1. Introduction

The upcoming Global Precipitation Measurement (GPM) mission will incorporate a constellation of existing operational and dedicated spaceborne microwave radiometers with the goal of studying global precipitation, a key component of the earth’s hydrological cycle. To achieve consistent global precipitation products among a number of sensors, a rainfall algorithm is needed to parametrically accommodate any passive microwave sensor in orbit. As improvements in both the number of channels, field-of-view resolution, and calibration of spaceborne radiometers have been made, algorithm development thus far has largely been focused on the latest sensor (Shin and Kummerow 2003). As such, efforts to produce consistent, homogenous geophysical products for long time series are impeded due to sensor-specific assumptions associated with sensor-specific algorithms. It is necessary, therefore, that algorithms be developed so that the issues of inconsistency in the geophysical parameter products from a diverse range of spaceborne sensors can be addressed.

Shin and Kummerow (2003) first made progress on this front by constructing a parametric rainfall retrieval algorithm within a Bayesian framework used in conjunction with a coupled Tropical Rainfall Measuring Mission (TRMM) precipitation radar (PR) and cloud- resolving model (CRM) a priori database. They retrieved the most likely hydrometeor profile with the assurance that simulated brightness temperatures corresponding to the retrieved profile agreed with observed (synthetic) brightness temperatures. Masunaga and Kummerow (2005) made further progress by allowing for variation in the drop size distribution (DSD) and ice density assumed by a TRMM PR algorithm to achieve the best agreement between simulated and observed brightness temperatures. However, brightness temperature discrepancies remain and uncertainties still exist.

Kummerow et al. (2006) quantified a number of these uncertainties using a simple rainfall algorithm. In a physical rainfall retrieval algorithm, rainfall estimation errors arise not only from assumptions in the a priori database, but also from errors in the nonraining scene. These errors are introduced from uncertainty in the nonraining parameters [column-integrated water vapor (TPW), surface wind (WIND), and integrated cloud liquid water (LWP)], as well as from errors arising from changes in rain–no-rain thresholds (Kummerow et al. 2006). Since a three-dimensional rainfall structure consistent with all brightness temperatures is desired, there must be assurance that the nonraining scene is also consistent with brightness temperatures, independent of the microwave sensor being utilized, since both errors in the retrieved precipitation profile and errors in the nonraining scene contribute to the total brightness temperature error. Just as there is a need for a parametric rainfall algorithm for GPM, so there is a need for a parametric nonraining algorithm.

The parametric algorithms being designed for GPM must also provide a full description of the errors in the retrieval. One such uncertainty arises during the rain–no-rain discrimination process. One needs to know the contribution to the brightness temperature from the nonprecipitating component of the cloud in order to assess the contribution from the precipitating component. The ability to detect rain within the radiometer-only retrieval framework, while most problematic over land, remains a significant issue over the ocean. Ferraro et al. (1998) discussed a number of discrimination methods involving brightness temperature thresholds, while acknowledging the difficulty of finding an algorithm that works equally well globally. Berg et al. (2006) noted that a liquid water path threshold is typically used in microwave radiometry for rainfall detection. They showed that this can lead to rainfall retrieval discrepancies in regions where nonraining, high LWP clouds may exist because of possible aerosol interaction delaying the onset of precipitation, such as over the East China Sea. There is, then, the desire to have an algorithm that has both the ability to assess the uncertainty in the nonraining parameters and the capability of discriminating raining clouds from nonraining clouds.

The objective of this study is not to develop a new nonraining retrieval, but instead to develop a framework that facilitates the merger of parametric background retrievals with rainfall retrievals allowing for a coherent atmospheric description to emerge within the GPM framework consistent with all observed brightness temperatures. To this end, the framework used for retrieving the nonraining parameters must not be dependent on a particular channel or empirical adjustments that would be unique to an algorithm optimized for a particular sensor. Instead, it should be designed such that it is portable to any microwave window channel radiometer. The physical algorithm is developed within the optimal estimation framework. The benefits of optimal estimation can be summarized as providing fully parametric retrievals necessary for GPM, associated error estimates, and retrieval diagnostics. It is important to note that while a unified algorithm for raining and nonraining scenes within an optimal estimation framework is desirable, separate algorithms are required as the use of an optimal estimation framework for global rainfall retrieval is currently not feasible, largely because of drastically increased computational expense.

This paper describes the implementation and results of a parametric, nonraining retrieval scheme for a number of currently orbiting microwave sensors (sections 3 and 4). In section 5, it is shown that there is a consistent response of the retrieval diagnostics to precipitation that can be used in the development of a new dynamic rainfall detection methodology for use in both the rain–no-rain determination process in passive microwave rainfall retrieval as well as the rainfall-screening process necessary for cloud property retrieval. Furthermore, at the end of the retrieval process, the brightness temperature discrepancies can be used to assess forward model–sensor calibration errors (also discussed in section 5).

2. Data

b. Ancillary data

Because of constraints in the information content available in radiance channels, not all parameters to which the forward model is sensitive can be retrieved. It is a common practice to retrieve those parameters that are of greatest interest and fix others that are needed for the forward model to simulate upwelling radiances.

For this study, parameters that are fixed include temperature lapse rates and vertical distributions of water vapor (through the use of a water vapor scale height). Temperature lapse rates and water vapor scale heights are generated using model output from the European Centre for Medium-Range Weather Forecasts (ECMWF) 40-yr reanalysis (ERA-40; additional details in Uppala et al. 2005). The temperature lapse rate is representative of the average temperature lapse rate from the surface to approximately 300 hPa, while the water vapor scale height is computed by fitting a function to the ERA-40 water vapor profiles assuming an exponential form given by

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where WV_Z is the ERA-40 water vapor magnitude at height Z, WV₀ is the water vapor magnitude at the surface, and SCLHT is the derived water vapor scale height.

At each satellite pixel location, the surface atmospheric temperature is assumed to be equal to the underlying sea surface temperature (SST). Weekly average SSTs from an optimum interpolation (OI) technique (Reynolds and Smith 1994; Reynolds et al. 2002) are produced by the National Oceanic and Atmospheric Administration (NOAA) and are used for the surface atmospheric temperature–ocean skin temperature. Additionally, daily AMSR-E retrieved Remote Sensing Systems (RSS; products can be found online at www.remss.com) SSTs are used to assess errors that arise from using weekly OI SSTs as opposed to daily retrieved SSTs, the details of which are described in section 5.

Near-surface rain rates from the TRMM PR 2A25 algorithm (Iguchi et al. 2000; algorithm hereinafter referred to as 2A25) are used in this study for evaluation of the rainfall identification methodology. The TRMM PR has a swath width of 215 km and a spatial resolution of 4.3 km at nadir. Additional details on TRMM PR can be found in Kummerow et al. (1998).

3. Algorithm development

Within the context of the general inversion problem, the relationship between the physical properties of the atmosphere and the measured radiometric quantities can be generalized with the following expression:

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where y is the measurements vector (such as radiances arriving at the spaceborne sensor); f(x, b) is the forward model describing the physics of the measurement system and radiative transfer through the atmosphere; x is the state vector containing the properties of the atmosphere (TPW, WIND, and LWP) to be estimated; b is the set of parameters not included in the state vector, but assumed to be known in the model atmosphere (sea surface and atmospheric temperatures, scale height for water vapor, height of cloud in atmosphere, absorption spectral line strengths and widths, etc.); and ε is an all-encompassing error term containing uncertainties due to the nature of the measuring instrument, errors in the forward model (f), and uncertainties in the forward model parameter (b) assumptions. Given y, the retrieval problem is an exercise in inverting Eq. (2) so that the state x may be inferred. The success of the inversion depends both on how well the model physics contained within f(x, b) represent nature and on the ability to properly characterize the errors associated with the instrument and model assumptions.

b. Inversion

In this retrieval problem, one must be able to invert f(x, b) (Eq. 2) in order to retrieve the most likely background state x given a set of observed microwave brightness temperatures y. To provide a reasonable error estimate associated with the retrieved state, a suitable treatment of both forward model errors and experimental errors is required in designing a retrieval framework. One can consider the treatment of errors within a probability density function (PDF) framework, whereby the true value of a parameter can be described by a PDF with an associated mean and variance σ². A distribution of this type, completely described by Gaussian statistics, is a representation typically chosen when information describing the shape of the PDF is limited primarily to the mean and variance of the parameter of interest.

In this study, the inversion methodology is based on the optimal estimation technique (Rodgers 1976; Rodgers 1990; Marks and Rodgers 1993; Rodgers 2000), an approach that allows one to map the PDF of the observation to the PDF of the state x. Since measurement noise may overwhelm the signal and lead to instability in the retrieval, one can further constrain the solution by incorporating prior information about the state x (i.e., a priori state PDF) within the optimal estimation framework. The a priori state is assumed to follow Gaussian statistics. This serves the purpose of making the retrieval problem more linear and, as a result, increases the speed of convergence in the algorithm (Rodgers 2000). The utility of the optimal estimation technique is dependent on the validity of the assumption that the state vector parameters follow Gaussian error statistics. It has been noted in the literature that the distributions of the state vector parameters (and their variability statistics) are likely either non-Gaussian [e.g., WIND PDF; see Monahan (2006a,b)] or lognormal [TPW and LWP PDFs; see Foster et al. (2006) and Liu and Curry (1993)]. For parameters that are physically restricted to positive values and have variance that is comparable to or greater than their mean value, Rodgers (2000) notes that the lognormal distribution (a type of Gaussian PDF) of the quantity may be more appropriate (i.e., transforming the state vector parameter such that it is the logarithm of the geophysical parameter of interest). A study of the sensitivity of the solution to a change in the assumed a priori state PDF (not provided; additional discussion on sensitivities in section 3c) has shown that the retrieved LWP PDF is most affected (up to a 10% change in the mean and variance of the retrieved global LWP compared to less than a 1% difference for TPW and WIND) when we incorporate the assumption of a lognormal PDF versus a simple-Gaussian PDF. Considering both this result and the lognormal argument of Rodgers (2000), we make the logarithm transformation (i.e., assume lognormal statistics) for the LWP parameter only in this study.

We proceed with the formulation of the optimal estimation framework through use of Bayes’s theorem. Within this probability framework, one is able to formally calculate the posterior PDF (the retrieved state and the associated error covariance) by updating the prior PDF with information from the measurement PDF. The conditional probability P(x|y) of a state x given a set of measurements y is given by

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where P(y|x) is the conditional probability of y given a state x, P(x) is the prior PDF of the state vector x, and P(y) is the prior PDF of the measurement vector. The optimal solution is the state vector that maximizes the conditional probability P(x|y) for a particular y. Considering P(y), a normalizing term independent of the state, the quantity P(y|x)P(x) is maximized when

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has reached a maximum. The first exponential term in Eq. (4) contains information on the Gaussian PDF of y given state x with the width of the distribution governed by 𝗦_y, the error covariance matrix representing the sum of both the measurement and model error covariances. Currently, only diagonal elements of 𝗦_y are considered, thus neglecting any possible error covariances. However, it is possible that both measurement error among the microwave channels and model error may be correlated (examples of correlated/systematic measurement error are given in section 5b). The second exponential term in Eq. (4) represents the PDF of the a priori state, the variance of which is described by 𝗦_a. Prior knowledge of the mean and variance of the state can typically be derived from climatology. Equation (4) is maximized when

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is minimized. Thus, a cost function Φ has been defined. The maximum probability solution of x is solved when the gradient of Φ with respect to x has reached a minimum. Linearizing the forward model about some base state x_i, the solution can be found in an iterative manner using Newton’s method (Rodgers 2000), given by

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where 𝗞 is the Jacobian or weighting function matrix representing the sensitivity of the linearized forward model to a change in the state parameter. The state x_i+1 is considered the optimal solution when the difference in states (x_i+1 − x_i) at successive iterations is an order of magnitude smaller than the estimated error variance of the state at the current iteration, defined by

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where n is equal to the number of parameters being retrieved and 𝗦_x is the retrieval error covariance, the details of which are described at greater length in the following section.

c. Optimal estimation diagnostics

1) Uncertainty estimation

The posterior PDF of the retrieved state is characterized by the most likely value for the state as well as an estimate of the uncertainty in the retrieved state, defined by Rodgers (2000) as

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The diagonal elements of 𝗦_x provide the error variance associated with each retrieved parameter while the off-diagonal elements provide information on correlations between the errors in the retrieved parameters. In this study, we will be focusing on the error variances associated with the state vector parameters.

From Eq. (8), one can see that the estimated error variance arises from two sources. First, uncertainties arise from the forward model assumptions and sensor noise (square root of the diagonal values of 𝗦_y). Uncertainties in brightness temperature arising from uncertainties in the forward model parameters can be found in Table 1. Contributions to 𝗦_y due to sensor noise are generally less than 1 K. Therefore, forward model parameter error is generally larger than instrument error. Information regarding specific magnitudes of sensor noise for each of the channels can be found in Kummerow et al. (1998) for TMI, Hollinger et al. (1990) and Colton and Poe (1999) for SSM/I, and Kawanishi et al. (2003) for AMSR-E. The uncertainty in the a priori state (square root of the diagonal values of 𝗦_a) derived from 1 yr of independent RSS AMSR-E retrievals amounts to roughly 50% of the a priori state values. While the channels used are largely sensitive to changes in the state vector over the range of parameter values seen in this study (Prigent et al. 1997), the sensitivity of the measurements to changes in the parameters is a function of the atmospheric regime. For example, the sensitivities decrease in regions of high TPW, high LWP, and low WIND (as seen in Prigent et al. 1997). Therefore, while we can consider that the a priori state serves as a general weak constraint on the retrieval (considering the amount of error on the a priori state in relation to the measurement/model error) with most of the information coming from observed brightness temperatures, this will not always be the case as the relative strength of the a priori state will increase as the sensitivities of the measurements to changes in the state space decrease.

In the Bayesian optimal estimation framework employing Gaussian statistics, it is an implicit assumption that all errors in 𝗦_y are of random nature. While radiometric noise is considered to be random (examples of systematic error can be found in section 5), model parameter error may be systematic. A number of forward model parameters were perturbed by an amount equivalent to σ_source in order to assess the effects of the parameters on brightness temperatures over a range of channels utilized in this study. The resultant change in brightness temperature is given by σ_TB and can be seen in Table 1.

Regarding sea surface temperatures, not only do algorithm specifics contribute to SST error, but also spatial and temporal averages contribute to the total parameter error. A first-order description of the error is derived by computing the SST standard deviation over a particular 7-day period within 1° × 1° grid boxes spanning the global oceans, using the following equation applied to each grid box over the globe for each consecutive week of a 12-month period:

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where σ^i,k is the standard deviation in grid box i over week k, N is the number of days in a week with valid daily RSS AMSR-E SST retrievals, SST^i,j is the daily RSS AMSR-E SST retrieval averaged over grid box i for day j of the week, and OISST^i,k is the 1° × 1° OI SST for grid box i and week k. Then, the global average SST standard deviation σ_SST is computed by averaging σ^i,k over all weeks and all 1° × 1° grid boxes, giving the contribution to the allotted forward model parameter error that arises from using an SST value that is both spatially and temporally averaged.

The errors in the computed temperature lapse rates (σ_LR) and water vapor scale heights (σ_SCLHT) as derived from ERA-40 were computed by finding the standard deviation of each parameter within each grid box over a 12-month period, given by

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where σ^k is the standard deviation of the ERA-40 derived parameter over 1yr in grid box k of the ERA-40 dataset, N are the number of days used in the computation, X^i,k is the temperature lapse rate or water vapor scale height computed daily for every ERA-40 grid box k, and is the annual mean temperature lapse rate or scale height for a particular grid box. The error source (σ_source) for both parameters is computed by averaging σ^k over all grid boxes. Thus, for the atmospheric lapse rate and water vapor scale height, σ_source is representative of the global and annual average standard deviation of the atmospheric lapse rates and scale heights and is used to describe the width of the Gaussian error distribution for these parameters.

For the placement of the cloud layer in the forward model, a standard deviation of 1 km was chosen for σ_CLDHGT. The temperature at which the cloud layer absorbs and emits changes as the altitude of the cloud layer is increased or decreased. Thus, the change in radiance (σ_TB) emanating from the cloud is due to an assumed change in the radiating temperature of the cloud layer. The value for σ_CLDHGT is difficult to estimate and is chosen somewhat arbitrarily. However, it is important to note that in this study, we found minimal impact on the forward model computed brightness temperatures (approximately less than a 0.20-K brightness temperature change per 1 km change in cloud height for the channels most sensitive to changes in liquid water content). Therefore, while we may not be appropriately estimating the uncertainty in cloud height, the sensitivities of the brightness temperatures to changes in the cloud height (seen in Table 1) are small enough overall that the assumption has a minimal impact on the retrieval.

As can be seen in Table 1, the sources having the largest impact on the simulated brightness temperatures include potential errors in sea surface temperatures and vertical distribution functions for the atmospheric temperature and water vapor. While it is not a goal of this study to reconcile residual brightness temperature discrepancies, it is important to note the effects that the model parameters have on computed brightness temperatures in addition to general model error and sensor noise as one begins to investigate the residual brightness temperature biases remaining at the end of the inversion process. A systematic bias in any of the parameters would lead to a channel-dependent bias in the simulated brightness temperatures.

2) Chi-square diagnostic

A standard chi-square (χ²) test is used to determine whether the retrieved state is valid. After a solution is found, if the forward computed brightness temperatures agree with the observations within the expected error variance range (𝗦_y), then the measurements term [second term of Eq. (5)] should roughly follow a χ² distribution with the degrees of freedom approximately equal to the number of channels in the observations vector. The degree to which forward model radiances [f(x, b)] for the solution match the observations vector (y) within the expected error variance range (𝗦_y) is related to how well the forward model assumptions and their assumed Gaussian error variances agree with the atmospheric scene being observed. If the scene under observation is not well represented by the forward model physics, then χ² will consistently be larger than expected. Overall then, χ² can be thought of as an indicator of general agreement between simulated brightness temperatures and sensor observations.

4. Geophysical parameter retrieval results

The developed parametric retrieval algorithm has been tested using AMSR-E, SSM/I, and TMI. For the TPW and WIND comparison, data from SSM/I and TMI from year 2000 are used. Additionally, the retrieval was also applied to data from January to August 2002 for TMI to note any differences resulting from the TRMM satellite boost in August 2001. Since there is only a 3-month overlap between AMSR-E and ERA-40 data availability, we are restricted to using data from June to August 2002 for an AMSR-E TPW and WIND comparison. For the comparison of AMSR-E LWP with MODIS LWP, we use the months of July–August 2002.

The retrieved TPW is matched to raob TPW if a satellite pixel overpass is within 0.50° of the raob location and within 1 h of the raob reported time. For the surface wind comparison, pixels are matched to the buoy location if the satellite pixel overpass is within 0.25° of the buoy location and within 5 min of the buoy-reported time. To have enough samples for a meaningful statistical comparison of retrieved versus in situ TPW, the requirements for matching retrieved TPW to raob TPW are much less stringent than they are for surface wind retrieval matchups. For the retrieved LWP from AMSR-E, data are averaged over the same 1° × 1° grid boxes used in the MODIS level-3 product. Since the operational retrieval for MODIS is applicable during daytime only, while the AMSR-E microwave retrieval can be done during day or night, we average AMSR-E LWPs over the 1° × 1° grid only if the computed solar zenith angle for the AMSR-E pixel is less than or equal to 80°. We have found that changing the solar zenith angle threshold requirements (from 80° to 60°) leads to only minor differences in the comparison between the two products. Additionally, since the microwave LWP is the true average LWP over the entire grid box (including clear sky), we rescale the optical estimate using the cloud fraction (also provided in the MODIS level-3 product) so that the average 1° × 1° optical estimate used in the comparison now includes clear sky.

A scatterplot of the retrieved TPW against radiosonde-observed (raob) TPW is shown in Fig. 1 for all sensors used in this study. ERA-40 TPW plotted against raob TPW is shown for additional comparison. The results for the surface wind speed retrieval are shown in Fig. 2. The black error bars (square root of the diagonal values of 𝗦_x) show the one-sigma standard deviation errors on the retrieved TPW. As was discussed, optimal estimation provides an error for each element of the retrieved state vector. For plotting purposes, data were binned, and the mean retrieval error was plotted for each bin containing more than 40 pixel matchups (5-mm bin size for TPW; 1 m s⁻¹ bin size for WIND).

From a qualitative inspection of Fig. 1, one can see that the scatter within the retrieved TPW dataset contributing to the total root-mean square (RMS) error far exceeds the optimal estimation retrieval error. This is not the case with the surface wind retrieval, as seen in Fig. 2. One possible reason for this is related to the coincident matchup requirements. As was mentioned, since the requirements for matching retrieved TPW to raob TPW are much less restrictive than they are for surface wind retrieval matchups, and because this is not accounted for in the calculation of 𝗦_x, this could lead to additional RMS error when comparing the TPW retrieval with in situ raob TPW.

Table 2 gives a breakdown of the statistics (mean bias, unadjusted RMS error) corresponding to the TPW and WIND retrievals for all five sensors used in this study. The number of in situ collocations for each sensor matchup is also provided in Table 2. As can be seen, for both TPW and WIND, biases (retrieved − observed) of differing magnitudes were found for each sensor.

For TPW, biases range from −0.04 mm (for TMI preboost) to +2.78 mm (SSM/I F15). Additionally, RMS errors range from 2.99 mm (AMSR-E) to 5.93 mm (SSM/I F13). ERA-40 TPW is in good agreement with raob TPW with a mean bias of −1.35 mm and an RMS error of 3.61 mm. It should be noted that ERA-40 assimilates these raobs and, thus, the good agreement is not very surprising. The retrieval error statistics are in reasonable comparison to other microwave algorithms available. Jackson and Stephens (1995) performed an algorithm intercomparison for TPW retrievals using SSM/I F08 and found RMS errors (relative to raobs) ranging from 4.66 mm [Alishouse et al. (1990) statistical algorithm] to 5.08 mm [Greenwald et al. (1995) physical algorithm], with biases in some algorithms approaching 1 mm. Vertical lines have been added to the TMI preboost and postboost panels to highlight that some of the discrepancies that exist when visually comparing the two periods arise from the fact that the very low TPW regimes are not sampled during the preboost period and the very high TPW regimes are not sampled during the postboost period. This is partly the cause for differences in the mean TPW bias between the two periods. Because raob overpass locations are not the same between the two periods, different regimes may be sampled and model bias may be making itself known. Aside from these issues, the cause of the additional discrepancy between the two periods is not known at this time.

For WIND, biases range from −0.64 m s⁻¹ (for TMI postboost) to +1.73 m s⁻¹ (SSM/I F14). RMS errors range from 1.40 m s⁻¹ (TMI postboost) to 1.87 m s⁻¹ (SSM/I F13 and F14). Additionally, ERA-40 compares reasonably well to the in situ data with a mean wind speed bias of −0.62 m s⁻¹ and RMS error of 1.62 m s⁻¹. The retrieval errors are also comparable to the 1.52 m s⁻¹ RMS error for a retrieval of TMI surface wind speed (compared to buoy) found by Connor and Chang (2000). Furthermore, Chang and Li (1998) used the operational Goodberlet and Swift (1992) surface wind speed retrieval algorithm for SSM/I F11 and found an RMS error of approximately 2.3 m s⁻¹. Quantification of the magnitudes and directions of the wind speed biases were not available in either study.

The LWP retrieval results for SSM/I and TMI are shown in the top panel of Fig. 3. For these sensors, the comparison is limited to both ERA-40 and an independent microwave retrieval algorithm used by RSS. The ERA-40 LWP global dataset is a 2.5° × 2.5° daily gridded product. In this study, since the LWP retrieval is computed at pixel resolution, the pixel-level LWP values are averaged to the same 2.5° × 2.5° resolution for more appropriate comparison. RSS LWP retrievals are treated in the same manner. The relative frequency (y axis) of a particular LWP is derived by computing the number of LWP occurrences in an LWP bin of arbitrary size and dividing by the total number of occurrences spanning all LWP bins. The maximum relative frequency occurs in the 0.00–0.05 kg m⁻² LWP bin for all products (ranging from 0.38 for ECMWF to 0.52 for RSS). There are few occurrences of LWP beyond the 0.20 kg m⁻² bin for each product. In general, the distribution of ERA-40 LWP is more significant at larger LWP values than are the others.

With regard to the microwave products, aside from differences in algorithms leading to systematic differences in results, there are also nonalgorithm effects (e.g., differences in resolution of TMI versus the SSM/I and the resultant cloud liquid water beam-filling issues, as well as potential systematic effects on the results due to diurnal sampling of TMI relative to SSM/I), all contributing to the net difference in results.

The comparison of the AMSR-E LWP retrieval with the optical LWP estimate is shown in the bottom panel of Fig. 3. Data were binned and the mean retrieval and one-sigma retrieval error (black error bars) were plotted for each bin containing more than 40 pixel matchups (approximately a 0.03–0.04 kg m⁻² bin size). Contours represent relative data densities and are chosen at arbitrary intervals. Optical estimates for LWP can be derived using cloud optical thickness and cloud droplet effective radius (Bennartz 2007; Horvath and Davies 2007; Lin and Rossow 1994). These two parameters can be retrieved from simultaneous solar reflectance measurements at a nonabsorbing visible and a water-absorbing near-infrared channel (Nakajima and King, 1990), both of which are available from the MODIS instrument. The errors associated with the optical technique can be found in King et al. (1997).

The microwave and optical techniques represent different, independent approaches for retrieving LWP from space. The mean LWPs from AMSR-E and MODIS are both approximately 0.04 kg m⁻², with the microwave retrieval slightly overestimating (+0.002 kg m⁻²) the optical retrieval in a mean sense. It would appear that the AMSR-E mean LWP should be noticeably lower than the optical mean LWP. However, we point out that the bias in LWP is a function of location in LWP space (i.e., microwave technique overestimates at the lowest LWPs while the optical technique overestimates beyond the 0.02 kg m⁻² LWP threshold). The competing effects between the high bias at the low end where much more data are available and the low bias at higher LWPs (data much more sparse) contribute to the two means being approximately the same.

Lin and Rossow (1994) also found that in an average sense, microwave and optical estimates of LWP agree well (both having a mean of 0.05 kg m⁻²) for overcast, warm, nonprecipitating clouds in calm sea environments, generally in agreement with our findings. Bennartz (2007), in studying boundary layer clouds in the South Pacific Ocean off the South American coast also found that while the average LWP is in agreement between the two estimates, the microwave algorithm consistently overestimates at low LWP values and underestimates at higher LWP values. The low sensitivity of the microwave technique to very low values of LWP makes it difficult for the microwave retrieval to distinguish between clear and cloudy scenes (Lin and Rossow 1994), adding to the error in the microwave retrieval. Horvath and Davies (2007) note that the MODIS cloud mask may have missed a number of the shallow cumulus cloud fields, therefore causing an underestimation in the subdomain cloud fraction, in turn resulting in an underestimation of the average LWP over the domain (and possibly explaining some of the high bias at the low end of the LWP spectrum). Bennartz (2007) notes that the reason for the low bias at higher LWPs is unclear. However, since the field of view for AMSR-E is much larger than the field of view for MODIS, beam-filling effects due to the nonlinear relationship between LWP and brightness temperature in conjunction with the larger AMSR-E footprint and likely inhomogeneities in the cloud field over the larger AMSR-E footprint are likely to lead to an underestimation in the microwave LWP (Bennartz 2007) when comparing the two products. Additionally, rain contamination can be an issue as well (Bennartz 2007), since the absorption/emission signal for lightly raining clouds (no ice) is similar to that of nonprecipitating, cloud water-only systems.

5. Additional applications

6. Conclusions

In preparation for the upcoming launch of the GPM mission, a parametric, nonraining physical retrieval applicable to a number of orbiting spaceborne microwave window channel sensors was developed. The development of the retrieval within the optimal estimation framework ensures that the forward model computed brightness temperatures corresponding to the retrieved geophysical parameters will be in agreement with observed brightness temperatures for nonprecipitating scenes regardless of the sensor being used, making this algorithm flexible enough so that future spaceborne microwave sensors can be accommodated. Because this physical retrieval does not require sensor-specific or scene-specific adjustment parameters other than channel frequencies/polarizations and incidence angles, the addition of radiative-transfer physics in precipitating scenes would facilitate the merging of cloud and precipitation retrievals, if so desired.

The retrieved TPW and WIND were compared with in situ measurements from radiosondes and buoys. RMS errors for TPW ranged from 2.99 to 5.15 mm. For surface wind, RMS errors ranged from 1.40 to 1.87 m s⁻¹. These were shown to be in agreement with the calculated RMS errors from other widely used nonprecipitating algorithms as well. Model TPW and WIND from ERA-40 also agree reasonably well with in situ results and are comparable in skill to the microwave retrievals. Additionally, LWP retrievals were compared to LWP from Remote Sensing Systems, MODIS, and ERA-40. There are noted differences between the microwave and optical retrieval products, as well as between the microwave retrievals and ERA-40, providing continued motivation for future studies of cloud property retrievals.

The retrieval χ² diagnostic responds in a consistent manner to atmospheric scenes that are not well described by the forward model assumptions, such as in the case of raining scenes. The diagnostic provides information on the scene being observed and indicates whether or not simulated brightness temperatures agree with observations. Because of the consistent, expected response to scenes of this nature, the ability exists to filter out regions of rainfall that would contaminate cloud property retrievals. Additionally, because the information contained within the optimal estimation diagnostics potentially provides a more powerful tool for identifying raining regions within the passive microwave algorithm framework as opposed to using a straight liquid water path threshold, a new capability for rainfall detection emerges that can be used in passive microwave rainfall retrieval. However, the algorithm is not yet mature enough to warrant the use of a χ² value that works equally well globally in identifying rainfall. Understanding the regional variations in the threshold as well as assessing the role of inhomogeneous scenes in increasing the χ² value is a subject of future research.

A retrieval framework of this nature allows one to assess and quantify potential calibration issues related to either the sensor or the forward model. However, since this approach does not solely address the source of the brightness temperature biases, additional exploration is required. As a first step in assessing the forward model assumptions, the change in brightness temperature biases resulting from a perturbation in the fixed assumptions (water vapor scale height and lapse rate, for example) of the forward model was investigated. Since the model brightness temperatures are sensitive to these two parameters, future work would include a rigorous error analysis to determine if lapse rates are reasonable and if the water vapor profiles can be described reasonably well through the use of a generalized scale height function. Additionally, an investigation of the information content of the available radiances could shed light on the possibility of retrieving one or both of these parameters.

Any potential impacts directly attributed to sensor calibration issues or other external influences affecting the sensor including variations in incidence angle have not been discussed. Sensor-specific issues would need to be addressed in future work in conjunction with forward model improvements so that a limit in retrieval performance would not be reached from a forward model perspective. The impact of differences in radiometer FOVs, both a function of sensor and diffraction limited, has not been assessed. While it can be assumed that TPW and WIND are relatively uniform over the largest sensor FOV, this may not be the case for cloud liquid water. Along with the nonlinear response of brightness temperatures to cloud liquid water, the impact of different radiometer FOVs on cloud liquid water retrievals would need to be explored in future studies. Additionally, since the error analysis in this study dealt solely with the variances of the state vector parameters, future work should entail a more detailed error characterization including the possible existence of correlations among the errors associated with the retrieved parameters, as well as their possible regime dependence.