The Improved Limb Atmospheric Spectrometer (ILAS), a sensor for stratospheric ozone layer observation using a solar occultation technique, was mounted on the Advanced Earth Observing Satellite (ADEOS), which was put into a Sun-synchronous polar orbit in August 1996. Operational measurements were recorded over high-latitude regions from November 1996 to June 1997. This paper describes the data processing algorithm of Version 5.20 used to retrieve vertical profiles of gases such as ozone, nitric acid, nitrogen dioxide, nitrous oxide, methane, and water vapor from the infrared spectral measurements of ILAS. To simultaneously derive mixing ratios of individual gas species as a function of altitude, the nonlinear least squares method was utilized for spectral fitting, and the onion peeling method was applied to perform vertical profiling. This paper also discusses in detail estimation of errors (internal and external errors) associated with the derived gas profiles and compares the errors with repeatability. The internal error estimated from residuals in spectral fitting was generally larger than the repeatability, which suggests either that some unknown factors have not been incorporated into the forward model for simulating observed transmittance data or that some parameters in the model are inaccurate. The external error was almost comparable in magnitude to the repeatability. Numerical simulations were carried out to investigate performance of the nongaseous correction technique. The results showed that the background level of sulfuric acid aerosols has little effect on the retrieved profiles, while polar stratospheric clouds (PSCs) with extinction coefficients of the order of 10−3 km−1 at a wavelength of 780 nm have nonnegligible effects on the profiles of some gas species. Despite the problems that require further investigations, it is shown that the ILAS Version 5.20 algorithm generates scientifically useful products.
 It is important to clarify the mechanisms and past trends of the ozone depletion caused by anthropogenic compounds, such as chlorofluorocarbons, and to make predictions for the future. Many satellite-borne sensors have therefore been developed and utilized to provide valuable information, including LIMS [Gille and Russell, 1984], TOMS [Heath et al., 1975], SAGE and SAGE-II [Chu and McCormick, 1979; Mauldin et al., 1985], HALOE [Russell et al., 1993], CLAES [Roche et al., 1993], MLS [Barath et al., 1993], ISAMS [Taylor et al., 1993], POAM II [Glaccum et al., 1996], and POAM III [Lucke et al., 1999]. Recognizing the importance of ozone depletion, the Environment Agency of Japan developed the Improved Limb Atmospheric Spectrometer (ILAS), which uses the solar occultation technique, and made ozone layer measurements [Sasano et al., 1999a] from the Advanced Earth Observing Satellite (ADEOS) developed by the National Space Development Agency of Japan (NASDA). The solar occultation technique was also used for SAGE/SAGE-II, HALOE, and POAM II/III. One of its advantages is that changes in the optical characteristics of the instrument do not usually affect measurement quality, because measurement signals are always calibrated by exoatmospheric signals in calculating transmittance. Another advantage is that it has a high signal-to-noise ratio because it uses the Sun as a light source. Despite of some disadvantages of the solar occultation technique compared to the limb emission sounding (e.g., low sampling frequency of two latitude bands per day, and no nighttime coverage), this technique has been regarded as most reliable and used to provide vertical profiles of stratospheric minor constituents.
 The ADEOS satellite was successfully launched from the NASDA Tanegashima Space Center, Japan, on 17 August 1996, and put into a Sun-synchronous subrecurrent orbit with an altitude of 800 km, an inclination angle of 98.6°, and a recurrent period of 41 days. The latitude bands that the ILAS covered are 57–72° in the Northern Hemisphere and 64–89° in the Southern Hemisphere [Sasano et al., 1999a]. The ADEOS satellite orbits the Earth in 101 min, with a descending equator crossing time of 10:41 am. This enabled ILAS to make about 14 occultation measurements in each hemisphere every 24 hours, with a longitude separation of about 25°. Unfortunately, the ADEOS satellite lost its power supply due to a solar panel failure, and consequently ceased operation on 30 June 1997. We successfully obtained data for 6679 ILAS occultation events during its 8 months of operation.
 The ILAS instrument consists mainly of a light-collecting telescope, a Sun-tracking device, a visible spectrometer, an infrared spectrometer, a Sun-edge sensor, and electrical circuits [Suzuki et al., 1995; Nakajima et al., 2002a]. The visible spectrometer covers the wavelength region of 753–784 nm that includes a molecular oxygen absorption band (O2A band), and it can derive vertical profiles of temperature and pressure [Sugita et al., 2001], and aerosol extinction coefficients [Hayashida et al., 2000]. The infrared spectrometer covers the wavelength region of 6.21–11.77 μm (1610–850 cm−1) with a 44-element pyroelectric detector (spectral resolution of 0.13 μm), and this provides vertical profiles of mixing ratios of various gases. The Sun-edge sensor is a linear array sensor to locate the top edge of the Sun, and this information is used to assign a tangent height for each measurement from geometric calculations [see Nakajima et al., 2002b]. The instantaneous fields of view (IFOVs) are rectangular for both spectrometers, and the centers of the IFOVs are controlled to point them toward the center of brightness of the Sun.
 The present paper describes in detail the Version 5.20 data processing algorithm, which incorporates the results of investigations since the release of the Version 3.10 products, and also discusses measurement errors. Primary differences between the Version 3.10 and Version 5.20 algorithms are the tangent height registration method, the smoothing filter, readjustment of instrument functions, consideration of the solar limb-darkening effect, and revision of the climatological data sets used in the retrieval.
 This paper consists of six sections and two appendices. Section 2 briefly describes the measurement principle, instrument outline, and data processing flow. Section 3 deals with preprocessing of infrared signal data. A theoretical forward model for calculating transmittance spectra is introduced in section 4, and a retrieval procedure for unknown parameters from transmittance spectra measured and theoretically calculated is described in section 5. Finally, section 6 discusses the error estimation procedure and error estimates.
Appendix A describes the atmospheric model used in processing the ILAS data. Appendix B supplements the description of the data processing algorithm (Version 4.20) for the aerosol extinction coefficient at 780 nm by Hayashida et al. .
2. Outline of ILAS Measurements
 The ILAS is a sensor that uses the solar occultation technique to derive vertical profiles of various gases and aerosols. This technique employs spectral measurements of the attenuation of sunlight as it passes through the atmosphere to derive the amounts of gases and aerosols responsible for the absorption. Measurements are made of various light paths passing through different atmospheric layers at different heights to obtain information on vertical distributions. To retrieve unknown parameters, spectral fitting of the theoretically calculated transmittance to the measured transmittance as well as vertical inversion techniques are employed.
 Let us denote wave number as ν, the solar spectrum outside the atmosphere as I0(ν), and the spectrum reaching the ILAS after passing through the atmosphere via path length s as I(ν). Then the Beer–Bouguer–Lambert law gives the following relationship:
where κ(ν) [cm2/molecule] is an absorption cross section, and ρ [molecules/cm3] is a number density of a gas. In equation (1), atmospheric emission is negligible and ignored here because of a strong intensity I0(ν).
 The output from the ILAS spectrometers is an integration of equation (1) over a wave number range for each detector element and over a solid angle Ω associated with the instrumental instantaneous field of view (IFOV) with a weight Ψk (ν, ω), where ω is a solid angle, because each element covers a finite wave number and has a finite IFOV. In the wave number region from νkL to νkU, the output signal fk for element number k (k = 1, …, 44, for the infrared channel) is
The weight Ψk (ν, ω) is called the instrument function for element k. Ψk (ν, ω) is an inherent characteristic of the spectrometer that should be precisely determined from theoretical optical simulations based on its design parameters and from laboratory experiments before launch. Cross-talk effects are also added to the output expressed by equation (2), details of which will be given in section 4. In the solar occultation measurements, spectra expressed by equation (2) are detected along the different solar ray paths determined by satellite movement. fk is denoted as Level 0 (L0) data.
 Denoting output corresponding to the exoatmospheric Sun as fk0, the atmospheric transmittance yk along a ray path for element k can be calculated as
yk is the pseudotransmittance for detector element k and is defined as Level 1 (L1) data. The theoretical atmospheric transmittance corresponding to the output from the ILAS is calculated and compared with the pseudotransmittance, to derive gas mixing ratio profiles, which are called Level 2 (L2) data.
 Each occultation event for a sunrise observed from the satellite takes the following data acquisition sequence: the signal from deep space , the signal corresponding to atmospheric measurements fk, the signal from the exoatmospheric Sun fk0, and the signal from deep space again (see Figure 1). The deep space signal is measured twice, and these measurements are used to estimate the 0% level trend with a linear regression. The 0% level trend is almost negligible. On the other hand, the 100% level trend is estimated from the signal change during the exoatmospheric measurement and extrapolated to the atmospheric measurement period. Since the 100% level trend is sometimes not negligible for some infrared elements, this extrapolation is necessary to estimate transmittances during atmospheric measurements. The situation of data acquisition sequence is almost the same for sunset events, but the order of the fk and fk0 measurements is reversed.
 During a measurement, signals are sampled at about 12 Hz, AD converted to 12 bits, and recorded as one unit of data, which is called a major frame, whose length is 5970 Bytes. Each major frame contains data from the infrared spectrometer, data from the visible spectrometer, and data from the Sun-edge sensor, in that order. The small differences in the timing of the data sampling by each detector are taken into consideration when processing the data.
2.2. Instrument Overview
 This section outlines the ILAS instruments, to clarify the data processing flow and its algorithm. Details are given by Nakajima et al. [2002a].
2.2.1. Infrared Spectrometer
 Infrared spectrometer data are used to derive vertical profiles of gases in the atmosphere. To detect gases such as ozone (O3), nitric acid (HNO3), nitrogen dioxide (NO2), nitrous oxide (N2O), methane (CH4), and water vapor (H2O), the spectrometer covers the wavelength range from the NO2 band at 6.2 μm to the HNO3 band at 11 μm. A 44-element pyroelectric detector is attached, covering the range from 6.21 to 11.77 μm (1610–850 cm−1). Output from the detector is passed to a preamplifier and a lock-in amplifier.
Figure 2 provides the atmospheric transmittances for 44 elements at a 20 km tangent height, theoretically calculated using the ILAS forward model (see section 4) with the gas profiles from the ILAS reference climatological data (see Appendix A) for January in the Southern Hemisphere at a latitudinal band of 62.5°S–67.5°S. This shows the absorption as a function of wavelength for different gases. There are nine target gases for retrieval from the infrared channel data in Version 5.20 algorithm. Six of them, O3, HNO3, NO2, N2O, CH4, and H2O are the gases for standard products. The other three, CFC-11, CFC-12, and carbonyl fluoride (COF2) are those for research products. Absorption effects by other minor gases, carbon dioxide (CO2), dinitrogen pentoxide (N2O5), chlorine nitrate (ClONO2), and CFC-14 (CF4) are also considered in the theoretical transmittance calculations. Note that the research products are the tentative retrieval targets, and the possibility of retrieving N2O5 and ClONO2 instead of COF2 is under investigation as an improvement of post Version 5.20 algorithm.
 Since the solar intensity decreases with wavelength in this infrared spectral region, according to Planck's law, the gains of the amplifiers are adjusted to compensate accordingly by dividing the 44 elements into 6 blocks, each having different gain. The signal-to-noise ratio decreases with wavelength, from about 1200 in the shorter wavelength region to about 500 in the longer region.
2.2.2. Infrared Spectrometer Instrument Function
 An almost linear relationship holds between element number and wavelength, and each element covers almost the same wavelength width of 0.13–0.14 μm. The output of each element is affected by cross-talk from adjacent elements; the main cause of this is considered to be optical, that is, scattering of the light generated between the detector surface and the optical filter that covers the detector array. The contribution of cross-talk was estimated to be about 2.25% from laboratory experiments before launch and from analysis of ILAS data from orbit.
2.2.3. Visible Spectrometer
 The visible spectrometer employs a 1024-element MOS (metal-oxide-semiconductor) photodiode array, and covers the wavelength region from 753 to 784 nm with a spectral resolution of 0.15 nm. Molecular oxygen (O2A band) causes absorption in this region, from which temperature and pressure information can be derived. Aerosol extinction coefficients can also be derived from the wavelength region outside the O2A band. Since validation and algorithm studies have not yet been completed for the temperature and pressure products from the ILAS measurements, UKMO stratosphere assimilation data [Swinbank and O'Neill, 1994] was used for ILAS infrared data processing and ray path calculation. The UKMO data was temporally and spatially interpolated to give data at the ILAS measurement points. Details of temperature and pressure from the UKMO data are given in Appendix A. The influence of temperature errors on the retrieved species is analyzed in section 6.
2.2.4. Sun-Tracking Device and Sun-Edge Sensor
 The Sun-tracking device consists of a coarse Sun sensor for solar acquisition, a fine Sun sensor for tracking, and a tracking mirror with two-axis gimbals. On-board software controls the gimbals, using feedback signals from the coarse and fine Sun sensors to point the spectrometer IFOVs toward the center of brightness of the Sun [Nakajima et al., 2002a]. A numerical digital filter is applied to the processing of the estimated Sun-edge location data in order to reduce the effects of Sun-tracking fluctuation with a period of 960 ms.
 The Sun-edge sensor employs the same detector array as the visible spectrometer. It measures the vertical distribution of the Sun's radiance on its surface. From this, the top edge of the Sun is detected as an angular distance from the IFOV direction. From information on the positions of the Sun, the Earth, and the satellite, the tangent height for each measurement is geometrically estimated by ray path calculation, taking into account the effects of atmospheric refraction. Details of tangent height assignment are discussed by Nakajima et al. [2002b].
2.2.5. Instantaneous Field of View and Vertical Resolution
 The instantaneous field of view (IFOV) of the spectrometer was 1′40″ vertically and 13′50″ horizontally for the infrared spectrometer, and 1′40″ vertically and 2′4″ horizontally for the visible spectrometer. This vertical width corresponds to about 1.6 km at the tangent altitude. On the other hand, the angular diameter of the Sun is about 32′ and it corresponds to about 32 km at the tangent altitude and changes seasonally subject to the distance between the Sun and the Earth. The signal was continuously sampled at 12 Hz, but the spatial sampling interval was modulated by atmospheric refraction. A numerical simulation taking into account refraction showed that the spatial sampling interval varied from about 0.11 km at an altitude of 15 km, 0.17 km at 25 km, 0.22 km at 35 km, 0.26 km at 45 km, to 0.27 km at and above 55 km. These values of vertical resolution are almost the same for all measurements because the ADEOS satellite has a Sun-synchronous orbit.
2.3. Data Processing Flow
 The data processing flow is as follows. First, the data required are extracted for further processing; then, outlying data are removed, and finally interpolation, deconvolution, and trend correction are employed. After this preprocessing, which is called L1 processing, the L1 data (pseudotransmittances) are generated, and smoothed to reduce random noise. Ray tracing calculations including refraction effects are performed to obtain ray path geometry and ray path length, measurement location (latitude and longitude), and tangent altitude for each major frame. Details are given in section 3.
 The next step consists of the theoretical calculation of atmospheric transmittance for each detector element, under the conditions of temperature, pressure, and optical ray path given by the process above, as described in section 4. The observed pseudotransmittance and theoretically calculated transmittance are used to retrieve gas profiles called L2 data. Section 5 describes the retrieval in detail.
 Finally, the measures of uncertainty for the L2 data products are determined, including the retrieval errors and predetermined external errors, information on the measurement locations and time, and data quality information. The derivation of statistics from the error analysis is reported in section 6.
3. Infrared Signal Preprocessing
3.1. Data Extraction
 The first step is to extract the data for further processing from the whole data stream of each occultation event. Data to be extracted are the major frames corresponding to deep space (fk,beforeoffset and fk,afteroffset, atmospheric measurements (fk), and exoatmospheric measurements (fk0) (see Figure 1). These data are defined as Level 0a (L0a) data.
3.2. Treatment of Outliers
 A median filter is used to detect spike noise in the L0a data. The difference between the median and the observed data at the midpoint data in the time sequence is calculated by applying the filter to the data of nine major frames. If the difference exceeds a certain critical value, the midpoint data are replaced with the median. The magnitudes of the instrumental noise level in L0a data are about 0.6 counts for the visible data, and about 0.9 counts for the infrared data. As the critical value, an empirically determined value of 20 counts is used for atmospheric measurements and 5 counts for other parts. In fact, a very small fraction of the data, (i.e., less than 0.4% for atmospheric measurements and less than 0.01% for other parts), were treated as spike noises in all of the ILAS measurement data. Level 0b (L0b) data are defined as the data after outlier processing.
3.3. Deconvolution of Infrared Data
 As mentioned earlier, the infrared output passes through a lock-in amplifier and is therefore affected by its time constant. Restoring the original instantaneous signal outputs requires deconvolution processing. The response to the stepwise input of the infrared detection system can be approximated as a third-order delay system using the following equation,
where t is time and t0 is a response delay time so that f(t) = 0 for the 0 ≤ t < t0 case. The time constants t0, T1, T2, and T3 were evaluated for each detector element from laboratory experiments and from the ILAS data from orbit. The average values over the detector elements for each time constant are used for data processing [Nakajima et al., 2002a]. The impulse response function is derived from differentiating equation (4) with respect to time t. For practical use, finite differences of equation (4) are calculated corresponding to 12 Hz data sampling to give an impulse response matrix K. Let us denote the ILAS output as y and the instantaneous signal input as x; then, in a matrix expression, we obtain
 With a smoothness factor α, which is optimally determined as the inverse of the signal-to-noise ratio, the deconvolution is implemented using the following equation [Tikhonov et al., 1995],
3.4. Trend Correction and Pseudotransmittance Calculation
 By expressing the 0% and 100% signal levels at time t and for element k by fkoffset (t) and fk0 (t), respectively, the pseudotransmittance yk(t) can be calculated with the following relation.
 This trend correction reduces the effects of short-term changes in instrument performance. The trend correction is applied to the data after the deconvolution processing to L0b data. yk(t)are defined as L1 data and stored for further use.
 A digital filter was designed to cut out high-frequency noise. The filter is a finite impulse response (FIR) low-pass filter with a 0.5 Hz cutoff frequency. It is a 20th-order inverse Chebyshev low-pass filter [Hamming, 1977], which has a 40 dB noise reduction performance at the first ripple point in the stop band. The half-width of this filter is about 10.1 data points, thus enlarging the field of view by 10.1 times the sampling interval. It is applied to 21 major frame data points of L1 data of the infrared spectrometer as well as the visible spectrometer and the Sun-edge sensor. Total effective vertical resolution, taking into account the size of the IFOV (1.6 km) and the effects of atmospheric refraction, is 1.9 km at 15 km, 2.5 km at 25 km, 3.0 km at 35 km, 3.4 km at 45 km, and 3.5 km at and above 55 km. The filter used previously (until Version 4.20) was a simple moving average for 11 data points (major frames). The autocorrelation analysis of the time series of the ILAS atmospheric measurement data revealed that the Sun tracking, and consequently the infrared and visible signals, fluctuated with an oscillation frequency of about 1.04 Hz. The filter effectively removed these fluctuations.
 The optical ray path calculation needed for retrieval is done by using temperature and pressure profiles from the UKMO data and information from the Sun-edge sensor data [see Nakajima et al., 2002b]. The refraction effects are estimated at every 1 km interval of the ray path using Snell's law, assuming spherically symmetric atmospheric layers. The tangent altitude registration method is a hybrid type with a geometrical registration and a number density matching of O2 molecules, which is described in detail by Nakajima et al. [2002b].
4. Forward Model
4.1. Layering and Optical Depth Calculation
 Methods of calculating the theoretical transmittance to compare with the pseudotransmittance derived from ILAS measurements are described below. Here, the spatial structure of the atmosphere is assumed to depend only on altitude and to be spherically symmetrical. For numerical calculations the atmosphere is divided into layers of finite thickness which are treated discretely and assumed homogeneous.
 Spherical symmetry and homogeneity are basic assumptions applied to solar occultation retrieval. In the ILAS data processing, the atmosphere from 7 to 71 km in altitude is divided into 1 km thick layers. The integral term in equation (1), which is the total optical depth along the ray path, can be approximated as the sum of the optical depths for each homogeneous atmospheric layer in a digitized expression.
Here, j (j = 1, …, J) is an index for the atmospheric layers through which the ray passes; we assign the value 1 for the outermost layer and increases to J (= 64) downward. κj(ν) and ρj are the representative values in the jth layer (cf. equation (19) in section 5). Lj is the ray path length in the jth layer.
 Since the real atmosphere contains multiple gases that may have absorption at the same wave numbers, the contribution from all the gases concerned, the number of which is denoted by g (g = 1, …, G), should be taken into account. The number density ρ(g), j of each gas can be expressed as a function of its volume mixing ratio x(g), j, atmospheric temperature Tj and pressure pj, and Boltzmann constant kB, as
The blackbody emission at the representative Sun surface temperature I0(ν) is taken from the data table of solar spectra in the MODTRAN subroutines [Berk et al., 1989; Anderson et al., 1995], which is based on a calculated solar irradiance. I0(ν) is given with a 1 cm−1 spectral resolution and includes the effects of absorption due to gases in the solar atmosphere.
4.2. Instrument Function of the Infrared Channel
 In Version 5.20, for a fast calculation of the theoretical transmittance, the integration over solid angle ω is replaced by a single ray passing through the geometrical center of the IFOV at the instrument position. Thus the instrument function Ψk(ν, ω) term in equation (2) can be expressed as Ψk(νw) with a digitized wave number νw (w = 1, …, Nw), where Nw is the number of divisions for the wave number region from νkL to νkU. The ILAS output can be expressed as equation (13) using the width of wave number Δν = (νkU − νkL)/Nw,
4.3. Cross-Talk Model of the Infrared Detector Array
 The cross-talk can be modeled as equation (14) with the coefficient c representing the ratio of incoming light from, and outgoing light to, the adjacent elements:
At both edges of the detector array, it is assumed that light only goes out and that none comes in.
where the solar limb-darkening effects described in section 5 are considered for calculating f′k0.
4.5. Cross-Section Calculation
 The absorption cross section k(g), j(ν) is calculated by the following three methods: a line-by-line (LBL) method with line parameters, a LBL method with pseudoline parameters, and cross-section interpolation. From the LBL calculation, k(g) · j(ν) is expressed, by using the line parameters of the absorption molecules and the line shape function, as
where N(g) is the number of absorption lines for gas (g), S(g), n(Tj) is the intensity of the nth absorption line including a molecular vibrational partition function, and h(g),n(ν, pj, Tj) is the line shape as a function of wave number ν, pj and Tj. The term h(g),n(ν, pj, Tj) is called as the Voigt function. The parameters needed to calculate S(g),n(Tj) and h(g),n(ν, pj, Tj) are provided by the HITRAN (high resolution transmission) database, 1996 version [Rothman et al., 1998]. The data used for our calculations from the HITRAN96 line parameters are the spectral line position, line intensity at 296 K, air-broadened half-width at 296 K, lower state energy, and coefficient of temperature dependence of the air-broadened half-width for each gas molecule, including isotopes. This LBL method with line parameters is used for O3, HNO3, N2O, CH4, H2O, NO2, COF2, and CO2. For gas parameters of CFC-11, CFC-12, and N2O5, only cross-section data are stored in the HITRAN96 database. Therefore, the pseudoline parameters (G. C. Toon, private communication, 1995) of these gases are preferred for ILAS operational data processing, enabling k(g),j(ν) to be calculated for any temperature and pressure.
 For ClONO2, and CF4, k(g),j(ν) is calculated from the infrared cross-section data measured at various temperatures provided in the HITRAN database; these are interpolated over wave number for ILAS operational data processing. This calculation uses the data obtained at the temperature closest to that of each atmospheric layer.
 It is computationally inefficient to integrate the Voigt function for each line and sum over all lines whenever an absorption cross-section calculation is required for each temperature and pressure. Therefore, to reduce the computation time, a table (P-T table) of coefficients was prepared in advance for calculating the absorption cross section at any temperature and logarithm of pressure for each gas and each discrete wave number. In routine data processing, two-dimensional spline interpolation is executed using the P-T table for fast computation, which also enables parallel processing of the theoretical transmittance calculation without any loss of calculation precision.
 Since the cross section varies rather smoothly on a P-T surface, but changes drastically on the wave number axis, interpolation is applied to the P-T surface and the wave number interval should be set to a small value. For the theoretical transmittance calculations, cross sections need to be calculated for a 1000 cm−1 wave number range at 0.002 cm−1 resolution, a total of 500,000 points. This resolution may look insufficient for upper atmospheres where the line broadening is small. However, it was confirmed for major species (e.g., O3) that the error in the transmittance of an infrared element caused by this resolution is less than 0.0001, which is less than the ILAS detectable transmittance resolution. To achieve high efficiency, this arithmetic calculation is distributed over 12 node workstations using high performance Fortran for parallel processing [Yokota et al., 1999].
 For LBL calculation of the Voigt function, the subroutine by Hui et al. , which adopts a fifth-order polynomial equation for rapid calculation, is employed to generate the P-T table. To set the ends of line wings to which the cross-section calculation is performed, the method proposed by Yokota et al.  was used for high precision and efficient calculation. No P-T tables were generated for ClONO2 and CF4, because the P-T conditions for the provided cross-sectional data were too small to generate the tables. Instead, cross-sectional data for them are interpolated directly and used in each radiative transfer calculation.
 The continuum absorption of water vapor is calculated with the algorithm CKD 2.1 of Clough et al.  and Thériault et al. . The wing ends for the LBL calculations of H2O are set at ±25 cm−1 from each absorption line center for adding the continuum contribution correctly to the LBL calculation values.
 The continuum absorption around 6.4 μm due to molecular oxygen is calculated using the expression of Thibault et al. . A constant mixing ratio of 0.209 for molecular oxygen is assumed at all altitudes.
 The ILAS retrieval scheme relies on the forward method to iteratively search for the least squares estimate of the unknown theoretical transmittance that gives the optimum coincidence to the pseudotransmittance (y) observed by ILAS. The parameters estimated are the vertical distributions of multiple gas mixing ratios. The onion peeling technique as well as the spectral fitting is utilized to derive the vertical distributions and to simultaneously estimate various gas mixing ratios from the top layer of the atmosphere downward. Details of the vertical inversion technique, spectral fitting, and other items are described below.
5.1. Vertical Inversion by Onion Peeling
 The onion peeling technique assumes that the spatial structure of the atmosphere, i.e., the gas distribution, depends only on altitude. Only the unknown parameters of the target layer need to be retrieved in the onion peeling procedure because the parameters of the layers above can be replaced by estimates retrieved in the previous stage of the procedure. In the ILAS data processing, the atmosphere is divided into layers 1 km thick, and the path length corresponding to each 1 km thick tangent layer is about 220–230 km.
5.2. Spectral Fitting
 The Levenberg–Marquardt technique [Marquardt, 1963] is used for iterative nonlinear optimization of spectral fitting to derive the gas mixing ratios for each layer. In the discussion below, we consider an atmospheric layer with an index m. Let R and G be the number of detector elements (R = 44 in this case) and the number of unknown parameters (G = 9 in this case) in each layer. Using the notation as a G × 1 vector expressing the unknown parameters, ym as the R × 1 observed pseudotransmittance vector, and as the theoretical transmittance for layer m, the residual χ2 of the spectral fitting can be expressed with the weights, Sε−1, given by the inverse of a R × R covariance matrix expressing the measurement errors of the system (see section 6.1), as follows,
where the exponent T means a transpose of a vector or matrix.
 After numerical investigations on convergence stability, Sε−1 is given as a diagonal matrix; the diagonal elements are estimated from the inverse of the signal-to-noise ratio for each detector element of the infrared spectrometer obtained in the exoatmospheric measurements.
 The iteration is stopped when the following two convergence conditions, (1) and (2), are satisfied: (1) the value of χ2 is continuously decreasing, or it becomes smaller than a given value; and (2) the amount of decrease of χ2 becomes smaller than a given value, or the rate of decrease χ2 of becomes smaller than a given value.
 Five or six iterations are enough to attain convergence in typical ILAS data processing. An example of spectral fitting between ILAS observed transmittance data and theoretical one at the altitude of 20 km is shown in Figure 3. The bottom part of Figure 3 shows fitting residuals at the altitude of 20 km for 14 occultation events, which were randomly selected during the ILAS operational period. Note that the scale of the transmittance residuals is enlarged to 100 times that of the transmittance in the upper figure. The residuals suggest some systematic biases in the infrared elements, although they are small.
5.3. Retrieval Parameters, Layer Representative Values, and Initial Profiles
 A nine-dimensional vector, , is simultaneously estimated; this corresponds to mixing ratios of the nine target gases of the ILAS infrared spectrometer data processing: O3, HNO3, NO2, N2O, CH4, H2O, CFC-11, CFC-12, and COF2. The theoretical transmittance calculations consider four other gases that have absorption in the infrared region (CO2, N2O5, ClONO2, and CF4), which cannot be neglected. The ILAS climatological atmosphere model (see Appendix A) is therefore used to provide these gases mixing ratios. Since the contribution of these four gases is small, discrepancies of these gas mixing ratios from those of the real atmosphere will not result in significant errors in the retrieved results.
 The unknowns to be retrieved are the mixing ratios of the gases at the altitudes of the boundaries between the layers, i.e., x(g),m for level m, and x(g),0 = x(g),1 at the outer boundary m = 0. On the other hand, ILAS measurements were not necessarily made along a path exactly at a layer boundary. Therefore, the representative ray path is selected just above, but close to, the boundary of interest (see Figure 4).
 Now, let us define parameters to represent the pressure, temperature, and gas mixing ratio in layer j for ray path m as , respectively, using the following equations:
where the weight coefficients cjL and cjU (cjU = 1 − cjL) are derived by calculating the contribution of each altitude level to the ray path in the layer, assuming linear distributions for the mixing ratio, temperature, logarithm of pressure, and spherical symmetry. The ray path is assumed to be straight, for simplicity. Above the layer of interest, the coefficients are approximated as cjU = cjL = 0.5, because their contribution is much smaller than that of the layer of interest.
 Using the parameters , , and thus obtained for ray path m and layer j, the optical thickness D(g),jm(ν) of ray path m due to gas g in layer j is approximated by
where Ljm is the ray path length for ray path m and layer j. From equation (19) and equations (12), (13), (14), and (15), the transmittance for ray path m can be calculated. The unknown parameters are x(g),m (g = 1, …,9). As the minimum altitude of measurements varies with the measurement conditions for each occultation event, representative ray paths are determined down to the lowest layer (m = M). Retrievals are also made down to layer M.
 To make the magnitudes of the unknown parameters of the same order, they are normalized with their representative values at each altitude as x′(g),m = (x(g),m/x(g),m0) − 1, where x(g),m0 is a representative value. The medians of climatological statistics calculated for each gas, as a function of month and altitude from UARS data provide representative values (see Appendix A). Using x(g),m0 as the initial guess for iteration, the normalized value for the unknown parameter is set to zero. Numerical simulations showed no dependency of the retrieved results on the initial values.
5.4. Limb-Darkening Effect
 Solar radiance decreases from the center to the edge of the Sun (solar limb darkening). On the other hand, although the size of the IFOV is constant, the apparent size of the IFOV relative to the Sun when projected onto the Sun's surface increases as the tangent height of measurements decreases, because refraction compresses the image of the Sun. Therefore, limb-darkening effects must be considered when calculating the theoretical transmittance, especially when the tangent height is lower than about 25 km. The current processing method uses an experimental expression described by Allen  to calculate the limb-darkening factor for a representative wavelength of 9 μm for the infrared. The IFOV expansion is estimated from the refraction calculations in the same way as the ray path calculation.
5.5. Nongaseous Correction Due to Aerosols and Other Bias Factors
 In the infrared wavelength region, aerosols have continuum-type absorption features that overlap the gas absorption spectrum (see Figure 5). As the ILAS infrared spectrometer has a low resolution of the order of 10–40 cm−1, it is not easy to separate the contribution of aerosols from the total transmittance data, but such separation is essential in order to retrieve gas profiles correctly. Aerosol information is also important for studies on ozone depletion and heterogeneous chemistry. In addition, careful investigation has shown that infrared signals are affected not only by aerosol absorption, but also by other continuum contributions (an offset in the absorption cross section), the causes of which have not yet been clearly identified. The contributions from aerosols and offset are hereafter called nongaseous contributions.
 As was done in Version 3.10, a simple technique is applied to correct the nongaseous contribution effects. Even with the low spectral resolution, elements 7, 16, 34, and 43, (counting from 0, Figure 2) suffer only small absorption due to gases. Their central wavelengths are 7.12, 8.27, 10.60, and 11.76 μm, respectively. These four elements are called “window elements.” Since the signal attenuation for these window elements can be regarded as due mainly to nongaseous contributions, climatological data sets for gas profiles are used to derive the remaining small attenuation due to nongaseous contributions as the first approximation, from which transmittance and then extinction coefficient profiles due to the nongaseous contribution at the window elements are estimated. Then, the extinction coefficients estimated at the four window elements are linearly interpolated and extrapolated with wave number throughout the whole range of the infrared channel to obtain the transmittance at other elements. The 44-element transmittance data thus estimated are used to correct for the nongaseous contribution when deriving gas profiles. The nongaseous correction is applied only below 40 km.
6. Error Analysis
 Errors in the retrieved gas profiles are classified as internal errors and external errors. Internal errors are statistically evaluated for each unknown parameter, for each altitude, and for each occultation event, using residuals of the spectral fitting, which are affected by various factors. Possible factors which may contribute to internal errors are uncertainties in the instrumental signals (signal-to-noise ratio), the shape of the instrument function, the coefficients of cross-talk, and other factors in the forward model (numerical scheme, linear interpolation in the nongaseous correction, other minor gases not considered, spectroscopic parameters, and solar spectrum data). Internal errors may contain both systematic and random components.
 External errors are defined as errors that are not estimated directly from the ILAS individual data during the retrieval process but are given externally from numerical simulations in advance. External errors are caused mainly by uncertainties in the input temperature profile for gas retrieval and by discrepancies between the climatological gas profiles used for the nongaseous correction and the gas profiles of the real atmosphere. Because repeating the retrieval processes while changing the temperature and model profiles of the gases in order to estimate the external errors for every routine retrieval would consume too much computation time, numerical simulations and trial estimations for typical events were carried out in advance to the routine processing. Thus, the magnitudes of the external errors were evaluated in common for each gas and for each altitude.
 Since the spectral residuals can be systematically affected by the approximation in the nongaseous correction, it may result in both internal errors and biases in the retrieved profiles, which will be discussed later. Note that impact of the external error sources on spectrum residual may appear partially as the internal errors. Another major source of errors is the uncertainty in determining tangent altitudes [see Nakajima et al., 2002b], which may be related to temperature and pressure errors. This error is not incorporated into the errors in Version 5.20 products.
6.1.1. Internal Error
 Internal errors are calculated from the residuals of equation (17) for spectral fitting using the nonlinear least squares method. An estimated G × G covariance matrix, , of the estimators for the G-dimensional unknown parameter vector, , at the mth atmospheric level, is calculated from the residuals of spectral fitting as
where are the diagonal elements of the Jacobian matrix. This is the noise propagation covariance matrix (KmmTSε−1Kmm)−1 scaled by the ratio of the observed χ2 (equation (17)) to the expected χ2 from noise only (R − G). Then, denoting the elements of equation (20) as , (i, j = 1, …, G), the standard error, σ(g),minternal, for each gas g in the mth layer is given by the gth diagonal element of as
in mixing ratio units.
6.1.2. External Error Due to Temperature Uncertainties
 As already mentioned in section 2.2, temperature and pressure information was obtained from UKMO assimilation data. Since information of the UKMO data is provided only once a day at 12 UT and the spatial resolution is coarse (3.75° in longitude and 2.5° in latitude), data that has been interpolated both temporally and spatially may not be sufficiently representative for the ILAS measurement points. Moreover, some biases in the UKMO stratospheric temperature data have been reported [e.g., Pullen and Jones, 1997]. To investigate these effects on gas profile retrieval results, numerical simulations were done by assuming temperature uncertainties of ±5K at 70 km, ±2K at 20 km, and linear interpolation between these altitudes. Polar summer and winter models were used for the temperature profiles, and the corresponding gas profiles from the ILAS reference atmosphere for the January Antarctic and June Antarctic profiles were employed (see Appendix A). Retrievals for a total of 4 cases (the two biased temperature models × the two (summer/winter) gas profile conditions) were numerically simulated. Denoting the true values as x(g) · m and the retrieved values as , the standard error σ(g),mT for a gas g was estimated by
in mixing ratio units.
6.1.3. External Errors From Gas Profiles Used For Nongaseous Correction
 As described in section 5, nongaseous correction is applied in the gas retrieval, where climatological gas profiles (medians) are used to compensate the gas contribution in the signal. The effects of discrepancies between gas profiles from climatological data and profiles from the real atmosphere were simulated to determine the standard error due to these uncertainties.
 First, we selected 32 samples of ILAS data randomly selected under the conditions of 2 occultation events for each month and each hemisphere, from the data set of the operational 8 months. Using the 10th percentile (x10), median (x50), and 90th percentile (x90) levels of the climatological data set as the retrieval inputs for the selected data, 96 results were obtained. We, then, calculated the median of [max(|x90 − x50|, |x10 − x50|)] for each gas and for each altitude. To reduce fluctuations with altitude, vertical smoothing was applied to the series of medians. The median thus obtained can be regarded as an estimate of the standard error, which gives an external error in question. As a result, we derived the standard error due to nongaseous correction and denoted it as σ(g),mN for a specified altitude level m.
6.1.4. External Error
 Using the expressions σ(g),mT and σ(g),mN, the external error σ(g),mexternal is defined as
in mixing ratio units.
6.1.5. Total Error
 The total error, σ(g),mTotal, which is provided as an item of L2 data, is defined as the root sum square of the internal and external errors, and is given by
in mixing ratio units.
6.2. Statistics From the Error Analysis
Figure 6 shows profiles of relative internal errors in percent, which were calculated from the average of the internal errors divided by the average gas profiles for all the data from November 1996 to June 1997. The same figure also shows profiles of the relative external errors, in which the values of the external errors were divided by the average gas profiles. The errors for the aerosol extinction coefficients at 780 nm are also included in the figure. Since the external errors for this quantity were evaluated for each occultation event, as described in Appendix B, the relative external error was calculated from the average external error for all occultation events.
 As can be seen in the figure, the dominant contribution to the total error is from internal error, except for the ozone and aerosol extinction coefficients at 780 nm. Systematic errors in spectral fitting are reflected in the internal errors, because the internal errors are calculated from the residuals in the spectral fitting. Therefore, the internal errors provided here may not necessarily represent measurement precision; the repeatability of profiles should be investigated empirically.
6.2.1. Measurement Repeatability
 According to BIPM et al. , the term “repeatability of measurement results” is defined as the “closeness of the agreement between the results of successive measurements of the same measurand, which means a particular quantity subject to measurement, carried out under the same conditions of measurement.” Hence, the repeatability of ILAS measurements was approximated empirically using the following procedure.
 Short periods (3–5 days) of quiet and calm atmospheric conditions were chosen, when the gases were likely to be homogeneously distributed along a latitudinal band, to evaluate measurement repeatability from the retrieved gas profiles. For selected periods in June for the Northern Hemisphere and in the latter half of March for the Southern Hemisphere, the variability of gas profiles from the average and its standard deviation σrep was calculated. Defining the relative standard deviation as , the smallest value for each altitude was selected and plotted (as a percentage) for each gas in Figure 6. All the values of εrep, except those for ozone and aerosol extinctions, were much smaller than the internal error. Since εrep includes the contribution from natural variability to some extent, this gives an upper limit of measurement repeatability. Comparisons among relative percentages of internal errors, external errors, and repeatability standard deviations for each species at 15, 20, and 30 km are shown in Table 1. Also listed in Table 1 are the averages of relative differences between the ILAS product and the validation data for each species. These averages of relative differences reflect the ILAS retrieval accuracy (i.e., systematic error).
Table 1. Summary of Internal Error, External Error, Measurement Repeatability, and Accuracy for the ILAS Standard Gas Products and Aerosol Extinction Coefficient at 780 nm
6.3. Discussion on Errors Due to Nongaseous Correction
 The linear interpolation approximation used in the nongaseous correction may cause systematic errors in gas profiles. For example, if the linearly interpolated extinction spectra of the sulfuric acid aerosols (used as background aerosols) or polar stratospheric clouds (PSCs) differ significantly from the true ones, it very likely results in systematic residuals in the spectral fitting. When large discrepancies overlap gas absorption wavelengths, the derived gas profiles have large errors. To evaluate possible systematic errors due to this, theoretical simulations were conducted using climatological gas profiles (see Appendix A), and using the theoretical extinction spectra for sulfuric acid aerosols with two different weight percentages of sulfuric acid as background aerosols and several types of PSCs.
Biermann et al.  gave the refractive indices for sulfuric acid aerosols (H2SO4/H2O binary solutions). The assumed weight percentages of sulfuric acid are 50 wt% at 200 K and 75 wt% at 230 K. The refractive indices for the supercooled ternary solution (STS) are also from the work of Biermann et al. . Four combinations of weight percentages for sulfuric acid and nitric acid are assumed here: 5 and 37 wt%, 33 and 15 wt%, 47 and 3 wt%, and 60 and 0.5 wt%, respectively. They correspond to temperatures of 188 K, 192 K, 194 K, and 200 K, respectively, under the assumption of 10 ppbv nitric acid and 5 ppmv water vapor [Carslaw et al., 1995]. The refractive indices for nitric acid trihydrate (NAT) and water ice (ICE) are taken from the work of Toon et al. . The parameters for the size distributions assumed for this calculation are shown in Table 2. Figure 5 depicts the extinction coefficient spectra calculated for various aerosol/PSC types. They are normalized to the extinction coefficient at a wavelength of 780 nm.
Table 2. Parameters of the Aerosol and PSCs Size Distribution for the Nongaseous Correction Simulation (rg: Mode Radius, and sg: Standard Deviation)
 As a background condition, the extinction coefficients at 780 nm for sulfuric acid aerosols are assumed to be 1 × 10−7 km−1 at 30 km, 1 × 10−4 km−1 at 20 km, and 1 × 10−3 km−1 at 10 km, with linear interpolation on a logarithmic axis between these altitudes. In addition, this profile is multiplied by factors of 2 to 6 to further simulate different conditions. For PSC conditions, the extinction profiles are assumed to consist of background aerosols and PSCs. For the background aerosols, the extinction coefficients are given as 1 × 10−7 km−1 at 40 km and 1 × 10−4 km−1 at 10 km, and linearly interpolated between these values on a logarithmic scale. A PSC layer with a Gaussian shape, a peak at 15 km, and a width of 3 km is added to this background aerosol layer. The extinction coefficient at the peak is such that the total extinction coefficient is 0.5, 1.0, or 2.0 × 10−3 km−1, generating three different conditions. Other PSC layers with peaks at 20 and 25 km were also simulated with total extinction coefficients of 0.5, 1.0, and 2.0 × 10−3 km−1. In total, nine cases were simulated with gas profiles corresponding to February and June in the Northern Hemisphere. Respective sets of temperature and pressure profiles were arbitrarily chosen from the UKMO data for February and June, in the Northern Hemisphere.
 The optical thickness due to gases is first calculated as a function of wave number from the data sets described above. The total transmittance spectrum due to gases and aerosols/PSCs is then evaluated as a function of wave number, which is converted to the expected 44-element ILAS output according to the procedure described in section 4 (forward model). Applying the operational data processing to this simulated data, including the nongaseous correction, the profiles of gases are retrieved. Figure 7 shows the differences in gas number density (not mixing ratio) between profiles derived in this manner and those initially assumed (i.e., regarded as true) as a function of the extinction coefficient at 780 nm. The slopes given by linear regression analyses are listed in Table 3. The differences (biases) in the mixing ratio at an altitude of 20 km, assuming typical numbers of atmospheric molecules, are also shown in Table 3, where we set the extinction coefficient value at 780 nm at an altitude of 20 km as 5.0 × 10−4 km−1 for sulfuric acid aerosols, and as 1.0 × 10−3 km−1 for PSCs.
Table 3. Estimates of Systematic Errors Remaining After the Nongaseous Correction for the ILAS Standard Gas Products
O3 (at an altitude of 20 km)
NO2 (at an altitude of 20 km)
HNO3 (at an altitude of 20 km)
The regression coefficient a is a proportional factor that relates the bias error of the number density of a gas to the 780 nm aerosol extinction coefficient, as shown in Figure 7. Typical values for bias errors in concentrations (cm−1) and volume mixing ratios (ppmv) at a tangent altitude of 20 km are also listed for cases of the extinction coefficient value at 780 nm as 5.0 × 10−4 km−1 for sulfuric acid aerosols and as 1.0 × 10−3 km−1 for PSCs. The component ratios are as follows: STS(a): 5 wt% H2SO4/37 wt% HNO3/H2O, STS(b): 33 wt% H2SO4/15 wt% HNO3/H2O, STS(c): 47 wt% H2SO4/3 wt% HNO3/H2O, and STS(d): 60 wt% H2SO4/0.5 wt% HNO3/H2O. S(50) and S(75) are sulfuric acid aerosols whose components are 50 wt% H2SO4/H2O and 75 wt% H2SO4/H2O, respectively.
−8.95E + 13
1.87E + 11
−4.48E + 10
9.88E + 10
1.40E + 09
4.94E + 07
3.36E + 11
1.65E + 09
1.68E + 08
2.23E + 14
1.47E + 12
1.12E + 11
1.53E + 11
6.97E + 08
7.65E + 07
1.12E + 11
2.32E + 09
5.61E + 07
1.50E + 15
4.37E + 13
1.50E + 12
−2.94E + 11
7.62E + 09
−2.94E + 08
2.14E + 11
2.35E + 10
2.14E + 08
−2.66E + 14
1.04E + 13
−2.66E + 11
−3.18E + 11
2.98E + 10
−3.18E + 08
−1.91E + 11
2.71E + 10
−1.91E + 08
2.54E + 12
1.42E + 12
2.54E + 09
−4.44E + 11
8.82E + 09
−4.44E + 08
−2.74E + 10
9.01E + 09
−2.74E + 07
2.67E + 14
1.13E + 13
2.67E + 11
3.70E + 11
1.12E + 10
3.70E + 08
9.07E + 10
1.57E + 09
9.07E + 07
2.89E + 14
1.19E + 13
2.89E + 11
5.29E + 11
1.51E + 10
5.29E + 08
1.90E + 11
7.24E + 09
1.90E + 08
2.40E + 14
8.77E + 12
2.40E + 11
3.25E + 11
9.11E + 09
3.25E + 08
2.28E + 11
7.23E + 09
2.28E + 08
N2O (at an altitude of 20 km)
CH4 (at an altitude of 20 km)
H2O (at an altitude of 20 km)
8.91E + 12
1.04E + 12
4.45E + 09
−3.04E + 14
5.00E + 12
−1.52E + 11
−1.54E + 14
1.74E + 12
−7.71E + 10
−3.67E + 12
3.72E + 11
−1.84E + 09
−1.88E + 14
3.79E + 12
−9.41E + 10
−1.25E + 14
1.57E + 12
−6.23E + 10
−6.73E + 11
1.07E + 12
−6.73E + 08
−1.10E + 13
6.35E + 12
−1.10E + 10
−3.56E + 14
6.73E + 12
−3.56E + 11
−8.75E + 13
4.38E + 12
−8.75E + 10
1.62E + 15
6.26E + 13
1.62E + 12
−2.89E + 15
3.96E + 13
−2.89E + 12
−1.55E + 13
1.49E + 12
−1.55E + 10
3.10E + 14
8.12E + 12
3.10E + 11
−1.15E + 15
3.67E + 13
−1.15E + 12
−8.23E + 11
5.38E + 11
−8.23E + 08
−1.20E + 14
5.39E + 12
−1.20E + 11
1.07E + 13
9.02E + 11
1.07E + 10
−1.74E + 12
6.43E + 11
−1.74E + 09
−2.22E + 14
9.05E + 12
−2.22E + 11
1.13E + 14
4.49E + 12
1.13E + 11
−5.69E + 12
8.13E + 11
−5.69E + 09
−1.44E + 14
3.33E + 12
−1.44E + 11
−1.22E + 14
1.44E + 12
−1.22E + 11
 The error analysis based on the simulation makes several important points. First, the nongaseous correction has little effect on any gas profiles for the background sulfuric acid aerosol cases. On the other hand, for a PSC layer of water ice with an extinction coefficient of 10−3 km−1 at 20 km, the retrieved ozone has a large artifact of a 0.9 ppmv overestimation. For a NAT PSC layer with extinction coefficients of the order of 10−3 km−1, gases other than ozone and nitric acid suffer from biases that are not negligible. This means that the type of PSC has both qualitative and quantitative effects on the retrieved gas profiles. Note that the effects of aerosols/PSCs due to the nongaseous correction are, in terms of gas number density, almost linearly related to the 780 nm extinction coefficient. Possible systematic errors in the retrieved gas profiles caused by the nongaseous correction suggest that great care be given when interpreting those profiles.
7. Summary and Concluding Remarks
 In this paper we have described details of the Version 5.20 processing algorithm used to derive vertical profiles of gas mixing ratio from data from the ILAS, which uses the solar occultation technique. The algorithm is based on the nonlinear least squares method. For vertical profiling, the onion peeling method is applied. Introducing parallel processing and a table look-up method for the absorption cross-section calculation (P-T table) results in rapid and accurate computation.
 Error estimation procedures for the product have been discussed in detail. The data products distributed to the public include an estimate of the total error generated by combining the internal and external errors. Internal errors are estimated from ILAS data for every occultation event, while external errors are separately estimated for typical cases and used in common. The internal errors are evaluated from the residuals in spectral fitting, so they may be overestimated in the presence of sources that produce bias errors in the simulated spectrum. In fact, when the internal errors were compared with the repeatability of product profiles estimated for a calm period in summer, the repeatability was much smaller than the internal error. This implies that the forward model may miss some important factors, or that some parameters in the model might not be correct. The external errors consist mainly of errors caused by temperature uncertainties and of differences between climatological gas profiles and the real atmosphere.
 Numerical simulations were used to show the influence of nongaseous correction on the bias errors in the derived profiles for several types of PSCs and aerosols. The results indicate that the effects of background sulfuric acid aerosols are small, and that PSCs with extinction coefficients of the order of 10−3 km−1 at a wavelength of 780 nm can significantly affect some of the derived gas profiles; the effects can be positive or negative, depending on the gas and PSC type. The magnitude of artifacts depends on the PSC type, and is proportional to the magnitude of the extinction coefficients at 780 nm in their realistic range. Care should be taken when interpreting products in the presence of PSCs with large extinction coefficients.
 We calculated the ILAS reference atmosphere data, which has been used to generate climatological gas profiles for nongaseous correction, by using published data mainly from the instruments MLS [Barath et al., 1993] and CLAES [Roche et al., 1993] onboard the UARS. This climatological data set was also utilized for the initial conditions in retrieving gas profiles. The gas species and the instruments that provided the climatological data are listed in Table A1. MLS and CLAES data obtained from observations made between January and December 1992 were used to generate climatological profiles for each gas species, for every month, on every 5° latitude band, and for altitudes from 7 to 71 km at 1 km intervals. ATMOS provided all the data measured during the ATLAS-3 mission period in 1994. On the basis of this, the CF4 mixing ratio was evaluated to have a constant value of 60 ppbv at all altitudes. A mixing ratio of 350 ppmv was used for CO2. The mixing ratio for COF2 was inferred from the MkIV experiment conducted on 8 May 1997 as part of the ILAS validation experiment (G. C. Toon, private communication, 1999).
Table A1. Data Sources for the ILAS Reference Atmosphere Model
Data source version of each gas is shown in the parentheses.
For the altitude range below the lowest level of MLS (>12 km), the ILAS Version 3.10 H2O profiles were used above 12 km and values were extrapolated using the average slope of SAGE-II H2O profiles for the range from 12 km down to 5 km.
 The temperature and pressure were the stratospheric assimilation data (3.75° longitude, 2.5° latitude, 22 isobaric surface levels between 1000 and 0.3162 hPa corresponding to the UARS standard pressure levels) provided by the UKMO [Swinbank and O'Neill, 1994] on a daily basis (12 UT). Spatial and temporal interpolations of the temperature and pressure profiles were made to every ILAS measurement point and time represented by the tangent altitude of 20 km. Above the UKMO data's highest altitude level, temperature and pressure data from the CIRA 1986 model [Rees et al., 1990] (for every month, every 5 km in altitude, and every 10° in latitude) were used by smoothly connecting this data with the UKMO data.
Appendix B:: A Brief Description of the Algorithm Used for Computing Aerosol Extinction Coefficients at 780 nm
 The algorithm (Version 4.20) used for deriving profiles of the aerosol extinction coefficients at 780 nm, σaero(780), from the visible spectrometer was described by Hayashida et al. . For readers' convenience, the algorithm used for Version 5.20 is described briefly here in comparison to that for Version 4.20.
 Values of σaero(780) are deduced at tangent altitudes with 1 km interval, using the visible spectrometer data with a 1024-element detector array. Figure B1 schematically shows a pseudotransmittance curve based on the data from the visible spectrometer at 20 km tangent altitude. As seen in the figure, the transmittance spectrum consists of the absorption spectrum due to oxygen molecules in the center and an undulating baseline component over the whole wavelength range. The baseline has contributions from aerosol extinction (Mie scattering), extinction by air molecules (Rayleigh scattering), and absorption by ozone (Wulf band). A σaero(780) value is derived from an average of 12 transmittance data around 780 nm y(780), as indicated by the gray bar in the figure. Detector elements that contain the solar Fraunhofer lines are excluded from averaging.
 The baseline shape in the transmittance spectrum is inferred from spectral data excluding the region of oxygen absorption by linear fitting of the following model function to the pseudotransmittances as a function of wavelength λ:
 ξ(λ) is an absorption cross-section spectrum of ozone. Here, laboratory data provided by Brion et al.  are used for ξ(λ), while the MODTRAN subroutine was used until Version 4.20. The term d ξ(λ) corresponds to the contribution from the ozone Wulf band. Eliminating the contribution of ozone leaves a vertical profile of the transmittance due only to aerosol (Mie) extinction and molecular (Rayleigh) scattering along the ray path. Inversion of this transmittance profile results in a profile of the total (Rayleigh + Mie) extinction coefficient per unit path length. Correction for solar limb-darkening effects is made at every altitude. Finally, by subtracting the Rayleigh component per unit length from the total extinction coefficient, the aerosol extinction coefficient σaero(780) is obtained. The Rayleigh contribution is theoretically calculated [Fröhlich and Shaw, 1980] using the UKMO temperature and pressure data.
 It should be noted that a digital filter is also applied to the visible spectrometer signals. Accordingly, the total effective vertical resolution, taking into account the size of the IFOV (1.6 km) and atmospheric refraction effects, is 1.9 km at 15 km, 2.5 km at 25 km, 3.0 km at 35 km, 3.4 km at 45 km, and 3.5 km at and above 55 km. These values are the same as for the gas retrieval.
 Major error sources associated with the derived aerosol extinction coefficients include (1) measurement errors due to instrument characteristics and performance, (2) errors in the absorption contribution by ozone, (3) errors in estimating the Rayleigh scattering component using the UKMO data, (4) errors in determining tangent altitude, (5) uncertainties in calculating light path length, and (6) errors in estimating the solar limb-darkening effects. As for the measurement errors (1), error propagation in interpolation and extrapolation at the 0 and 100% levels is taken into account, as well as the data smoothing effect of the digital filter. The error term (2) is also considered in Version 5.20. An internal error provided in the Version 5.20 data products is defined as the square root of the sum of squares of terms (1) and (2).
 The errors in the Rayleigh scattering component (3) come from uncertainties in the temperature and pressure data that are interpolated spatially and temporally for the ILAS measurement point using the UKMO data. Numerical simulations were used to estimate this term. Assuming that the uncertainties in temperature were ±2K at 10 km and ±5K at 70 km, the variation in Rayleigh scattering (error term (3)) was calculated for each measurement event, and defined as an external error. The total error reflected in the data products, is defined as the square root of the sum of squares of the internal and external errors. As in Version 4.20, other error factors (4)–(6) were not considered in Version 5.20 either.
 The authors would like to thank Pat McCormick, C. Camy-Peyret, and Andrew Matthews for their continuous support and encouragement to the ILAS project team in algorithm development, validation experiments, and scientific analyses. The authors greatly thank the suggestion of R. Blatherwick regarding the O2 continuum near 6 μm and the support of P. Varanasi by continuously supplying the latest information on cross-section data for CFCs and other minor gases. The authors also thank G. C. Toon for the pseudoline parameters, S. A. Clough for information on the H2O continuum calculation (CKD2.1), L. S. Rothman and other HITRAN development members for information on the line parameter database, G. Anderson and her colleagues for information on FASCODE and MODTRAN, and R. Swinbank for the UKMO assimilation data of temperature and pressure. We are also grateful to Fujitsu F.I.P. Co. for their contribution in processing the ILAS data at the ILAS Data Handling Facility (ILAS-DHF) of the National Institute for Environmental Studies. The ILAS project has been sponsored by the Ministry of the Environment, the former Environment Agency of Japan.