The Effects of Orbital Precession on Remote Climate Monitoring

4330 JOURNAL OF CLIMATE VOLUME 14

STEPHEN S. LEROY* Danish Meteorological Institute, Copenhagen, Denmark

(Manuscript received 22 January 2001, in ®nal form 30 May 2001)

ABSTRACT The effect of the diurnal cycle when monitoring the climate from low earth orbit is examined brie¯y. Equations are derived that relate the harmonics of the diurnal cycle to temporal sampling error and drift rates in that error. Special attention is given to nodal precession of satellite orbits. Using an insolated blackbody as a simple model for the diurnal cycle, roughly simulating subtropical desert surface temperature, the effects of orbital precession are examined numerically. From an initial con®guration, wherein satellites are evenly spaced in nodal crossing time, minor differences in precession rates lead to biases proportional to the amplitude of the semidiurnal cycle and inversely to the square root of the number of satellites. Overall biases for a single mission can be dramatically reduced by ¯ying in a formation wherein the satellites' orbits are evenly distributed in their equator-crossing times. To monitor surface temperature, it is suggested that at least six satellites be ¯own in formation and that their precession rates be controlled to well within 25 min. The tolerance for monitoring any other variable can be scaled according to the size of its semidiurnal cycle.

1. Introduction Oceanic and Atmospheric Administration weather satellites, notably the Microwave Sounding Unit (MSU; Several organizations throughout the world are con- Spencer and Christy 1992a,b). Individual MSUs have templating placing instruments into low Earth orbit in been shown to measure hermispheric mean temperatures an attempt to monitor trends in the climate system. For with a precision of 0.01 K, but the accuracy of the MSU example, the European Organisation for the Exploitation measurements is not completely understood. In the fu- of Meteorological Satellites (Eumetsat) Polar System ture, the Metop and NPOESS series of satellites are will monitor the climate by means of a series of Me- expected to monitor the climate, and alternative, lower- teorological Operational (Metop) satellites, and the U.S. cost instruments also are being considered. (Goody et Interagency Program Of®ce (IPO) will implement the al. 1998). These alternatives are a constellation of or- National Polar-Orbiting Operational Environmental Sat- biting interferometers obtaining high-resolution infrared ellite System (NPOESS) series for similar purposes. The spectra and a constellation of orbiting global positioning tightest constraint on any climate observing system must system (GPS) receivers collecting occultations of the be that it can measure either very small climate trends atmosphere (Melbourne et al. 1994). Two constellations over the course of its lifetime or that it has negligible of GPS occultation receivers are being considered: the systematic bias so that it can be directly compared with Constellation Observing System for Meteorology, Ion- future measurements using the same technique. As an osphere, and Climate of Taiwan (Rocken et al. 2000), upper limit, it would be desireable to monitor the warm- and the Atmospheric Chemistry Explorer mission of the Ϫ1 ing of the climate, with an accuracy of ഠ0.01 K yr , European Space Agency (ESA) (Hoeg and Leppelmeier typical of model predictions (Kattenberg et al. 1996). 2000). In the past, many attempts to monitor trends in tro- Whenever one formulates a time series by averaging pospheric temperature were done with instruments in data derived from satellite measurements over short in- the Television Infrared Observational Satellite (TIROS) tervals of time, one must contend with the inability of Operational Vertical Sounder suite aboard the National that satellite to measure at all times of day at every location on the globe. In geostationary orbit, a tempo- * Permanent af®liation: Jet Propulsion Laboratory, California In- rally continuous dataset can be obtained, but spatial uni- stitute of Technology, Pasadena, California. formity cannot be obtained. Conversely, in low earth orbit (LEO), a satellite can obtain spatial uniformity but not a temporally uniform dataset. In the latter case, ev- Corresponding author address: Stephen S. Leroy, Danish Mete- orological Institute, Lyngbyvej 100, DK-2100 Copenhagen é, Den- ery position on the globe is sampled 2 times daily sep- mark. arated by almost exactly 12 h, once when the satellite E-mail: [email protected] is on the ascending branch of its orbit and then when

᭧ 2001 American Meteorological Society

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solar time must be obtained to suppress sampling error due to the diurnal cycle. There are other effects related to the harmonics of the diurnal cycle that complicate the temporal sampling error. For one, even the 24-h-period ¯uctuations might contribute to sampling error biases and trends at high latitudes because in these regions the twice-daily sampling is not separated by exactly 12 h for sun-synchronous orbits, whose inclinations are typically 98Њ. Fur- thermore, all odd-number harmonics of the diurnal cycle are expected to be removed by twice-daily sampling, but all even-numbered harmonics are expected to contribute to the temporal sampling error. Because the diurnal cycle as forced by insolation can be expected to be systematic over monthly timescales, the temporal sampling error of measurements by a single-LEO measuring system is examined given that the ¯uctuations are those due to a solar-forced diurnal cycle. Whether the sampling error can be suppressed if multiple LEOs are used is also examined. Last, the effects FIG. 1. Plots of the hour displacement as a function of latitude for of both single LEO and multiple-LEO sampling systems various inclinations. A sun-synchronous orbit has an inclination of 98.2Њ corresponding to the solid curve. in con®gurations that precess even only slightly are examined. Here the term ``climate monitoring'' refers to using satellite data as a kind of thermometer of the earth's the satellite is on its descending branch. With only these climate system. The equations presented here apply to measurements, when constructing a bias-free time series any geophysical scalar variable of the climate, mainly one would have to assume that temporal ¯uctuations those geophysical variables that are likely to contain a that occur exactly when the measurements are taken do signature of the diurnal forcing by solar insolation. Oth- not affect a long-term average of the measurements of ers have examined the sampling error in measurements that quantity. Fluctuations do occur, though, and dom- of outgoing longwave radiation (OLR) likely to be in- inant among systematic temporal ¯uctuations that might curred by single-satellite systems (Salby 1982a,b). A affect a twice-daily measuring system are those asso- geophysical variable whose diurnal signature strongly affects OLR is cloud cover (Bergman and Salby 1996). ciated with the diurnal cycle of the quantity being mea- Stratospheric trace species also have a strong diurnal sured. signature that hampers their monitoring by the Upper- A LEO can largely eliminate ¯uctuations that occur Atmosphere Research Satellite (Salby 1987). with a 24-h period, but it cannot eliminate ¯uctuations In this paper, an analytic formulation is given for that occur with a 12-h period. Because measurements translating a diurnal cycle into the biases it would pro- are separated by 12 h, the ¯uctuation from a true daily duce through temporal sampling error of an arbitrary mean of one observation is exactly the opposite of the multisatellite climate sampling system. This formulation ¯uctuation from a true daily mean of the other obser- can be applied to analyze the likely impacts of a diurnal vation, provided that its period is 24 h. If the ¯uctuation cycle in midtropospheric temperature in the MSU da- has a period of 12 h though, the ¯uctuation of the ®rst taset; however, this would require a season-long con- observation has the same sign as the ¯uctuation of the tinuous record of midtropospheric in situ temperature second observation; thus, the twice-daily cycle of the data at intervals of minutes, a project to be done in ¯uctuation adds a systematic bias onto the true climate another paper. In the next section, the formulation is mean, and this bias is a major component of the tem- applied to a simple model of the diurnal cycleÐthat of poral sampling error of this particular sampling pattern. surface temperature in a tropical desert regionÐand (In truth, temporal sampling error can also arise from compare the sampling error as produced by a single- undersampling cycles other than the diurnal cycle, but satellite system to that produced using a multisatellite in this paper only the diurnal cycle is directly ad- observing system. In the ®nal section, a summary and dressed.) The sampling error can also take the form of discussion of the implications of this work is given and a trend because of aliasing of the diurnal cycle by nodal recommendations for future climate monitoring exper- precession of satellites' orbits. Thus, because even the iments are made. most stable sun-synchronous orbits precess, the temporal sampling error can be exhibited as a trending bias 2. Formulation in any climate monitoring system. Should a multisatel- The most general con®guration assumed is m different lite system be used, however, enough coverage in local satellites in orbits with different inclinations, ascending

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FIG. 2. Filter function for three evenly spaced orbits with inclination 98.2Њ, time t ϭ 0. Squares indicate the real part of the ®lter and diamonds the imaginary part. The latitude is (a) 23.45ЊN, (b) 66.55ЊN.

2 nodes, and nodal precession rates. Their nodal preces- 2␲ 3nae J2 ⍀ϭϪkk Ϫ cosi , (2) sion rates are ⍀j in sun-®xed coordinates. When ⍀j ϭ (1 sidereal yr) 2a222(1 Ϫ e ) 0, satellite j is in a sun-synchronous orbit. If all satellites Ϫ3 sample only at nadir, then nearly equal numbers of where J 2 ϭ 1.08 ϫ 10 is the oblateness coef®cient soundings at a point on the earth's surface take place for the earth,n is the mean motion of the satellite, ae when the satellite is in the ascending branch of its orbit is the earth's equatorial radius, a is the radius of the and in the descending branch. In fact, the sounding hour orbit, and e is its eccentricity (Seeber 1993). For an altitude of 700 km, a sun-synchronous orbit has an in- angles ␩ (␩a for the ascending branch of the orbit and clination i Ӎ 98.2Њ. The sounding offset time ␺ (␾)is ␩d for the descending branch) are given by k given by ␩akϭ⍀t ϩ⌽ kk ϩ␺ (␾) and tan␾ ␩dkϭ⍀t ϩ⌽ k ϩ␲ Ϫ ␺ k(␾), (1) ␺k(␾) ϭ arcsin (3) ΂΃tanik where the time t is measured on monthly timescales, ⌽k is an initial ascending node crossing time (with respect and is interpreted as the LST at which satellite k with to midnight local time) for satellite k, and ␺k is a func- orbital inclination ik crosses geocentric latitude ␾ if it tion of geocentric latitude ␾ given solely by the incli- crosses the equator on its ascending branch at 0000 LST. nation ik of the orbit. Note that geophysical variables Figure 1 shows a family of such hour displacement are usually mapped on grids of geodetic latitude rather curves for different inclinations i. Except for purely than geocentric latitude, so care must be taken con- polar-crossing orbits (i ϭ ␲/2), the polar regions remain cerning which is used. The hour angles ␩a and ␩d can unsampled, and ␺k is unde®ned. For a sun-synchronous be measured in hours as local solar time (LST) or ra- orbit, soundings on the ascending (descending) branch dians, 2␲ radians being equivalent to 24 h of LST. The of the orbit occur approximately 1 h earlier (later) in nodal precession of the orbit with respect to the sun is the day at 60ЊN than at the equator [(see Eq. 1)]. a function of the elements of the orbit through Of primary concern in this paper is the temporal sam-

Unauthenticated | Downloaded 09/30/21 03:13 PM UTC 15 NOVEMBER 2001 LEROY 4333 pling error deriving from the harmonic series of the in m F (t, ␾) ϭ exp[in(⍀ t ϩ⌽)] cos n[␺ (␾) Ϫ ␲/2]. diurnal cycle. The sampling error can take the form of nkkk͸ m kϭ1 a bias or a trend when the satellites' orbits precess. The diurnal cycle can be Fourier analyzed as (6) ϱ The temporal sampling error is the difference between d ϭ d ϩ ᑬ d exp(in␩) (4) 0 ͸ n the average of measurements of a geophysical variable Ά·nϭ1 over time and the true time mean of that variable. Thus, in which ᑬ{ } takes the real part of the quantity in in this formulation, it is the second term on the right of braces, d 0 is the true temporal mean of geophysical var- Eq. (5). The ®lter function gives a dimensionless mea- iable d, and dn is the complex amplitude of the harmonic sure of the degree to which individual harmonics of the n, d1 being the amplitude of the diurnal tide, d 2 being diurnal cycle contribute to the temporal sampling error. the amplitude of the semidiurnal tide, and so on. Both This contribution by the diurnal cycle can manifest itself ␩a and ␩d can be inserted for ␩. The satellites sample as an overall bias induced simply by the con®guration only at the times of day given by Eq. (1). As time t of a set of sun-synchronous satellites or as an arti®cial progresses, the sampling times of precessing satellites trend induced by one or more precessing orbits. If the also evolve, and the diurnal-cycle time series [second diurnal cycle is nonstationary in time, the harmonics of term on right of Eq. (4)] is aliased to timescales asso- the diurnal cycle dn themselves become functions of ciated with nodal precession rates. time. In this paper, the diurnal cycle is considered to be In assembling an observational mean, assume that the stationary in time, which holds reasonably well over number of data from each satellite is the same and that monthly timescales. Over longer timescales, the d will each also samples evenly between ascending and de- n scending branches of the orbit. This is a simplistic meth- evolve, but the low-n components are not expected to od of computing a climate mean from satellite data. The change by more than a small fraction. observational mean ͗d͘ at time t is then approximated by inserting the two sampling times ␩ and ␩ given in a d a. Sun-synchronous, evenly distributed orbits Eq. (1) and averaging over the m satellites: ϱ As a special case, consider a constellation of satellites ͗d͘ϭd ϩ ᑬ dF(t, ␾) , (5) 0 ͸ nn with the same inclination and equator-crossing times Ά·nϭ1 evenly distributed [⌽k ϭ (k Ϫ 1)␲/m and ⍀k ϭ⍀]. The where the ®lter function Fn is given by ®lter function becomes exactly

 cos(n␺k) for n ϭ 2lm F (t, ) exp(in t) 0 for other even n (7) n ␾ ϭ ⍀ ϫ  Ϫsin(n␺k) csc(n␲/2m) exp(in␲/2m)/m for odd n

in which l is an integer. From the leading term on the harmonics at two latitudes (23.45Њ and 66.55ЊN) for right, it is clear that the nth harmonic of the diurnal three evenly spaced sun-synchronous orbits (i ϭ 98.2Њ). cycle will alias to a frequency given by n⍀. Orbits typ- At both latitudes, the second and fourth harmonics are ically drift on the order of 2 h over a 10-yr period, completely suppressed. At the lower latitude, the odd meaning that the diurnal tide would appear as a 120-yr harmonics are signi®cantly but not completely sup- cycle in the observed mean, the third harmonic would pressed. At the high latitude, though, the odd harmonics appear as a 40-yr cycle in the observed mean, and so are only modulated by a complex factor of order unity. on. For faster orbital precession rates, the bias occurs Considering that the sixth harmonic of the diurnal cycle on shorter timescales. is very small, the largest bias in the observational mean For a purely polar orbit, ␺k ϭ 0 for all latitudes, and the ®lter function only passes n ϭ 2lm harmonics of comes from the ®rst harmonic of the diurnal cycle at the diurnal cycle. For instance, with four orbits evenly high latitudes. distributed in equator-crossing time, then the only harmonics that can bias a mean measurement are the 8th, 16th, 24th, and so on, harmonics. Because there is little b. Differential precession from sun-synchronous systematic amplitude in the harmonics of the diurnal orbits cycle, a four-satellite observation system would essen- tially introduce no bias due to the diurnal cycle. Although a constellation of satellites may be intended In Fig. 2 is shown the ®lter function for different to be evenly distributed in equator-crossing time in sun-

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synchronous orbits, they in fact will drift in equator- mean⍀⍀ , ␦⍀k ϭ⍀k Ϫ and ␦⍀k has a variance of 2 crossing time slowly. In such a case, apply Eq. (6) with ␴⍀. In the approximation that n␴⍀t K 1, the ®lter func- ␾k ϭ (k Ϫ 1)␲/m as in the previous section, and ⍀k has tion is approximately

 m cos(n␺ )(1ϩ in␦⍀ t) exp[in(k Ϫ 1)␲/m] for even n  kk͸  kϭ1 Fn(t, ␾) ഠ exp(in⍀t)/m ϫ m (8) i sin(n␺ )(1ϩ in␦⍀ t) exp[in(k Ϫ 1)␲/m] for odd n.  kk͸ kϭ1

For the terms with n even, the zeroth-order term in n␦⍀kt sums to zero, and the ®rst-order term adds randomly. For the terms with n odd, the zeroth-order term in n␦⍀kt adds to a nonzero sum. Thus, the ®lter function is approximately  cos(n␺k) for n ϭ 2lm F (t, ␾) exp(in⍀t) ϫ cos(n␺ )(n␴ t) exp(i␳)/ m for other even n (9) nkഠ  ⍀ ͙ Ϫsin(n␺k) csc(n␲/2m) exp(in␲/2m)/m for odd n

Ϫ1 in which exp(i␳) is a random phaser that depends on Teq ϭ 24.0 K and ␤ ϭ 0.25 h , the nonlinear solution the particular sequence ␦⍀k. is Fourier transformed to obtain the coef®cients dn. Equation (10) is the same as Eq. (8) except that all Because datasets with temporal resolution over cli- even harmonics can contribute linearly in time t. This mate timescales are scarce, this simple model will have means practically that the large second harmonic of a to suf®ce to estimate the overall effect of the diurnal diurnal cycle can contribute when even a well-con®g- cycle in biasing climate monitoring systems in low earth ured constellation has slight differential precession, orbit. even though the second harmonic does not initially contribute to an overall bias. Also, note that the second a. Nadir-sounding systems harmonic is suppressed by the square root of the number of satellites m. The Earth Observing System [National Aeronautics and Space Administration (NASA)], NPOESS (IPO), and the Metop (ESA) satellites are all major endeavors 3. Example: Diurnal cycle of an insolated whose goal in part is to monitor the climate. Other cli- blackbody mate monitoring systems involving several coordinated A simple model of the diurnal cycle of surface tem- satellites have been suggested, such as a constellation perature is that of a blackbody with radiative time con- of GPS occultation receivers (Rocken et al. 2000; Hoeg stant ␶ that is heated by the sun during the day and not at all at night. The differential equation describing the diurnal temperature cycle of such a body is

dTÃÃ␤␲cos ␨ Ϫ T 4 in daylight ϭ (10) dt 4 ΆϪTÃ 4 at night in which ␨ is the solar zenith angle, TÃ is temperature nondimensionalized by Teq, and ␤ is the inverse of the radiative time constant of the slab. The ``equilibrium'' temperature Teq is the time-average temperature of the linearized version of this model, but it must take into consideration the effects of the atmosphere. In this case, the atmosphere strongly damps the ¯uctuations of the diurnal cycle largely because of its greenhouse char- FIG. 3. Diurnal cycle for a desert region. The thin curve shows the acteristics, and so here Teq is tuned to match data taken average diurnal cycle obtained from the weather station at the Jet from a weather station in an arid region. That station is Propulsion Laboratory of the California Institute of Technology for at the Jet Propulsion Laboratory (JPL) of the California July 2000. The thick curve is a ®t of the radiating-slab model to the data from the JPL weather station. The temperature constant Teq was Institute of Technology, Pasadena, California, and the set to 24.0 K, and the radiative time constant was set to 4 h. The data are taken from July 2000 (see Fig. 3). After ®tting diurnal mean has been subtracted.

Unauthenticated | Downloaded 09/30/21 03:13 PM UTC 15 NOVEMBER 2001 LEROY 4335 and Leppelmeier 2000) and a constellation of small in- As a consequence, the initial bias (␶ ϭ 0) is Ϫ0.01 KÐ frared interferometers (Goody et al. 1998). The temporal a signi®cant reduction from the single-satellite systemÐ sampling error for a single satellite with a precession but the bias trend is about Ϫ0.015 K yrϪ1. As time ␶ rate of 2.5 min yrϪ1 with an inclination of 98.2Њ is shown progresses, the constellation drifts farther out of its as a function of time in Fig. 4. Also shown is the bias evenly spaced con®guration and the overall bias in- for a six-satellite constellation in sun-synchronous orbits creases to amplitudes similar to those of the single- with nodal precession rates normally distributed by a satellite system but divided by͙6 . random number generator with a standard deviation of For each system, the bias drift is almost entirely due 2.5 min yrϪ1. to the n ϭ 2 component of the diurnal cycle. Because Figure 4 shows the temporal sampling error over cen- the nth harmonic of the diurnal cycle is aliased by sat- tury timescales to illustrate the biases and bias trends ellite nodal precession ⍀ to a period of n⍀, the semi- that can be expected from two systems that might mon- diurnal tide for the single-satellite aliases to a bias with itor surface temperatures in a desert. Theoretically, the a period of 288 yr. The constellation system, with its overall bias depends on the con®guration of the ob- range of precession rates, shows bias cycles with periods serving system. Because the progression of time sees of 535, 1036, 610, 202, 131, and 228 yr, each a semi- changes in the nodal crossing times due to nodal pre- diurnal cycle aliased by precession. cession, different times in the plot correspond to different con®gurations. For the single-satellite system, b. Cross-track scanning systems there is no con®guration that simultaneously eliminates the bias and the trend in the bias. At time ␶ ϭ 0, the In reality, any orbiting instrument intended to monitor bias is maximized at about ϩ1.7 K, which persists for the climate will sample over a wider swath than just the approximately 20 yr. If this system is to be used as a nadir. For instance, the Atmospheric Infrared Sounder benchmark, then follow-on single-satellite systems (AIRS) of NASA's Earth Observing System (on board should ¯y with the same nodal crossing time or risk a Aqua) will scan a swath Ϯ47Њ of the nadir. Also, GPS substantial difference in bias. The maximum trend in occultation instruments effectively ``scan'' to the limb the bias occurs at about ␶ ϭ 210 yr in the plot with an because it is a limb-sounding measurement. Certainly, amplitude of ϩ0.038 K yrϪ1, which easily exceeds pre- the ®lter function is modi®ed for a cross-track scanner. scan dicted trends in global warming, which generally are Deriving the scanning ®lter functionF n (t, ␾) is done Ϫ1 ഠ0.01 K yr (Kattenberg et al. 1996). The constellation by integrating Eq. (6) over a continuum of ␺k ranging system for which sampling error is produced in Fig. 4 from ␺k Ϫ ␦␺k/2 to ␺k ϩ ␦␺k/2, ␦␺k being the full begins in orbit planes evenly distributed in nodal-cross- scanning range of the instrument as a function of lati- ing time and then slowly drifts out of that con®guration. tude:

in m 1 ␺kkϩ␦␺ /2 Fscan(t, ␾) ϭ exp[in(⍀ t ϩ⌽)] ϫ cos n(␺ЈϪ␲/2) d␺Ј. (11) n ͸ kk m kϭ1 ␦␺k ͵ ␺kkϪ␦␺ /2

Evaluating this integral gives the following expression It is noticeable that biases are not signi®cantly sup- for the scanning ®lter function: pressed by scanning across the satellite ground track. The reason is that the dominantly contributing diurnal in m harmonicÐthe secondÐis not signi®cantly reduced by Fscan(t, ␾) ϭ exp[in(⍀ t ϩ⌽ )] cos n(␺ Ϫ ␲/2) nkkk͸ m kϭ1 the scanning. This is particularly true for the single- satellite system simply because the full range of the ϫ sinc(n␦␺k /2), (12) diurnal cycle is not sampled even with the scanning. the offset time ␺k and swath width ␦␺k both being func- For the constellation, nearly the entire range of the di- tions of latitude, and sinc x ϭ (sin x)/x. The main con- urnal cycle is sampled (6 ϫ 3.43 h) but it is done un- sequence of cross-track scanning is that larger swaths evenly in local time. For the constellation, however, the

␦␺k increasingly suppress higher-order harmonics n. bias trend in the ®rst 10 yr of the simulation is reduced The ®lter function of Eq. (12) is applied to the two to less than 0.01 K yrϪ1. systems described in the previous section. Again, the In this paper it has been assumed that a simple mean sounding latitude is 20ЊN. The single satellite is assigned of the observations suf®ces as a computation of the an AIRS-like swath of 1650 km (1.05 h in ␦␺), and the climate mean. The uneven sampling by the constellation six-satellite constellation is assigned a limb sounding presumably can be taken into consideration in a more swath of ഠ5721 km (3.43 h in ␦␺), characteristic of sophisticated computation of the climate mean. One pos- GPS occultations. The results are shown in Fig. 5. sible method would be to analyze not only a climate

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FIG. 4. Precession rates for the six-satellite constellation are ran- FIG. 5. Bias simulations for cross-track scanning systems (the same domly selected from a normal distribution with std dev of 2.5 min as Fig. 4 except that the single-satellite system scans over 1.05 h in yrϪ1, and the values are 1.346 12, Ϫ0.694 781, 1.179 78, 3.568 62, local time and the six-satellite system scans over 3.43 h in local time). 5.484 36, and Ϫ3.158 97 min yrϪ1.

stellation of more than one satellite can initially suppress mean (as a function of latitude and longitude) but also the second harmonic, but after even a small amount of the coef®cients of the ®rst few harmonics of the diurnal differential nodal precession occurs, the second har- cycle. Such an analysis would weight the data in a way monic begins to contribute. The second harmonic is that would distinguish the diurnal cycle from the mean, aliased to 2 times the frequency of the nodal precession but it would only work given suf®cient coverage in hour rate, whereby the period of the cycle is in the range of angle/LST. 100±300 yr for nominal precession rates of about 2.5 min yrϪ1. Third, the breadth of cross-track scan angle does not 4. Summary and recommendations signi®cantly affect the sampling error induced by the Equations describing the evolution of sampling error diurnal cycle. It is the second harmonic of the diurnal incurred by climate monitoring satellites with nodal pre- cycle that dominates the sampling error, and as a con- cession are presented. Furthermore, these equations are sequence a cross-track scanning instrument would need applied to a simple model of a diurnal cycle in surface to scan across the semidiurnal cycle to eliminate it from temperature with a peak-to-peak temperature difference a mean measurement. This is impractical. Even cross- of 14 K. The model is representative of a subtropical track scanning by a constellation of satellites does not desert. Three main points come from the simulation of signi®cantly suppress sampling errors caused by the sampling biases. Three recommendations also arise semidiurnal cycle. from this work. The ®rst recommendation is that it is most important First, the more satellites used in a climate monitoring to control the equator-crossing times of a climate mon- system, the more the temporal sampling error is reduced. itoring instrument. In the simulation, the six-satellite The overall bias and its drift rate in any con®guration constellation allowed a drift in the sampling bias of 0.1 are reduced proportionally to the inverse square root of K within a decade of the initial con®guration when drift the number of satellites. The overall bias is even more rates are random but on the order of 2.5 min yrϪ1. This strongly reduced when the satellites are ¯own in a for- suggests an error of less than 0.1 K after about 25 min mation such that the equator-crossing times are evenly of drift. An upper limit on the precession rate should spread. When the nodes slowly precess, however, the then be substantially less than 25 min. drift in the sampling error is again inversely proportional The second recommendation is that several satellites to the square root of the number of satellites. In the be ¯own in formation to monitor the climate. The sam- simulation presented here, the six-satellite constellation pling error drift rate theoretically scales as the inverse saw a drift in the sampling error over the ®rst 10 yr of square root of the number of satellites, but the overall less than 0.01 K yrϪ1 in surface temperature. bias can be reduced even more strongly if a number of Second, it is the second harmonic of the diurnal cycle satellites are ¯own in formation. The formation should that contributes most strongly to sampling error. The be such that the satellites are evenly spaced in equator- ®rst harmonic is mostly eliminated because every sat- crossing time. In the simulation presented here, the ini- ellite samples as frequently during the ascending branch tial con®guration yields a bias of less than 0.1 K. The of its orbit as during the descending branch. As one single-satellite system yielded a bias greater than 1 K. moves to higher latitudes, though, the ®rst harmonic If a single-satellite system is used, then subsequent mis- contributes more to the sampling error. Also, a con- sions to monitor the climate must be placed in precisely

Unauthenticated | Downloaded 09/30/21 03:13 PM UTC 15 NOVEMBER 2001 LEROY 4337 the same orbit as the ®rst. Six satellites were simulated or model the diurnal cycle of this quantity before the in this paper, but possibly four may be satisfactory de- theory presented herein is applied. pending on the variable to be monitored. The third recommendation is that analysis of the data Acknowledgments. This work was performed at the should involve analysis of the diurnal cycle in addition Danish Meteorological Institute and was supported by to analysis of climate means. In this paper, only the the GRAS Science Application Facility of Eumetsat. simplest possible determination of the climate mean was Thanks to Richard Goody for suggesting the topic. Thanks also to Alejandro Levi for making data from assumed. If the diurnal cycle is analyzed as well, it JPL's weather station archives available. should theoretically be possible to separate the effects of the diurnal cycle from longer-term climate trends if the data cover enough hours of the day. REFERENCES In the future, it is expected that the method of ana- Bergman, J., and M. Salby, 1996: Diurnal variations of cloud cover lyzing how the diurnal cycle is aliased to climate time- and their relationship to climatological conditions. J. Climate, 9, 2802±2820. scales can be directly applied to temporal sampling error Goody, R. M., J. Anderson, and G. R. North, 1998: Testing climate in the MSU record of tropospheric temperatures. This models: An approach. Bull. Amer. Meteor. Soc., 79, 2541±2549. would be done by developing a model of the diurnal Hoeg, P., and G. Leppelmeier, 2000: ACE: Atmospheric Climate Ex- cycle of upper-air temperatures and validating it with periment. Danish Meteorological Institute Scienti®c Rep. 00-01, Copenhagen, Denmark, 51 pp. an appropriate dataset. Whatever dataset is used, it must Kattenberg, A., and Coauthors, 1996: Climate modelsÐProjections record upper-air temperatures with an approximately of future climate change. Climate Change 1995: The Science of hourly frequency on a global spatial scale. One possible Climate Change, J. Houghton and L. G. Meira Filho, Eds., Cam- bridge University Press, 285±357. dataset is a time series from geostationary satellites, Melbourne, W. G., and Coauthors, 1994: The application of space- such as the Geostationary Operational Environmental borne GPS to atmospheric limb sounding and global change Satellites. Another possible dataset would be a com- monitoring. NASA JPL Publication 94±18, Jet Propulsion Lab- posite of lidar data taken from stations around the world. oratory, California Institute of Technology, Pasadena, California, 138 pp. After validation, one can obtain the coef®cients dn as Rocken, C., Y.-H. Kuo, W. S. Schreiner, D. Hunt, S. Sokolovskiy, functions of time and space, implement appropriate sat- and C. McCormick, 2000: COSMIC system description. Terr. ellite inclinations and precession rates, and then produce Atmos. Oceanic Sci., 11, 21±52. biases such as those presented in Figs. 4 and 5. Salby, M., 1982a: Sampling theory for asynoptic satellite observations. Part I: Space±time spectra, resolution, and aliasing. J. A simple model of the diurnal cycle of surface tem- Atmos. Sci., 39, 2577±2600. perature in an arid region was used to illustrate how the ÐÐ, 1982b: Sampling theory for asynoptic satellite observations. diurnal cycle gives rise to a temporal sampling error by Part II: Fast fourier synoptic mapping. J. Atmos. Sci., 39, 2601± 2614. a satellite observing system. This diurnal cycle in tem- ÐÐ, 1987: Irregular and diurnal variability in asynoptic measure- perature is anticipated to be representative of those in ments of stratospheric trace species. J. Geophys. Res., 92, 14 the lower troposphere, infrared sounding systems also 781±14 805. Seeber, G., 1993: Satellite Geodesy: Foundation, Methods, and Ap- registering diurnal components of cloudiness. In the up- plications. Walter de Gruyter, 531 pp. per troposphere, though, diurnal temperature variations Spencer, R. W., and J. R. Christy, 1992a: Precision and radiosonde are not as large, and consequently the temporal sampling validation of satellite gridpoint temperature anomalies. Part I: error of this quantity is not as large. To decide the sat- MSU channel 2. J. Climate, 5, 847±857. ÐÐ, and ÐÐ, 1992b: Precision and radiosonde validation of satellite con®guration needed to minimize temporal sam- ellite gridpoint temperature anomalies. Part II: A tropospheric pling errors for a given quantity, one must ®rst estimate retrieval and trends during 1979±90. J. Climate, 5, 858±866.

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