1.2. Mathematical Formulation of the MOPITT CO Retrieval Algorithm
 A nonlinear optimal estimation algorithm [Rodgers, 2000; Pan et al., 1998] and a fast radiative transfer model [Edwards et al., 1999] are used to invert the measured A- and D-signals to determine the tropospheric trace gas concentrations. Retrievals of CO may involve up to six channels (12 measured signals) in two distinct bands: two channels in a solar reflectance band near 2.3 microns (channels 2 and 6), and four channels in a thermal emission band near 4.7 microns. As described in Table 1, the thermal band channels include two LMC channels (channels 1 and 5) and two PMC channels (channels 3 and 7). The eight thermal band signals are sensitive to thermal emission from the Earth's surface as well as atmospheric absorption and emission. The solar band signals are sensitive to atmospheric CO through absorption processes only. It is implicitly assumed in the following that only clear-sky radiances (i.e., radiances uncontaminated by clouds) are fed to the retrieval algorithm. A detailed description of the MOPITT cloud-detection algorithm has recently been published [Warner et al., 2001].
Table 1. Thermal-Channel Modulator Characterististicsa
|Channel||Modulator Type||Cell Pressure, mbar||Enabled Signals|
 The general concepts underlying the MOPITT CO retrieval algorithm have been discussed in detail previously [Pan et al., 1998]. Recently, however, significant changes have been made to the MOPITT CO retrieval algorithm in order to improve the quality of the retrieval products and make them more useful to potential data users. In addition, after MOPITT became operational, the retrieval algorithm was reconfigured in response to observed noise and bias characteristics of the operational (in-orbit) MOPITT radiances. The purpose of this paper is to describe the current operational retrieval algorithm used to process the MOPITT radiance data in detail, describe the MOPITT CO retrieval products (with reference to the retrieval averaging kernels), and finally present some selected actual retrieval results which demonstrate the capabilities of this new tool for global atmospheric chemistry studies.
 In atmospheric remote sensing, the common problem of inverting a set of measured radiances to determine aspects of the atmospheric state (temperature profile, trace gas mixing ratio profiles, etc.) is often ill-conditioned, meaning that no unique solution exists. Thus additional information of some type is usually required to constrain the retrieval to fall within physically reasonable limits. The CO retrieval algorithm used for MOPITT exploits the maximum a posteriori (“MAP”) solution which is a specific type of optimal estimation technique [Rodgers, 2000]. The general strategy of such techniques is to seek the solution most statistically consistent with both the measured radiances and the typical observed patterns of CO vertical profiles as represented by the a priori. The methodology for generating the a priori (i.e., both the a priori mean profile and the a priori covariance matrix) is described in detail in section 2.
 The equation relating the true atmospheric state and the measured radiances can be written as
where y is the measurement vector (i.e., the observed MOPITT radiances), x is the state vector (i.e., all the desired retrieved variables), b represents all other forward model parameters (i.e., all parameters needed to calculate the MOPITT radiances not explicitly included in the state vector), F(x, b) represents the forward radiative transfer model [Edwards et al., 1999], and Nϵ is the radiance error vector. The goal of the retrieval algorithm is to estimate the true state vector x from the measured radiances y and the associated measurement errors.
 In the MOPITT CO retrieval algorithm, the measurement vector y is formed solely from the calibrated satellite radiances (also called the “Level 1 Product”). For the thermal band signals, SiA and SiD are used to represent the A and D signals for the ith radiometer and are included in the measurement vector y directly. A different strategy is employed for the solar band channels. For these channels, the ratio SD/SA is employed because it greatly reduces the effect of the generally unknown and highly variable surface reflectivity. This ratio is still a valid indicator for CO, however, because of the much greater sensitivity to CO exhibited by the solar D channel than by the solar A channel. Forward model studies also indicate that the contaminating effects of gas species other than CO are reduced in the ratio SiR = SiD/SiA [Pan et al., 1995]. For these reasons, the solar band channels are represented in the measurement vector only through the solar ratio signal SiR.
 The thermal band signals depend not only on the atmospheric CO distribution but also on various other atmospheric quantities (such as the atmospheric temperature and water vapor mixing ratio profiles) and surface parameters (surface temperature Tsfc and longwave emissivity ϵsfc). Accurate values for all of these geophysical parameters must be obtained to produce accurate retrievals. Atmospheric temperature and water vapor profiles are obtained by spatially and temporally interpolating reanalysis profiles from NCEP to the location and time of each MOPITT pixel. However, sources of geophysical data such as NCEP are unable to provide accurate values of surface temperature and emissivity (both of which are highly variable) at the temporal and spatial resolution demanded by the MOPITT retrievals. Fortunately, information contained in the MOPITT thermal band signals allows retrieval of the surface temperature and emissivity along with the CO profile, and makes external data sources for these quantities necessary only for providing a priori and initial guess values.
 Thus, rather than assuming fixed values for Tsfc and ϵsfc, both parameters are included in the retrieval state vector x along with the elements of the CO profile. (A detailed inspection of the radiative roles of Tsfc and ϵsfc reveals that their effects on the thermal-band signals are often nearly indistinguishable. Therefore MOPITT radiances do not always contain sufficient information to retrieve both parameters independently. Both parameters are included in the retrieval state vector because (1) they represent physically different sources of radiance variability and (2) assuming fixed values for either parameter would unnecessarily constrain the CO retrieval.)
 Using these notations, the most general measurement and state vectors for a given pixel can be written, respectively, as
where qi represents the CO mixing ratio at the ith pressure level of the predefined retrieval grid. The seven levels in the current operational retrieval grid include the surface, 850, 700, 500, 350, 250, and 150 mbar.
 The MAP solution combines two independent estimates of the same vector quantity (i.e., the state vector determined solely from the measurement vector y and the “virtual” measurement represented by the a priori state vector xa) inversely weighted by their respective covariances. The MAP solution for this problem is written
where Ca is the a priori covariance matrix (described below), K is the weighting function matrix defined by
which thus describes the model-calculated sensitivity of the each of the measurement vector elements to each of the elements of the state vector (described further below), KT is its transpose, and Cϵ is the radiance error covariance matrix. Cϵ can be used to represent errors from sources including (but not limited to) instrumental noise, forward model error, and ancillary error (e.g., errors in the assumed temperature and water vapor profiles) [Rodgers, 2000]. The corresponding covariance for the MAP solution is
and describes the uncertainty in the retrieved state vector .
 However, equation (4) for the maximum a posteriori solution cannot be used directly to retrieve the CO profile because the weighting function matrix K is itself a function of . Rather, an iterative form of the MAP solution is needed. The method of Newtonian iteration is therefore used, in which
where n is the order of iteration, and Fn is the theoretical radiance vector based on n as calculated by the forward model. The initial guess state vector 0 need not equal the a priori state vector xa. For example, during processing of a swath of MOPITT radiances, the initial guess CO profile for a particular pixel might be efficiently generated from the retrieval of a nearby, previously processed pixel. (In current operational processing, however, CO initial guess profiles are provided by monthly-mean output from the chemical transport model known as “Model for OZone And Related Tracers,” or MOZART [Hauglustaine et al., 1998].) In any case, the a priori state vector xa always represents the “best guess” CO profile and surface parameters excluding any analysis of the MOPITT radiances. The calculation and implementation of the a priori is discussed in section 2.3 below. After each iteration of the maximum a posteriori solution, the solution is checked for convergence. Currently, the convergence test is based on the fractional change in the CO profile relative to the previous iteration. By definition, convergence occurs when the root-mean-square (RMS) value of the fractional change in the seven levels of the CO profile decreases to 5% or less. Retrievals typically converge in three or four iterations. Increasing the number of retrieval iterations (i.e., decreasing the convergence threshold) produces a negligible effect on the retrieval results.
 Retrievals of the CO profile consist of a “floating” surface-level retrieval (tied to the pixel-dependent surface pressure value) and retrievals at up to six fixed pressure levels including 850, 700, 500, 350, 250, and 150 mbar. (In elevated areas where one or more of the fixed pressure level values exceed the actual local surface pressure, that part of the retrieved state vector is filled with the missing-value identifier.) As demonstrated in section 3, the retrieval grid resolution is typically finer than the vertical resolution of the actual retrieved CO profile. We expect that many users of MOPITT CO retrievals will attempt to compare MOPITT CO profiles with products (either from chemical transport models or from other observations) with higher intrinsic vertical resolution. The pressure grid on which the retrieval results are reported was selected to facilitate these comparisons by minimizing errors caused by interpolating or extrapolating MOPITT results. Moreover, future schemes for processing MOPITT radiances will likely incorporate more signals and exhibit finer vertical resolution than the current scheme. The current retrieval grid will be capable of accommodating the additional information contained in these retrievals.
 The MOPITT CO “Level 2 Product” consists of retrieved values and estimated uncertainties of the CO profile, CO total column, surface temperature, and surface emissivity. For the CO profile, the retrieved error covariance matrix is also provided. Although this covariance matrix may be useful in and of itself, it is also a necessary element of averaging kernel calculations (as described below).