnded uncertinties for derived dewpoint temperture UT d nd derived reltive humidity U RH in ech system cn be clculted s [ ] [ ] d d d UT k u (T ) u (RH) nd (4) d URH k u (T ) u (T d). (4b) The prtil derivtives in Eq. (4) re often referred to s sensitivity coefficients nd cn be obtined nlyticlly. From Eq. (3), u(t ), u(t d ), nd u(rh) re the stndrd uncertinty ssocited with the mnufcturer s specifictions for ech sensor. The vlue of k is s described bove. To illustrte the uncertinty of derived dewpoint temperture nd derived reltive humidity, we ssume tht the ccurcy specifictions from the mnufcturers re. or.3 C for ir temperture, % or 5% for reltive humidity, nd.3 or.5 C for dewpoint temperture. For the derived dewpoint temperture clcultions, we took the known mbient temperture for rnge extending from 5 to 5 C nd the known mbient reltive humidity extending cross rnge from % to %. For derived reltive humidity clcultions, the mbient temperture rnge ws set between 5 nd 5 C nd the dewpoint temperture depression (difference between mbient ir temperture nd dewpoint temperture) ws limited to rnge of 5 C. 1/ 1/ TABLE 1. Uncertinty of derived dewpoint temperture. Inputs Uncertinty of derived T d ( C) T ( C) RH (%) (. C, %)* (.3 C, 5%)* * The first term is for T s stted ccurcy, nd the second term is for RH s stted ccurcy. 3. Results nd discussion. Uncertinty of derived dewpoint temperture Figure 1 illustrtes the vrition of the uncertinty of derived dewpoint temperture with mbient temperture nd reltive humidity. The uncertinty of derived dewpoint temperture (U T d ) increses s the known reltive humidity decreses. With decresing mbient ir temperture, the uncertinty of derived dewpoint temperture decreses. When the known reltive humidity is %, the uncertinty of derived dewpoint temperture rnges from 1.8 to C, resulting in system with n ir temperture ccurcy of. C nd reltive humidity ccurcy of % (Tble 1). From Fig. 1 nd Tble 1, in principle, for ny dewpoint temperture instrument or sensor bsed upon the reltive humidity mesurement, the uncertinty of dewpoint temperture mesurement cnnot fll below.5 C unless the mnufcturer uses more ccurte sturtion vpor pressure eqution thn tht provided by the WMO in Eq. (1). Furthermore, if the known mbient temperture nd reltive humidity hve n ccurcy of.3 C nd 5%, respectively, this combintion cretes rnge of uncertinty for derived dewpoint temperture of C (Tble 1). In relity, the ccurcy combintion of.3 C nd 5% for the mbient temperture nd reltive humidity is relistic expecttion when qulity sensors re deployed in remote field conditions. b. Uncertinty of derived reltive humidity The uncertinty of derived reltive humidity is illustrted in Fig.. In generl, the uncertinty of derived reltive humidity exponentilly increses with decresing dewpoint temperture depression or n incresing reltive humidity. When the dewpoint temperture depression is reltively smll, incresing the mbient

4 84 JOURNAL OF APPLIED METEOROLOGY VOLUME 43 TABLE. Uncertinty of derived reltive humidity. T ( C) Inputs T T d ( C) (. C,.3 C)* Uncertinty of derived RH (%) (.3 C,.5 C)* * The first term is for T s stted ccurcy, nd the second term is for T d s stted ccurcy. temperture decreses the uncertinty of derived reltive humidity (Fig. nd Tble ). Under the ccurcy combintion of. nd.3 C for mbient ir temperture nd dewpoint temperture depression, the uncertinty of derived reltive humidity rnges from.4% to 6.6% (Tble ). The uncertinty of derived reltive humidity increses when n ccurcy combintion of.3 nd.5 C is used for mbient ir temperture nd dewpoint temperture depression (Fig. ). In similr wy, the ltter combintion of mesurements is more resonble expecttion for users using modern sensors nd prctices under field conditions. Thus, the uncertinty of derived reltive humidity should rnge from.6% to.9% (Tble nd Fig. ). The results in Figs. 1 nd reflect the imperfections in the WMO eqution under certin conditions when humidity vribles re derived. However, even when other equtions such s the Wexler (1976) eqution or Americn Society of Heting, Refrigertion, nd Air Conditioning Engineers (ASHRAE 1997) eqution were used to clculte sturtion vpor pressure, similr results were produced. In ddition, Gtes (1994) presented the expected error in derived dewpoint temperture using mesured dry-bulb temperture nd reltive humidity nd the ASHRAE eqution. Note tht the expected error or root-men-squre error used in Gtes (1994) indictes one stndrd devition in the units of mesure ( 68% confidence level for norml distribution). However, the method used by Gtes to simulte the results ws bsed on the RSS rther thn the expected error or the root-men-squre error (Gtes 1994). The results presented in our study reflect the uncertinties of derived dewpoint temperture nd derived reltive humidity t the 95% confidence level. To the uthors knowledge, the ccurcy combintions shown in the third column of Tbles 1 nd (top pnel of Figs. 1 nd ) re chievble in the lbortory, but the results illustrted in the fourth column re more relistic of the observtions from remotely deployed sensors (bottom pnel of Figs. 1 nd ). 4. Conclusions Significnt uncertinties exist when dewpoint temperture nd reltive humidity re derived using dt from utomted wether sttions, climte dtbses tht include derived humidity vribles (T d or RH), nd humidity sensors designed to mesure one vrible nd progrmmed to output nother. The mgnitude of the uncertinty for derived dewpoint temperture not only increses with incresing mbient temperture, but lso increses with the flling mbient reltive humidity. The mgnitude of uncertinty for derived reltive humidity increses with decresing dewpoint temperture depression. It lso chnges with mbient temperture, especilly when mbient reltive humidity is high, which in relity is common condition. We recommend tht one should use, when possible, the ctul direct mesurements for ir humidity relted observtion. However, if derived ir humidity dt must be used, the uncertinty should be clculted, especilly for derived dewpoint tempertures when the reltive humidity is low nd for derived reltive humidity when the reltive humidity is high. The hope for reduction in uncertinty of derived humidity vribles lies in exmining the bis reltionship between mesured humidity vribles nd other vribles (e.g., mbient temperture of sensors nd mbient wind speed). Trnsfer functions tht remove this bis will reduce the input uncertinty in the uncertinty nlysis nd will result in lower uncertinty of the derived vribles. We intend to study this spect more in the future. Acknowledgments. This study ws supported in prt by the Ntionl Climtic Dt Center (NCDC) in our project of Performnce Study of Air Humidity/Wter

5 MAY 4 NOTES AND CORRESPONDENCE 85 Vpor Monitoring Systems for the U.S. Climte Reference Network (USCRN). We thnk Nebrsk Stte Climtologist Allen L. Dutcher nd Dr. Qi Hu for their vluble comments nd suggestions on the originl mnuscript. This pper hs been pproved s Journl Series No. 147 by the Agriculturl Reserch Division, University of Nebrsk t Lincoln. REFERENCES ASHRAE, 1997: ASHRAE Hndbook Fundmentls. Americn Society of Heting, Refrigerting nd Air-Conditioning Engineers, 576 pp. Buck, A. L., 11: New equtions for computing vpor pressure nd enhncement fctor. J. Appl. Meteor.,, Gtes, R. G., 1994: Dew point temperture error from mesuring dry bulb temperture nd reltive humidity. Trns. ASAE, 37, Goff, J. A., nd S. Grtch, 1946: Low-pressure properties of wter from 16 to 1 F. Trns. Amer. Soc. Het. Vent. Eng., 5, ISO, 1993: Guide to the Expression of Uncertinty in Mesurement. Interntionl Orgniztion for Stndrdiztion, 1 pp. NIST, 1994: Guidelines for evluting nd expressing the uncertinty of NIST mesurements results. Ntionl Institute of Stndrds nd Technology Tech. Note 197, pp. Wexler, A., 1976: Vpor pressure formultion for wter in rnge to C A revision. J. Res. Ntl. Bur. Stnd., 8A, WMO, 1996: Guide to Meteorologicl Instruments nd Methods of Observtion. 6th ed. WMO-No. 8, World Meteorologicl Orgniztion, 51 pp.