The first Lagrangian point, L-1, is a point between the Earth and the Sun where a mass will be in equilibrium between the gravitational pulls of Earth and Sun and its centrifugal force about the center of mass of the Earth-Sun system. DSCOVER, the first spacecraft planned for Earth observation near L-1, includes a radiometer to measure shortwave and longwave flux from the full disc of the Earth. The latitude of the subsatellite point is near the solar declination, and its longitude is near that for local noon. The spacecraft is always over the tropics. Moreover, it is always over the summer hemisphere. Both of these effects will cause the full-disc measurement of Earth-emitted radiation to be higher than the global mean. In this paper we compute the longwave flux near L-1 using deconvolution theory for wide field-of-view radiometer measurements of Earth-emitted radiation, which makes the computations simple and provides a theoretical framework for understanding the results. The bias due to being over the summer hemisphere will peak twice during the year, so that a semiannual cycle will occur. The global mean Earth-emitted radiation has an annual cycle due to the variation of Earth-Sun distance. If the spacecraft orbit is less than 10° from the Earth-Sun line, the bias to the global mean is computed to be about 8 W m−2 for the annual average with a semiannual cycle of about 1.5 W m−2. In addition to the seasonal cycles, there is a diurnal cycle in the measurement at L-1 as the Earth rotates.
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 Lagrange demonstrated two centuries ago that there are three points on the Earth-Sun axis where a mass will be in equilibrium between the gravitational pulls of Earth and Sun and its centrifugal force about the centre of mass of the Earth-Sun system [Moulton, 1902]. The first Lagrangian point L-1 is between the Earth and the Sun on this axis, as indicated in Figure 1. The second Lagrangian point is on this axis but beyond the Earth from the Sun and the third is on the axis on the far side of the Sun. An object located at L-1 will remain there until disturbed, as the equilibrium is unstable. However, the instability is sufficiently slow that the station-keeping requirements are not excessive. The first spacecraft built for Earth observation near L-1, DSCOVER, includes a radiometer [Rice et al., 1999] to measure shortwave and longwave flux from the full disc of the Earth as the Earth turns beneath it. There is a circle of exclusion of approximately 2° within which the solar irradiance overpowers any signal from the spacecraft in all spectral ranges. Thus data can be received from a spacecraft only outside this circle. The DSCOVER spacecraft does not have a recorder, but would transmit its data continuously. Thus a “halo” orbit has been selected for DSCOVER, so that the spacecraft will orbit the L-1 point as far away as 15° Earth-central angle.
 Because L-1 is on the Earth-Sun line, the subsatellite point for a spacecraft at L-1 coincides with the subsolar point. The latitude of the subsatellite point is then the solar declination and its longitude is that for local noon. A spacecraft at L-1 is always over the tropics. Moreover, it is always over the summer hemisphere. Both of these effects will cause the full-disc measurement of Earth-emitted radiation to be higher than the global mean. The bias due to being over the summer hemisphere will peak twice during the year as the subsolar and subsatellite point cross the equator, so that a semiannual cycle will occur. It is known that the global mean Earth-emitted radiation has an annual cycle due to the annual variation of Earth-Sun distance; this cycle will be observed.
 In addition to the seasonal cycles, there will be a diurnal cycle as the Earth rotates. In the mean, the Earth-emitted flux is greatest at noon [Harrison et al., 1988; Smith and Rutan, 2003], so that there will be a bias at L-1 due to this effect. Each day as the deserts (both land and maritime) cross noon, their large longwave radiant fluxes will cause peaks in the Earth-emitted irradiance at L-1. Likewise, as the deep convective regions over the tropics pass noon, they will cause minima in the LW flux at L-1. Thus the effect of diurnal cycle of heating will be greatest when Africa and Eurasia are near noon. The full-disc view of the measurement will cause these effects to cancel somewhat.
 At any one time, there will be a number of weather systems over the Earth's disc. Although these cause large variations of flux locally, their net effect will be to cancel each other in the full- disc flux at L-1. The result will be a low-level synoptic noise on the Earth-emitted irradiance at L-1. Interannual variations due to climatic features such as El Niño will be observed at L-1, although the integrated effect at L-1 will be greatly diminished. This paper presents a method of evaluating these biases of measurements near L-1 relative to the global mean irradiance due to outgoing longwave radiation (OLR). The approach is to use the deconvolution theory for retrieving maps of Earth-emitted radiation from wide field-of-view radiometer measurements [Smith and Green, 1981]. This technique reduces the computations considerably compared to numerical simulations and provides a theoretical framework for understanding the results. This approach applies only to the longwave irradiance. The computation of solar radiation reflected from the Earth requires computer evaluation because of the complexity of the anisotropy of the reflected radiation as described by Smith  and is not treated here.
 The theory relating the distribution of Earth-emitted irradiance to the measurement is first presented, after which the bias of the measurement near L-1 due to the distribution of Earth-emitted radiation and its variation with season is evaluated. Finally the effect of the diurnal cycle of heating of the Earth will be quantified as it depends on the location of the spacecraft.
2. Analytical Method
 The radiometer will measure the broadband longwave irradiance due to radiation emitted from the entire Earth disc as a single measurement. Thus the measurement m over colatitude and longitude ϕ is the integral of the broadband radiance L over the portion of the Earth within the field of view:
where α = nadir angle at the radiometer, β is the azimuth angle and Ω is the solid angle subtended at the radiometer. The function g(α) describes the angular response of the radiometer to incoming radiance and for a flat plate or cavity radiometer, g(α) = cos α. The broadband radiance L is related to the broadband flux M at the “top of the atmosphere” TOA at latitude θ' and longitude ϕ′ by use of a broadband limb-darkening function R, which will be taken to be a function only of view zenith angle ϑ. The limb-darkening function satisfies a normalization condition 1 = 2R(ϑ)sinϑcos ϑd ϑ. Equation (1) is rewritten to give the measurement in terms of the flux at TOA:
The zenith angle ϑ is a function of nadir angle α and spacecraft altitude. Smith and Green  showed that the eigenfunctions of the operator on the right-hand side of equation (1) are spherical harmonic functions Ynm of degree n and order m, i.e.,
with eigenvalues given by
where Pn is the Legendre function of degree n, γ is the Earth-central angle to the point at TOA which is defined by α, and the upper limit of integration αh is the nadir angle to the limb of the Earth. As a consequence of the normalization condition of the limb-darkening function,
where re is the radius of the Earth and rSC is the radius from the centre of the Earth to the spacecraft. Thus the global mean measurement varies inversely as the inverse square of the distance, as is expected.
 The limb-darkening function depends on the temperature and humidity profile of the atmosphere and the temperature of the lower boundary, whether surface or cloud. For the present purpose R is taken to be the global mean broadband limb-darkening function. The global mean OLR is the term of order zero and is not affected by the directionality of radiances. It was shown by Smith  that the measurement of terms of order 1 and 2 in the expansion of the flux are insensitive to geographic variations of the limb-darkening function.
 The original objective of this theoretical development was to solve equation (2) so that the map of the flux distribution M(θ′, ϕ′) can be computed in terms of wide field-of-view measurements m(θ, ϕ). For the present case, these relations are applied to compute the full-disc measurements at large distances with the Earth-emitted flux map as input. The OLR field of the Earth has been represented by Bess et al.  as
where the Cnm are coefficients determined from monthly mean maps of satellite measurements of OLR. By use of equation (3), full-disc measurements can be written as
The eigenvalues λn are normalized here for convenience by use of the λ0, so that
Figure 2 shows the variation of the kn in terms of
For a measurement at the top of the atmosphere, i.e., for U = 1, kn = 1 for all n. As the spacecraft altitude increases so that U decreases, the kn diminish rapidly with increasing n, indicating that features which are described by high-degree terms of the spherical harmonic expansion have less influence on the measurement. The high-degree terms describe small-scale features which are smoothed out by the WOFV measurement. At very large distance, i.e., as U approaches 0, then kn = 1, 2/3 and 1/4 for n = 0, 1 and 2. All terms of degree higher than 2 become 0 for U = 0. These normalized eigenvalues are appropriate near L-1.
3. Seasonal Cycles
 The method described in section 2 is used first to compute the annual cycle of the measurements. As the Earth rotates under the spacecraft, the Earth-emitted irradiance varies as warmer and cooler regions appear at the subsatellite point. Also, the Earth-emitted irradiance from a given location varies throughout the day as the surface and atmosphere heat and cool during the day. For this analysis the diurnal effect will be partitioned into a diurnal mean term and the diurnal cycle during the day for the given point. The diurnal mean term will be included in this section and the diurnal cycle about this value will be treated in the next section.
 After the Earth has made one full rotation, the daily mean of the measurements will be the zonal mean. The zonal mean is expressed by the zeroth-order terms in equations (5) and (6). The monthly mean Earth-emitted radiation distribution has been computed by Bess et al.  and Bess and Smith [1987a, 1987b] on the basis of WFOV ERB measurements aboard the Nimbus 6 and 7 spacecraft and expressed in terms of spherical harmonic coefficients through order and degree 12. Because the measurement eigenvalues vanish for n greater than 2, equation (4) for the daily average measurement reduces to only 3 terms:
The C0 term is the global mean and varies during the year, primarily because of response of the global mean temperature to the annual cycle of the insolation caused by the eccentricity of the Earth's orbit. The C1 term describes the pole-to-pole variation of OLR over the Earth and its annual cycle is the major part of the Earth's seasons. The C2 term describes the equator-to-pole variation and has only a small seasonal cycle. The Nn are normalization constants for the Legendre polynomials and Nn = [2n + 1]1/2. The C1 and C2 terms introduce biases from the global mean irradiance into the measurement and will be evaluated.
The Cn(0) are the annual mean values of the Cn, the Cn(1) are the amplitudes of the annual cycles and ηn is the phase of the annual cycle. These parameters are listed in Table 1 for n = 1 and 2, from Bess et al.  for Nimbus 6 ERB measurements between July 1975 and June 1976.
Table 1. Coefficients Defining Annual Cycles of C1 and C2
Cn(0), W m−2
Cn(1), W m−2
 For a spacecraft at L-1, the subsatellite point is the same as the subsolar point, so that its latitude is the solar declination and the longitude is that of noon. For this case, cosθ = sinδ. The sine of solar declination can be written approximately as
where ηS is the time of northern summer solstice (in months) times 2π/12.
 With these relations, the C1 and C2 terms can be evaluated for an instrument at L-1. Figure 3 shows these terms individually and their sum through the year. The C1 term is dominated by its annual cycle, and the P1(cos Θ) term reduces to sin δ, so that two cosine of time terms multiply, resulting in a bias and a semiannual term as Figure 3 shows. This bias is due to the instrument moving through the year with the Sun so as to be over the summer hemisphere at all times. The C2 term shows the effect of being over the tropics at all times, and has a semiannual cycle due to the movement as L-1 follows the Sun over the equator twice each year. When the equator crosses the L-1 subsatellite point at the equinoxes, the C2 term is maximum and the C1 term nearly vanishes. Thus the two terms are nearly out of phase and the semiannual parts largely cancel, leaving a small semiannual cycle of approximately 1.5 W m−2. The small annual cycle is mainly due to the annual mean part of the C1 term. The biases for both terms are positive and sum to a net bias of approximately 8 W m−2.
 Because the spacecraft is not at L-1 but in a halo orbit around L-1, the latitude of the subsatellite point must be considered as distinct from the subsolar point. Given the spacecraft location, the expected measurement can be computed by use of equation (8). Figure 4 shows the measurement from various latitudes through the year. As a spacecraft in a halo orbit 15° about L-1 moves during the year, it will have a subsatellite latitude ranging from the equator to 23° + 15° = 38° both north and south. Figure 4 shows the change of measurement and bias from the global mean irradiance as the latitude of the subsatellite point changes.
4. Evaluation of Diurnal Heating Cycle
 During the day as the Sun heats the surface and to a lesser degree the atmosphere, the radiation emitted increases. This diurnal variation of OLR was first quantified by Harrison et al. . The analysis will now be extended to include this effect.
Smith and Rutan  described the diurnal heating cycle in terms of latitude, longitude and local time for each of the 4 seasons as
where ψj is the jth empirical orthogonal function and Φj is the jth principal component. The average of Φj(τ) over a day is defined to be zero. Most of the power of the diurnal cycle of OLR is described by the first 2 terms.
 The effect of the diurnal heating cycle on the irradiance at L-1 can be evaluated as follows. The empirical orthogonal functions (EOF) are expressed in terms of spherical harmonics and the principal components in terms of Fourier series in local time. A radiometer measuring irradiance from the full disc of Earth at large distances will view a range of local times. To account for this fact, local time is measured in radians (2π × hours/24), and is expressed as
Here τG is Greenwich time and (with solar declination) describes the orientation of the Earth relative to the subsolar point. The diurnal heating is thus given as a function of latitude, longitude and Greenwich time. The product of the Fourier series in longitude and spherical harmonics must be rewritten in terms of spherical harmonics alone for the latitude and longitude dependence as in equation (1) in order to evaluate the irradiance at large distances from Earth by using equation (2). The contributions to the C1 and C2 terms introduce biases into the measurement. These coefficients are functions of the Greenwich time. Thus the principal components are written as
The integral is over the entire Earth. The daily average of the diurnal cycle is zero, thus hj0 = 0 for all j at every location, so that the p = 0 terms in equation (11) are all zero, i.e., An0m = 0 for all m and n.
 The results of Smith and Rutan  cover only the Earth between 60°N and 60°S. The diurnal cycle diminishes with latitude because of the reduced solar elevation angle. Also, at high latitudes in summer, the increase of daylight hours reduces the diurnal cycle, and conversely the reduction of daylight hours in the winter hemisphere reduces the diurnal cycle. Finally, the solid angle subtended by an area at high latitude is small compared to an equal area at low latitude. Thus it is assumed that ψj is negligible for regions poleward of 60°. The effect of the diurnal cycle of OLR on the measurement is
As before, the summation includes n = 0, 1 and 2. Also, any terms of higher than wave number 2 vanish in the viewing from large distance, so that terms for ∣p∣ > 2 are zero in equation (12).
Evaluation of the terms in equation (13) shows that the major influence is due to the first EOF, which describes most of the heating and cooling cycle of the desert areas. Its principal component is a half-sine, with its maximum at local noon. As a consequence, this term is maximum at L-1. There is no opportunity for this effect to be averaged out. The second principal component is nearly skew-symmetric about local noon, and the second EOF does not have major long-range structure, thus the second EOF and principal components have only small first and second-degree terms describing their effect and their influence averages out near L-1. Consequently, the summation of equation (12) is dominated by the first term (j = 1).
Figure 5 shows the effect on the irradiance measurement of the diurnal cycle at a large distance from Earth for the subsatellite point at local noon (as earlier with the factor of the square of distance from the spacecraft to the centre of the Earth taken into account). The diurnal cycle of OLR causes a bias of 14 W m−2. This effect must be added to the result computed using the Cnm tabulated for the monthly mean OLR. Figure 6 shows the bias for a spacecraft over local noon considering the both the daily mean and the diurnal cycle terms.
 The bias from the global mean irradiance due to the diurnal heating cycle when averaged over a day is given by integrating equation (13) over all longitudes of Earth as it spins under the satellite. Only the terms for p = m remain after the integration, giving
Figure 7 shows the daily mean bias term as a function of spacecraft position in terms of local time and latitude of the subsatellite point. The bias has a maximum of 11 W m−2 near local noon and at 10 to 20° north latitude, and a minimum of −8 W m−2 about an hour after local midnight at low latitudes. Because An0m = 0, in particular An00 = 0; the zonal coefficients are zero and the daily mean bias is zero at the poles. This is because as the Earth turns for one day, any given location is viewed with the same weighting throughout its diurnal cycle and the diurnal cycle averages out for each point on Earth.
 The biases due to the C1 and C2 terms are an order of magnitude smaller than the global mean OLR. These terms can easily be taken into account, so that the measurement of global mean OLR from an orbit near L-1 can be very precise. Spatial sampling issues are of no consequence in this measurement and its interpretation. Conversely, no information is given about spatial distribution of OLR or processes for degree higher than 2.
 The values for C1 and C2 are tabulated by Bess and Smith [1987a, 1987b, 1991] for 1975 to 1987. The tabulated values of C1 and C2 vary from year to year by up to 1 W m−2. It is not clear whether these changes are physical or are due to measurement problems. If we consider C2 to be uncertain to 1 W m−2, the uncertainty of the global mean of Earth-emitted radiation based on measurements with the subsatellite point near the equator computed using equation (8) is 0.28 W m−2. At the equator, C1 does not affect the measurement. If the spacecraft is on the Earth-Sun line at the solstice so that the latitude of the subsatellite point is 23°, then an uncertainty of 1 W m−2 in C1 results in an error in the computed global mean OLR of 0.45 W m−2. If these uncertainties are combined as independent errors, the root-sum-square result is 0.5 W m−2.
 The Cnm tabulated by Bess and Smith [1987a, 1987b, 1991] were computed from measurement maps from Earth Radiation Budget instruments aboard the Nimbus 6 and 7 spacecraft, which were in Sun-synchronous orbits crossing the equator near noon and midnight. In order to compute a daily mean OLR, the day and night values were simply averaged because of lack of information about the diurnal cycle of OLR, which is a potential source of error of the Cnm. The Earth Radiation Budget Satellite was flown to get measurements of the diurnal cycle of OLR in order to account properly for this effect in computing daily mean maps of OLR [Barkstrom and Smith, 1986; Harrison et al., 1988]. The ERBE data are processed to include the observed diurnal cycle of OLR [Brooks et al., 1986]. More recently the Clouds and Earth Radiant Energy System (CERES) aboard the Terra and Aqua spacecraft provide data which similarly account for the diurnal cycle [Barkstrom et al., 2000; Wielicki et al., 1996]. The ERBE and CERES data should be used to compute the Cnm for low degrees n.
 The C2 is a measure of variation of OLR from equator to pole which is based on the entire globe. Likewise, C1 is a measure of pole-to-pole variation of OLR. The parameters are expected to be quite stable from year to year. These quantities are basic climate variables and should be monitored for subsequent years using Earth radiation budget satellite data. By use of C1 and C2 values based on radiation measurements taken by a satellite in low-Earth orbit which are contemporary with the full-Earth disc measurements near L-1, the above uncertainties can be reduced.
 The measurement at L-1 will vary during the day as deep-convective regions leave the view and deserts come into view. These diurnal variations of the measurement can be computed by using the tesseral terms, i.e., the terms not of order zero. The tesseral terms are also tabulated by Bess et al.  and Bess and Smith [1987a, 1987b, 1991]. These variations will average out over the course of a day.
 A second diurnal variation effect is due to the heating and cooling of regions during the day. This effect can be evaluated and added to the previous terms to give the overall bias of the measurement relative to the global mean Earth-emitted irradiance.
 The discussion thus far has all pertained to a spacecraft near L-1. The measurement equation (4) applies to a measurement taken at any altitude. If the altitude is large compared to the radius of the Earth so that U = radius of Earth/radius from Earth center to spacecraft is small, the implication of the summation reducing to only terms through second order applies. In particular, a spacecraft could also be placed near L-2 so as to observe the night side of Earth. At L-2 the Earth would be a small spot on the face of the Sun, so that it would be necessary to use a halo orbit in order for the spacecraft to receive signals from the Earth away from the intense solar power. The analysis presented here will apply to that case as well.
 It is demonstrated that a measurement of the full-disc longwave flux from the Earth at a large distance (twenty or more Earth radii) will have a bias from the global mean due to the pole-to-pole and equator-to-pole variations. This bias can be easily computed and taken into account. If the inclination of the spacecraft orbit is small (10° or less), the annual mean bias is approximately 9 W m−2 and it has a semiannual cycle of approximately 1 W m−2, and a smaller annual cycle. Over North Africa (40°N), the bias is up to 18 W m−2. Because of the diurnal heating cycle there is an additional bias of 15 W m−2 as North Africa passes through local noon. These results do not apply to the shortwave case, which must be treated separately.
pth Fourier term in time for spherical harmonic expansion of diurnal cycle of OLR, W m−2.
spherical harmonic coefficients for OLR distribution, W m−2.
diurnal heating cycle of OLR, W m−2.
directional response of radiometer.
pth Fourier coefficient in time for jth principal component of diurnal cycle of OLR, W m−2.
normalized eigenvalues of measurement equation.
longwave radiance, W m−2 sr−1.
measurement of longwave flux at spacecraft, W m−2.
OLR at “top of atmosphere,” W m−2.
normalizing coefficients for Legendre polynomials.
Legendre polynomial of degree n.
radius, in terms of Earth radii.
surface area of Earth, sr.
time of year, months.
inverse of radius to spacecraft in terms of Earth radii.
spherical harmonic function of order m and degree n.
nadir angle of incoming radiance, rad.
azimuth angle of incoming radiance, rad.
phase angle, degrees.
nth eigenvalue of measurement equation.
colatitude of spacecraft.
colatitude of point at top of atmosphere.
local time, radians.
longitude of spacecraft.
longitude of point at top of atmosphere.
jth principal component describing time variaation of diurnal cycle of OLR, W m−2.
jth empirical orthogonal component describing geographical distribution of diurnal cycle of OLR.
solid angle of incoming radiance at radiometer, sr.
 This work was supported by the Science Mission Directorate through the Science Directorate of Langley Research Centre and through the Scripps Institute of Oceanography.