Abstract

Two methods of computing the time-mean divergence and vorticity from satellite vector winds in rain-free (RF) and all-weather (AW) conditions are investigated. Consequences of removing rain-contaminated winds on the mean divergence and vorticity depend strongly on the order in which the time-average and spatial derivative operations are applied. Taking derivatives first and averages second (DFAS_RF) incorporates only those RF winds measured at the same time into the spatial derivatives. While preferable mathematically, this produces mean fields biased relative to their AW counterparts because of the exclusion of convergence and cyclonic vorticity often associated with rain. Conversely, taking averages first and derivatives second (AFDS_RF) incorporates all RF winds into the time-mean spatial derivatives, even those not measured coincidentally. While questionable, the AFDS_RF divergence and vorticity surprisingly appears qualitatively consistent with the AW means, despite using only RF winds. The analysis addresses whether the AFDS_RF method accurately estimates the AW mean divergence and vorticity.

Model simulations indicate that the critical distinction between these two methods is the inclusion of typically convergent and cyclonic winds bordering rain patches in the AFDS_RF method. While this additional information removes some of the sampling bias in the DFAS_RF method, it is shown that the AFDS_RF method nonetheless provides only marginal estimates of the mean AW divergence and vorticity given sufficient time averaging and spatial smoothing. Use of the AFDS_RF method is thus not recommended.

1. Introduction and motivation

Scatterometer vector winds provide practically the only observational means to estimate the dynamically important quantities of time-mean surface divergence and vorticity and wind stress curl and divergence over the ocean. Investigators are rarely explicit about the precise order of calculations leading to these estimates, however, and we show here that the order can have a profound impact on the end result.

In this study, we highlight the widely used surface vector wind dataset measured by the SeaWinds scatterometer on the QuikSCAT satellite. Rain degrades QuikSCAT wind measurement accuracy (e.g., Huddleston and Stiles 2000; Portabella and Stoffelen 2001; Stiles and Yueh 2002; Draper and Long 2004; Weissman et al. 2002, 2012; Fore et al. 2014). The commonly used QuikSCAT wind datasets [such as those produced by Remote Sensing Systems (RSS) and the Jet Propulsion Laboratory (JPL), which are described later] contain rain flags marking the time and location of rain events, and wind measurements coincident with these rain flags are often discarded prior to further manipulation of the winds. Excluding wind estimates in rain, while improving the overall accuracy of the wind dataset, significantly influences the time-mean divergence and vorticity fields. As we show here, the specific consequences of rain flagging on the mean divergence and vorticity depend strongly on the order in which the time-averaging and spatial differentiation operations are applied. In the case of uniform sampling in space and time, time-averaging and spatial differentiation are interchangeable, since both are linear operations. QuikSCAT winds, however, have nonuniform sampling as a result of flagging of rain-contaminated wind measurements. Additionally, all satellite wind measurements have nonuniform spatial sampling because of their limited swath width.

In the first order of operations, spatial derivatives are first computed from instantaneous vector wind measurements and then time averaged (e.g., at monthly, annual, or decadal periods). We refer to this method as derivatives first, averages second (DFAS). We also use tags to indicate whether this method has been applied to rain-free (RF) or all-weather1 (AW) winds. If any one of the four required wind measurements is rain flagged, the divergence and curl calculations are not executed. When applied to rain-free winds, the DFAS_RF method would appear to be the correct way to estimate the time-mean divergence or vorticity, but it discards even high-quality wind measurements if they are on the edge of a rain event.

The second order of operations involves time averaging the vector wind components first, then calculating the spatial derivatives of these time-mean zonal and meridional wind components. We refer to this method as time averages first, spatial derivatives second (AFDS). In contrast to the DFAS_RF method, the AFDS_RF method uses all rain-free wind measurements in the spatial wind derivative computation. This method thus allows wind measurements unmatched in time at neighboring grid points into the time-mean divergence and vorticity estimates. At first, it is difficult to see how this could produce valid estimates of the time-mean divergence and vorticity, but we show that the AFDS_RF method produces mean divergence and vorticity fields that surprisingly appear qualitatively consistent with those expected based on our knowledge of the large-scale atmospheric circulation.

To motivate this analysis, the time-mean divergence and vorticity fields computed using each of the two methods are shown in Fig. 1 from rain-free QuikSCAT winds for the 10-yr period November 1999–October 2009. Major differences between the two methods are readily apparent. For instance, the DFAS_RF divergence and vorticity fields are much more divergent and anticyclonic than the AFDS_RF fields throughout much of the global oceans, as seen from comparing Figs. 1a and 1b for divergence and Figs. 1c and 1d for vorticity. This is especially apparent in the midlatitude storm tracks and the tropical oceans.

Fig. 1.

Time-averaged divergence and vorticity computed from rain-free QuikSCAT winds over the 10-yr period November 1999–October 2009 (10−5 s−1): (a) DFAS_RF divergence, (b) AFDS_RF divergence, (c) DFAS_RF vorticity, and (d) AFDS_RF vorticity. The QuikSCAT winds used here are from the JPL v3 QuikSCAT dataset (Fore et al. 2014) gridded onto a 0.25° spatial grid.

Fig. 1.

Time-averaged divergence and vorticity computed from rain-free QuikSCAT winds over the 10-yr period November 1999–October 2009 (10−5 s−1): (a) DFAS_RF divergence, (b) AFDS_RF divergence, (c) DFAS_RF vorticity, and (d) AFDS_RF vorticity. The QuikSCAT winds used here are from the JPL v3 QuikSCAT dataset (Fore et al. 2014) gridded onto a 0.25° spatial grid.

The AFDS_RF method is appealing because it appears to reproduce large-scale features of the AW divergence and vorticity over most of the ocean, at least qualitatively. We demonstrate this by comparing the QuikSCAT AFDS_RF divergence and vorticity with the all-weather divergence and vorticity from the National Centers for Environmental Prediction (NCEP) 1°, 6-h analyses for the 3-yr period 2010–12 (comparing Figs. 1b,d and 2b,d).The QuikSCAT AFDS_RF divergence and vorticity fields qualitatively account for the sign and approximate position of most of the main large-scale features of the surface atmospheric circulation over the ocean compared to the NCEP all-weather divergence and vorticity. The apparent qualitative resemblance between the divergence and vorticity from the QuikSCAT AFDS_RF fields and the NCEP all-weather fields is puzzling and nonintuitive, as winds in raining conditions were not included in the QuikSCAT calculation. Why this is the case and whether we can really trust these AFDS_RF estimates are open questions, which we attempt to answer.

Fig. 2.

As in Fig. 1, but for NCEP divergence and vorticity computed using (a),(b) the NCEP rain-free winds and the DFAS_RF method and using the (c),(d) all-weather winds averaged for the 3-yr period January 2010–December 2012. The NCEP 1°, 6-h analyses were used.

Fig. 2.

As in Fig. 1, but for NCEP divergence and vorticity computed using (a),(b) the NCEP rain-free winds and the DFAS_RF method and using the (c),(d) all-weather winds averaged for the 3-yr period January 2010–December 2012. The NCEP 1°, 6-h analyses were used.

The DFAS_RF method, conversely, produces biases in the mean fields that are easy to understand dynamically. For instance, Milliff et al. (2004) first pointed out that, since rain occurs typically in convergent and cyclonic winds, failing to measure winds during rain naturally biases the mean divergence and vorticity toward the divergent and anticyclonic. These conditional sampling biases are also demonstrated in the NCEP DFAS_RF fields shown in Figs. 2a,c. The QuikSCAT DFAS_RF divergence and vorticity are highly correlated with these fields (comparing Figs. 1a,c and 2a,c). The conditional sampling bias of Milliff et al. (2004) can thus be replicated in the independent NCEP fields, even though the NCEP analysis does not resolve typical scales of convective rain events.

We provide a specific example where the differences between the two methods suggest fundamentally different relationships between atmospheric circulation and rainfall. The 2-yr averaged QuikSCAT DFAS_AW divergence is shown in the northwest Atlantic in Fig. 3a using the all-weather JPL QuikSCAT winds. The DFAS_AW divergence shows a band of convergence on the seaward edge of the Gulf Stream SST front. The DFAS_RF divergence in Fig. 3c, however, does not have this convergence band and instead shows widespread divergence, consistent with conditional sampling of divergent winds in rain-free conditions (Milliff et al. 2004). In contrast, the AFDS_RF divergence (Fig. 3b) shows a strong convergence band similar to that in the DFAS_AW divergence (Fig. 3a) but about twice stronger in magnitude. Even though both the AFDS_RF and DFAS_RF methods are based on the same rain-free wind measurements, the AFDS_RF divergence (Fig. 3b) indicates a strong convergence band, while the DFAS_RF divergence (Fig. 3c) shows none. Convergence and rainfall thus appear strongly linked in the DFAS_RF divergence, since excluding winds in rain effectively removes the mean convergence band. In contrast, convergence and rainfall appear weakly linked in the AFDS_RF divergence, since strong mean convergence exists in rain-free conditions. It is unclear how to reconcile these two contradictory conclusions without knowing why switching the order in which the time-average and spatial derivative operations are applied produces such different depictions.

Fig. 3.

Time-averaged divergence (10−5 s−1) from QuikSCAT computed using the (a) DFAS_AW, (b) AFDS_RF, and (c) DFAS_RF methods. These fields were averaged over the 2-yr period August 2007–July 2009 using the JPL v3 QuikSCAT winds.

Fig. 3.

Time-averaged divergence (10−5 s−1) from QuikSCAT computed using the (a) DFAS_AW, (b) AFDS_RF, and (c) DFAS_RF methods. These fields were averaged over the 2-yr period August 2007–July 2009 using the JPL v3 QuikSCAT winds.

These figures raise several intriguing questions that we attempt to answer: Does switching the order of time averaging and spatial differentiation, when applied to rain-free winds, really provide a valid method to estimate the all-weather time-averaged divergence and vorticity? If so, how is this accomplished? Is the perceived causal link between convergent, cyclonic winds and rainfall accurate? Are there systematic issues with the accuracy of the QuikSCAT rain flag? In this paper, we evaluate the AFDS_RF method to determine whether it is indeed a plausible method for obtaining estimates of the all-weather time-averaged divergence and vorticity fields, while providing a consistent explanation for why switching operation order (DFAS_RF vs AFDS_RF) produces such different results.

Note that we consider only the explicit case of spatial derivatives computed using centered finite differences. Bourassa and Ford (2010) use a more sophisticated method to estimate divergence and vorticity from scatterometer vector winds via circulation and divergence theorems at arbitrary grid spacings and variable spatial resolutions. Nonetheless, the rain flagging described in this analysis will affect divergence and vorticity computed using their method in a similar manner.

Earlier versions of QuikSCAT winds were particularly uncertain in rain (e.g., Portabella and Stoffelen 2001; Weissman et al. 2002; Portabella and Stoffelen 2002; Chelton and Freilich 2005; Weissman et al. 2012), but a more recent version processed through JPL (Fore et al. 2014) has significantly reduced, but not eliminated, QuikSCAT wind measurement uncertainties associated with rain (as discussed in section 6). Despite the recent availability of this dataset, documenting the impacts of these details on the time-mean divergence and vorticity is important for a variety of reasons. First, we believe it is necessary to highlight the sensitivity of the scatterometer-derived time-averaged divergence and vorticity to rain flagging and the order of operations. This will help ensure the accuracy and correct interpretation of these fields, particularly in previous studies. Second, while estimates of winds in rain from satellites have improved considerably, there remain some rain-induced uncertainties that may require removal of rain-contaminated winds. Finally, it is of significant scientific interest to understand these calculation details, as it will further our understanding of the interactions between surface winds and rainfall.

We focus here on the 10+-yr QuikSCAT data record, the observations of which are described in section 2. We proceed in section 3 by first demonstrating the effect that the order of operations has on the divergence and vorticity estimated from rain-free scatterometer winds. In section 4, we use a mesoscale weather model over the northwest Atlantic to compare methods of computing the time-averaged divergence and vorticity from rain-free winds, and compare these estimates with their all-weather counterparts. We quantify the accuracy of the AFDS_RF and DFAS_RF methods, while also showing the sensitivity of the accuracy of the AFDS_RF method on averaging period and degree of spatial smoothing. In section 5, we utilize the model’s all-weather winds to determine the root cause of the differences between the AFDS_RF and DFAS_RF methods. Finally, we end in section 6 with an analysis of four scatterometer wind datasets that contain estimates of winds in raining conditions. Here, we determine how well the AFDS_RF methods approximate the all-weather divergence and vorticity fields over other regions. We also discuss the order of operations applied to the satellite all-weather winds, which, because of the finite width of satellite measurement swaths, means that the AFDS_AW and DFAS_AW methods are not equivalent.

2. Description of satellite observations

a. QuikSCAT wind datasets

This analysis utilizes surface wind observations from the SeaWinds scatterometer on QuikSCAT (referred to as simply QuikSCAT) over the ice-free global oceans. QuikSCAT is an active radar that transmits radiation and measures power scattered back from the wind-roughened ocean surface. Backscatter measurements are processed through an empirically derived geophysical model function (GMF) to retrieve surface wind vectors on a spatial grid with nominal spacing of 12.5 or 25 km. All satellite-based surface wind retrievals, including QuikSCAT, are calibrated to the so-called equivalent neutral wind (ENW) at 10-m height.

The primary QuikSCAT dataset used here is the JPL version 3 (v3) dataset (Fore et al. 2014) distributed through PO.DAAC (SeaPAC 2014). JPL has recently reprocessed most of the QuikSCAT data record with several algorithm updates. The improvements compared to previous versions include the following: a correction to rain-contaminated winds using an autonomous neural network–based technique (Stiles and Dunbar 2010), which significantly improves the accuracy of vector wind retrievals in rain; a procedure to minimize cross-track biases of retrieved wind speeds; adjustments to the DIR threshold, which reduces grid-scale directional noise; and processing changes to the backscatter measurements that reduce noise while enhancing spatial resolution.

The retrieved winds are distributed in level-2B format at approximately 12.5-km grid spacing. Winds in rain in the level-2B data are included by default. For this analysis, we spatially interpolated each vector wind component to a uniform 0.25° spatial grid using a two-dimensional loess smoother with a half-power radius of 40 km.

At the 13.8-GHz frequency at which the QuikSCAT radar operates, the atmosphere is nearly transparent, except in the presence of precipitation. Precipitation contaminates the emitted and returned radar signal while also generating surface roughness features unrelated to the local surface wind [see the recent review by Weissman et al. (2012) and references therein]. Winds in grid cells where either one of two methods indicates rain are flagged. In the first method, referred to as the multidimensional histogram (MUDH) rain probability (Huddleston and Stiles 2000), the scatterometer wind retrievals are analyzed for how well the raw scatterometer backscatter measurements from multiple viewing geometries fit the geophysical model function. This method relies only on the scatterometer itself with no other outside information. The second method utilizes columnar-integrated rain-rate estimates from four passive microwave radiometers on separate platforms collocated with the scatterometer wind measurements. At each grid point, a wind vector observation was flagged as missing when either the scatterometer MUDH flag or the accompanying collocated radiometer rain flag indicated precipitation. We use the rain flag distributed in the RSS QuikSCAT dataset for this analysis. We also analyze the RSS QuikSCAT dataset further in section 6.

Rain flags in the QuikSCAT dataset remain uncertain for a variety of reasons, including sensitivity to rain. The radiometer rain flags are uncertain because of temporal and spatial collocation criteria, which can incorrectly identify times and locations of moving precipitation systems, and the intermittent appearance of rain, such as would occur in evolving convective systems. Additionally, the spatial response functions of passive radiometers differ from those of the QuikSCAT radar itself, which can add further uncertainty of the exact geographical location of rain.

b. Distribution of rain-flagged observations

Rain frequency is highly variable over the global ocean, as shown in Fig. 4. This shows the frequency of rain-flagged QuikSCAT observations during the 10-yr period November 1999–October 2009, expressed as a percentage of total possible observations. Along the intertropical convergence zone (ITCZ) and South Pacific convergence zone (SPCZ), more than 25% of the observations were rain flagged. In the northwest Atlantic and northwest Pacific, more than 15% of scatterometer wind observations were rain flagged. In the Southern Ocean poleward of 35°S, more than about 10%–15% of observations were rain flagged. In midlatitudes, precipitation is typically more prevalent along the midlatitude SST frontal zones associated with western boundary currents and their associated extensions into the interior of the ocean basin. Rain flagging is also more frequent in the tropical Atlantic and Indian Oceans but nearly negligible along most of the major eastern boundary current regions and in the western Arabian Sea.

Fig. 4.

Map of the QuikSCAT rain-flag frequency (%) over the 10-yr period November 1999–October 2009.

Fig. 4.

Map of the QuikSCAT rain-flag frequency (%) over the 10-yr period November 1999–October 2009.

3. Ordering of time-average and spatial derivative operations

We consider two methods of computing time-averaged divergence and vorticity fields, both of which use centered finite differences of gridded zonal and meridional wind components. The first, and possibly most common, is to time average the individual wind components before computing spatial derivatives (the AFDS method). Considering only the divergence explicitly, the divergence of the time-averaged u- and υ-wind components at the grid point (where i is the zonal index and j is the meridional index) is

 
formula

where and are the zonal and meridional grid spacing, respectively; , , , and are the number of observations at the adjacent eastern, western, northern, and southern grid points, respectively; t is the observation time; and the overbar represents the time mean.

The second way we consider to compute time-averaged divergence is to compute the spatial derivatives first and then time averaging (the DFAS method):

 
formula

where is the number of measurement times for which nonmissing data are available at all four grid points. With this order of operations, the time-averaged derivatives at the grid point are not calculated if an observation at any one of the four adjacent grid points is missing. Milliff and Morzel (2001), Milliff et al. (2004), and Chelton et al. (2004) referred to spatial derivatives computed in this way as “in-swath,” since observations not averaged in time are needed. Without any missing data, the AFDS and DFAS methods are mathematically identical.

Figures 5a,b show the QuikSCAT divergence and vorticity differences, respectively, between the two computation methods with contours of the rain frequency overlaid. Differences here are defined as AFDS_RF minus DFAS_RF, averaged over the 10-yr analysis period. As expected, these differences are highly correlated with rain frequency and show the strong consequences that data dropouts caused by rain have on the two estimation methods. Ultimately, the AFDS_RF method produces a much more convergent and cyclonic view of the wind field than does the DFAS_RF method throughout most of the world’s oceans.

Fig. 5.

Differences of the 10-yr averaged QuikSCAT divergence and vorticity (10−5 s−1) from the two computation methods shown in Fig. 1 with overlaid contours of the percentage of rain-flagged QuikSCAT observations from Fig. 14a. The contour interval is 5% with integral values of 10% are in bold. The differences are defined as AFDS_RF minus DFAS_RF.

Fig. 5.

Differences of the 10-yr averaged QuikSCAT divergence and vorticity (10−5 s−1) from the two computation methods shown in Fig. 1 with overlaid contours of the percentage of rain-flagged QuikSCAT observations from Fig. 14a. The contour interval is 5% with integral values of 10% are in bold. The differences are defined as AFDS_RF minus DFAS_RF.

4. Numerical simulation

A mesoscale numerical simulation was undertaken to analyze the sources of differences in the divergence and vorticity from switching the order of time-averaging and spatial differentiation operations applied to rain-free winds. One advantage of using a model for this purpose is that it provides a dynamically consistent and realistic representation of all-weather atmospheric variability free from observational uncertainty. In particular, there are uncertainties in the satellite rain flag that can conceivably affect our results but that are not present in the model simulation.

a. Simulation design

The high-resolution, mesoscale model Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS; Hodur 1997)2 was developed to inform U.S. Naval operations on weather events from regional to mesoscale (sub kilometer). The model has been implemented for research purposes and field experiments since the late 1980s. Specifically, COAMPS rainfall and low-level wind forecasts have been evaluated in several modeling studies (Kindle et al. 2002; Haack et al. 2005; Pullen et al. 2006, 2007; Hong et al. 2011), and the coupling of wind stress curl and divergence to spatial variations in SST has been documented in the U.S. West Coast region by Haack et al. (2008). These studies have demonstrated that COAMPS is capable of predicting all-weather surface winds with good accuracy and with high spatial and temporal resolution.

For this study, the model was run twice per forecast day over the northwest Atlantic at a grid spacing of 9 km with 50 vertical levels.3 The analysis in this section encompasses the 1-yr period of 2009. We utilized COAMPS 12-h wind forecasts (at 0600 and 1800 UTC) at the lowest model level (10 m) to approximately match the measurement time and frequency of the QuikSCAT measurements in the northwest Atlantic. We only consider grid points over the ocean for the ensuing analysis. Subgrid-scale turbulent mixing was parameterized using Mellor–Yamada level 1.5-s-order closure. Convective parameterization typically used in coarser COAMPS simulations (Kain–Fritsch) was turned off for this simulation.

We chose the northwest Atlantic because it is one of the more meteorologically active regions of the World Ocean, with strong and frequent storms and vigorous air–sea interaction along the Gulf Stream. Most importantly, this region has strong and frequent precipitation from synoptic-scale disturbances (e.g., Kelly et al. 2010; Rudeva and Gulev 2011; Booth et al. 2012), enhanced precipitation associated with air–sea interaction along the Gulf Stream frontal zone (e.g., Minobe et al. 2008; Kuwano-Yoshida et al. 2010), and regions of strong divergence and vorticity perturbations associated with the boundary layer response to SST (Doyle and Warner 1993; Chelton et al. 2004; Park et al. 2006; Song et al. 2006; Small et al. 2008; Joyce et al. 2009; O’Neill et al. 2010; Kilpatrick et al. 2014). Regions that are likely different from the northwest Atlantic include tropical convergence zones and monsoon regions, which have precipitation from tropical disturbances different in character than that associated with midlatitude frontal disturbances (e.g., Li and Carbone 2012; Kilpatrick and Xie 2015). In section 6, we investigate these other regions of the World Ocean using AW wind estimates from satellite.

b. Evaluation of COAMPS 10-m vector winds

The 10-m COAMPS vector winds were compared with 11 moored buoys in the northwest Atlantic, which are part of the National Data Buoy Center (NDBC) and Canadian Department of Fisheries and Oceans (CDFO) observational arrays (Fig. 6a). We adjusted the in situ buoy winds to 10-m height according to the standard procedure described briefly in  appendix A. These buoy observations were not included in any data assimilation into the model. The modeled 10-m wind speed and direction compare relatively well with the buoys, with a root-mean-square difference (RMSD) of 1.9 m s−1 for speed and 34° for direction. After restricting the comparison to wind speeds greater than 3 m s−1, the direction RMSD reduces to 25°, while the speed RMSD remains unchanged. The mean wind speed difference, defined as COAMPS minus buoy, is −0.4 m s−1, indicating that COAMPS slightly underestimates the wind speed on average in this simulation. The time-mean COAMPS minus buoy direction difference is 0°.

Fig. 6.

Maps denoting the locations of the buoys (red squares) used to evaluate (a) the COAMPS winds and (b) the satellite winds. The grayscale shows the QuikSCAT rain-flag frequency (%) over the 2-yr period August 2007–July 2009.

Fig. 6.

Maps denoting the locations of the buoys (red squares) used to evaluate (a) the COAMPS winds and (b) the satellite winds. The grayscale shows the QuikSCAT rain-flag frequency (%) over the 2-yr period August 2007–July 2009.

c. Evaluation of COAMPS rain frequency

Rain frequency for the 1-yr COAMPS simulation is shown in Fig. 7, along with rain frequency estimates from four satellite passive microwave radiometers, including the Advanced Microwave Scanning Radiometer for Earth Observing System (AMSR-E), the Special Sensor Microwave Imager (SSM/I) on F16 and F17, and WindSat, and from the Tropical Rainfall Measuring Mission (TRMM) 3B42 merged rain-rate estimates. The individual satellites estimated rain rate between 0 and 2 times per day, while the TRMM 3B42 analysis was constructed at 3-h intervals. All five datasets are distributed on a 0.25° spatial grid. We defined rain frequency for the satellites in terms of the relative occurrence of a nonzero rain rate. For the COAMPS simulations, we defined rain frequency by the relative frequency of nonzero surface 1-h accumulated precipitation. The main spatial features of rain frequency agree between COAMPS and satellite estimates, including maxima along the Gulf Stream frontal zone and a broader maximum further downstream. In terms of amplitude, the COAMPS rain frequency is a few percent larger than is apparent in the satellite observations. This difference may be due to our definition of rain frequency in COAMPS, since 1-h rain accumulations potentially capture more incidence of rain than the near-instantaneous satellite measurements. Another source of difference between the different satellite estimates is that each measures rain rate at different times of day. Additionally, there are also known biases in satellite rain rates that can account for some of the differences between different satellites (e.g., Hilburn and Wentz 2008; Lin and Hou 2008; Bowman et al. 2009; Fisher and Wolff 2011), although no studies appear to evaluate rain frequency directly. Regardless of these differences, we found that the results of this analysis were not sensitive to the definition of rain frequency in COAMPS. Using an alternative definition of rain frequency in COAMPS based on the instantaneous rain mixing ratio at the surface grid point yielded similar results.

Fig. 7.

Rain frequency (%) estimates for 2009 from (a) COAMPS simulation, (b) TRMM 3B42 merged dataset, (c) AMSR-E, (d) SSM/I on F16, (e) SSM/I on F17, and (f) WindSat.

Fig. 7.

Rain frequency (%) estimates for 2009 from (a) COAMPS simulation, (b) TRMM 3B42 merged dataset, (c) AMSR-E, (d) SSM/I on F16, (e) SSM/I on F17, and (f) WindSat.

d. Comparison of rain-free and all-weather time-averaged derivative wind fields

1) Differences in annual average fields

The annual averaged COAMPS divergence and vorticity over the northwest Atlantic is shown in Figs. 8a,e using the all-weather winds. These show a ribbon of convergent, cyclonic winds over the approximate location of the Gulf Stream. The annual average COAMPS AFDS_RF and DFAS_RF divergence and vorticity fields are also shown in this figure, which are all consistent with the time-averaged QuikSCAT fields shown in Fig. 1. The COAMPS simulation is able to qualitatively replicate the salient features associated with switching the orders of operations that are apparent in the QuikSCAT fields. Furthermore, the AFDS_RF fields qualitatively resemble the AW fields, consistent with the comparison made with the NCEP fields in Fig. 2.

Fig. 8.

COAMPS (top) divergence and (bottom) vorticity fields averaged for 2009: (a),(e) AW; (b),(f) DFAS_RF; (c),(g) AFDS_RF; and (d),(h) SM_AFDS_RF.

Fig. 8.

COAMPS (top) divergence and (bottom) vorticity fields averaged for 2009: (a),(e) AW; (b),(f) DFAS_RF; (c),(g) AFDS_RF; and (d),(h) SM_AFDS_RF.

Grid-level variability in the AFDS_RF fields in Figs. 8c,g becomes more apparent when magnifying this smaller region. In contrast, the DFAS_RF fields shown in Fig. 8b,f do not exhibit this level of noise. This variability arises from taking spatial derivatives of time averages composed of winds that were not all measured at the same time as a result of rain flagging. This noise can be essentially eliminated in the AFDS_RF fields by applying a spatial low-pass filter, which attenuates variability with wavelengths smaller than 125 km, as shown in Fig. 8d,h; this procedure is referred to hereafter as the SM_AFDS_RF method. Although we lose small-scale information in the smoothing process, the SM_AFDS_RF method visually improves the comparison to the AW fields relative to the unsmoothed AFDS_RF fields. Because of this apparent improvement, we also consider the SM_AFDS_RF method in the following comparisons.

Figures 9a,d shows maps of the percent relative error of the COAMPS SM_AFDS_RF divergence and vorticity for the annual average of 2009. Only relative errors for absolute magnitude of the AW divergence and vorticity greater than 0.25 × 10−5 s−1 are shown. For this annual average, the SM_AFDS_RF method overestimates the magnitude of the convergence and cyclonic vorticity over the mean path of the Gulf Stream by 50%–100%. The relative error shows strong seasonal variability, as indicated from comparison of the individual monthly averages of January and July 2009 in Figs. 9b,c,e,f. During January, the relative errors are between 100% and 200% over most of the Gulf Stream region but decrease significantly during July. This seasonality is due to the strong annual cycle of precipitation.

Fig. 9.

Percentage relative error in the COAMPS SM_AFDS_RF (top) divergence and (bottom) vorticity. (a),(d) The 1-yr average for 2009 and the 1-month averages of (b),(e) January 2009 and (c),(f) July 2009. The relative error is shown only for absolute values of the mean AW divergence and vorticity fields >0.25 × 10−5 s−1. The black contour shows the AW divergence (vorticity) contour of −1.0 × 10−5 s−1 (1.0 × 10−5 s−1).

Fig. 9.

Percentage relative error in the COAMPS SM_AFDS_RF (top) divergence and (bottom) vorticity. (a),(d) The 1-yr average for 2009 and the 1-month averages of (b),(e) January 2009 and (c),(f) July 2009. The relative error is shown only for absolute values of the mean AW divergence and vorticity fields >0.25 × 10−5 s−1. The black contour shows the AW divergence (vorticity) contour of −1.0 × 10−5 s−1 (1.0 × 10−5 s−1).

2) Sensitivity to averaging period and spatial smoothing

(i) Time averaging

We assess the sensitivity of the accuracy of the AFDS_RF and DFAS_RF methods to the period of time averaging through domain-integrated statistics, including the root-mean-square error (RMSE), cross correlation, and bias. Figure 10 shows these statistics computed as functions of averaging period for all grid points in the 1-yr COAMPS simulation (green curves, AFDS_RF; pink curves, DFAS_RF). The RMSE and bias were defined relative to the “true” COAMPS AW divergence and vorticity fields. For longer time averages, both methods produce more accurate estimates of the AW derivative fields, as evident from the lower RMSE and higher correlations for longer averaging periods (Figs. 10a,c). The DFAS_RF method is generally more accurate in terms of the RMSE and correlations but has a larger bias magnitude compared with the AFDS_RF method, which is a reflection of the conditional sampling bias discussed in the introduction. The SM_AFDS_RF method significantly improves the RMSE and correlation statistics. For annual averages, the RMSEs for all three methods are about 0.5 × 10−5 s−1. To put the RMSE into context, we normalized the RMSE values by the interquartile range of the AW divergence and vorticity, which provides a robust estimate of the range of variability in the AW fields (Fig. 10b); the normalized RMSE (NRMSE) is an estimate of the error-to-signal ratio. The NRMSE for the AFDS_RF method, expressed as a percentage, varies between roughly 120% for 1-week averages and 60% for annual averages. The DFAS_RF and SM_AFDS_RF NRMSEs vary between 30% and 50% and vary less with averaging period than the AFDS_RF NRMSE. Because of this strong dependence of accuracy on period of time averaging, we will also consider monthly periods, which will allow resolution of seasonal variability.

Fig. 10.

Statistics of the difference between the RF divergence (solid) and vorticity (dashed) fields and the AW fields as a function of time-averaging period from the 1-yr COAMPS simulation: (a) RMSE; (b) NRMSE, normalized by the interquartile range of the AW derivative field and expressed as a percentage such that the NRMSE provides context for the RMSE in terms of the range of variability exhibited by the AW fields; (c) cross-correlation coefficient; and (d) bias, or mean difference, defined as AFDS_RF minus AW (green curves) and DFAS_RF minus AW (pink curves). The statistics were computed for all the grid points in the simulation domain. The blue curves represent a spatially smoothed version of the AFDS_RF method, denoted as SM_AFDS_RF; the spatial smoothing applied to the fields attenuates spatial variability on spatial scales <125 km. Note, in (d), the blue curves lie over the top of and obscure the green curves.

Fig. 10.

Statistics of the difference between the RF divergence (solid) and vorticity (dashed) fields and the AW fields as a function of time-averaging period from the 1-yr COAMPS simulation: (a) RMSE; (b) NRMSE, normalized by the interquartile range of the AW derivative field and expressed as a percentage such that the NRMSE provides context for the RMSE in terms of the range of variability exhibited by the AW fields; (c) cross-correlation coefficient; and (d) bias, or mean difference, defined as AFDS_RF minus AW (green curves) and DFAS_RF minus AW (pink curves). The statistics were computed for all the grid points in the simulation domain. The blue curves represent a spatially smoothed version of the AFDS_RF method, denoted as SM_AFDS_RF; the spatial smoothing applied to the fields attenuates spatial variability on spatial scales <125 km. Note, in (d), the blue curves lie over the top of and obscure the green curves.

(ii) Spatial filtering

Figure 11 shows the RMSE, NRMSE, and correlation between the SM_AFDS_RF and AW fields for various low-pass filter cutoff radii (y axis) and time-average periods (x axis). Spatial filtering effectively decreases the errors using the AFDS_RF method. The RMSE is minimized and correlations maximized for smoothing radii between 100 and 150 km. Spatial smoothing is more effective at reducing the errors in the AFDS_RF method for shorter time-averaging periods. Our choice of 125-km spatial smoothing in Figs. 8d and 8h was made to minimize the RMSE and maximize the correlations in the SM_AFDS_RF method. Error reduction in the SM_AFDS_RF method, however, comes at the expense of spatial and temporal resolution.

Fig. 11.

Statistics of the difference between the COAMPS SM_AFDS_RF and AW (left) divergence and (right) vorticity as functions of the period of time averaging (x axis) and low-pass filter cutoff radii (y axis; km): (top) RMSE, (middle) NRMSE in percent, and (bottom) cross-correlation coefficient. The zero cutoff radius indicates no spatial smoothing.

Fig. 11.

Statistics of the difference between the COAMPS SM_AFDS_RF and AW (left) divergence and (right) vorticity as functions of the period of time averaging (x axis) and low-pass filter cutoff radii (y axis; km): (top) RMSE, (middle) NRMSE in percent, and (bottom) cross-correlation coefficient. The zero cutoff radius indicates no spatial smoothing.

3) Differences in monthly averaged fields

The COAMPS AFDS_RF divergence and vorticity distributions are much broader than the AW distributions, as shown by the histograms in Fig. 12. These are separated into all months (Figs. 12a,b), December–January (DJF; Figs. 12c,d) and June–August (JJA; Figs. 12e,f) to highlight seasonal variations. All of the RF and AW distributions are shifted toward divergence and anticyclonic vorticity, but the DFAS_RF (red curves) more strongly so. Moreover, the DFAS_RF curves are narrower than the much wider AFDS_RF distributions.

Fig. 12.

Histograms of COAMPS monthly averaged (left) divergence and (right) vorticity computed from all ocean grid points in the simulation domain. Each color represents one of the fields discussed. (a),(b) All months during 2009; (c),(d) DJF 2009; and (e),(f) JJA 2009.

Fig. 12.

Histograms of COAMPS monthly averaged (left) divergence and (right) vorticity computed from all ocean grid points in the simulation domain. Each color represents one of the fields discussed. (a),(b) All months during 2009; (c),(d) DJF 2009; and (e),(f) JJA 2009.

Differences between the AFDS_RF and AW distributions also appear more significant during winter than during summer (from comparison of the red and black curves), while the differences between the DFAS_RF and AW distributions exhibit less seasonal variability. The distributions of the monthly averaged SM_AFDS_RF divergence and vorticity (blue curves) are in much better agreement with the AW distributions than the unsmoothed AFDS_RF distributions during all seasons, particularly for divergence. The SM_AFDS_RF vorticity shows slight shifts toward stronger anticyclonic vorticity. Statistical significance of the differences in distributions of monthly averaged quantities is discussed in  appendix B. It is shown there that the AFDS_RF and DFAS_RF have less than 5% probability of being equivalent to the corresponding AW distributions, while the SM_AFDS_RF distributions have greater than 5% probability of being equivalent to the AW distributions in 40% of grid points.

Large seasonally varying errors are evident in all three methods, as evident from time series of RMSEs and cross correlations in Figs. 13a–d). The statistics were computed for all grid points in the spatial domain shown in Fig. 8. Higher RMSEs and lower correlations occur during the winter months when precipitation is more frequent. For example, the RMSE for the AFDS_RF method varies from about 4.0 × 10−5 s−1 in December to about 1.2 × 10−5 s−1 during July and August. The SM_AFDS_RF fields reduce the RMSE by a factor of roughly 2–3 relative to the AFDS_RF method (green curves), while improving the correlations. However, spatial smoothing does not fully mitigate the seasonal cycle in the AFDS_RF error.

Fig. 13.

Time series of (top) RMSE and (bottom) cross correlations of the various forms of the COAMPS monthly averaged wind derivative fields computed over all grid points of the 12-month simulation. The AFDS_RF method is shown by the green curves, the DFAS_RF method by pink curves, and the SM_AFDS_RF method by blue curves: (left) divergence and (right) vorticity.

Fig. 13.

Time series of (top) RMSE and (bottom) cross correlations of the various forms of the COAMPS monthly averaged wind derivative fields computed over all grid points of the 12-month simulation. The AFDS_RF method is shown by the green curves, the DFAS_RF method by pink curves, and the SM_AFDS_RF method by blue curves: (left) divergence and (right) vorticity.

This analysis indicates that both the AFDS_RF and DFAS_RF methods have significant discrepancies from the AW fields. The discrepancies appear especially apparent during the winter months, when precipitation is more frequent. Averaging for longer time periods and spatial smoothing reduce errors in the AFDS_RF method. For annual averages and 125-km spatial smoothing, the SM_AFDS_RF method still has RMSEs of ~0.5 × 10−5 s−1, which is 30%–50% of the total variability of the AW fields. The SM_AFDS_RF method overestimates the convergence and anticyclonic vorticity over the Gulf Stream by 50%–100%.

5. Relationship between the AFDS_RF and DFAS_RF methods

a. Conditional sampling of the divergence based on presence of rain

We now diagnose the mathematical differences between the COAMPS AFDS_RF and DFAS_RF methods. Throughout the remainder of this section, we only show the development for the divergence since the conclusions for vorticity are similar. We will also consider only annual averages to simplify the presentation. We begin by conditionally sampling the COAMPS AW divergence at each grid point based on whether all four of and at each time are in grid cells that are rain free (RF) or rain only (R), or consist of times when exactly 1, 2, or 3 of and are in raining grid cells at the same time [rain–rain-free mixture (M)]; this latter condition is satisfied in grid cells along the perimeters of rain. In this nomenclature, these three components are designated as DFAS_RF, DFAS_R, and DFAS_M, respectively. The time-mean AW divergence is the sum of the means of these three conditionally sampled components weighted by their relative occurrences:

 
formula

where .

Each term in Eq. (3) from the 1-yr COAMPS simulation is shown in Figs. 14a–d. The DFAS_RF divergence, weighted by its relative frequency of occurrence [i.e., ], is shown in Fig. 14b. This weighting does not generate notable differences from the unweighted DFAS_RF fields shown in Fig. 8b.

Fig. 14.

Components of the time-mean COAMPS divergence field over the northwest Atlantic for 2009: (a) all weather ; (b) weighted mean coincident RF component ; (c) weighted mean coincident rain-only component ; (d) weighted mean mixed rain–rain-free component (e) coincident component of AFDS_RF divergence; (f) uncoincident component of AFDS_RF divergence; (g) sum of the weighted mean rain-only and mixed rain–rain-free divergence components from panels (c) and (d); and (h) contribution to the uncoincident AFDS_RF divergence component independent of differences in the number of data points between grid cells, divided by 20. The fields in (b)–(d) are defined as in Eq. (3).

Fig. 14.

Components of the time-mean COAMPS divergence field over the northwest Atlantic for 2009: (a) all weather ; (b) weighted mean coincident RF component ; (c) weighted mean coincident rain-only component ; (d) weighted mean mixed rain–rain-free component (e) coincident component of AFDS_RF divergence; (f) uncoincident component of AFDS_RF divergence; (g) sum of the weighted mean rain-only and mixed rain–rain-free divergence components from panels (c) and (d); and (h) contribution to the uncoincident AFDS_RF divergence component independent of differences in the number of data points between grid cells, divided by 20. The fields in (b)–(d) are defined as in Eq. (3).

The weighted DFAS_R divergence, , is shown in Fig. 14c. As expected, it is strongly convergent over the whole domain, since rain occurs in convergent environments on average. There is some evidence of convergence enhancement near the Gulf Stream, but it is not as sharp as the convergence ribbon observed in the AW divergence.

A key to understanding the improvement of the AFDS_RF method over the DFAS_RF method in approximating the AW divergence and vorticity comes from the mixed rain–rain-free divergence DFAS_M. As shown in Fig. 14d, it is convergent over most of the domain and strongest in magnitude along the Gulf Stream. It has some similarities and differences from the weighted DFAS_R divergence shown in Fig. 14c. Like the rainy divergence, it is mostly convergent throughout the domain and is enhanced over the mean path of the Gulf Stream; however, Gulf Stream convergence enhancement from the DFAS_M component is roughly half the magnitude of the DFAS_R divergence in Fig. 14c. Perhaps not surprisingly, this indicates that convergence at the edges of rain is weaker than within the interiors of raining systems. The DFAS_M divergence also shows stronger and more widespread convergence south of 35°N, whereas the DFAS_R convergence weakens to the south. While the DFAS_M divergence contributes little to convergence north of the Gulf Stream downstream of Cape Hatteras, the DFAS_R component contributes relatively strong and widespread convergence in these regions. Convergence along the Gulf Stream off Florida and Georgia and into the Gulf of Mexico is stronger from the DFAS_M component as well.

b. AFDS_RF divergence

The AFDS_RF divergence is a specific case of Eq. (1) applied only to rain-free winds. All rain-free winds are included in the time averages in the AFDS_RF method, in contrast to the DFAS_RF method, where only those wind measurements coincident in time at all four adjacent grid cells are included [cf. Eq. (2)]. We isolate the effect that these uncoincident winds have on the AFDS_RF method by separating the time means of u and υ into coincident and uncoincident means, where the coincident time mean is defined over the set of coincident times where rain-free , , , and all exist. Thus, for example,

 
formula

where is the number of rain-free observations at grid point , composed of coincident observations and uncoincident observations. The time-mean rain-free coincident winds are denoted as , while denotes the time-mean rain-free uncoincident winds. With this notation, Eq. (1) can be expanded as

 
formula

Grouping the like coincident and uncoincident components together and rearranging yields

 
formula

From this equation, we can deduce that the AFDS_RF divergence depends not only on the time-mean winds, but also on the spatial distribution of the rain-free observations. In the limit of no rain-flagged observations (and hence no missing data), , and terms involving the uncoincident means vanish so that the AFDS and DFAS methods are formally identical. This can be seen by casting the DFAS_RF divergence in terms of the mean coincident winds:

 
formula

The uncoincident RF mean winds in Eq. (6) (i.e., the terms containing and ) represent the time-averaged vector wind components at the neighboring nonraining grid cells bordering rainy grid cells. As seen from comparing Eqs. (6) and (7), the uncoincident RF mean winds provide additional information to the AFDS_RF method that is not incorporated into the DFAS_RF method and is thus the key difference between them.

The contributions to the AFDS_RF method from the RF coincident mean component in Eq. (6) is shown in Fig. 14e from the COAMPS simulation. It is similar to the coincident RF divergence shown in Fig. 14b, except for the division of each wind component by a spatially varying N; thus, spatial variability in the number of RF data points biases the coincident component of the AFDS_RF method relative to the DFAS_RF method.

The apparent improvement of the AFDS_RF method in approximating the AW fields depends largely on how well the term containing the uncoincident means in Eq. (6) approximates the sum of the DFAS_R and DFAS_M components. A comparison between the two shows that the uncoincident term bears good resemblance to the sum of the DFAS_R and DFAS_M components, at least in terms of spatial structure, as seen by comparing Figs. 14f and 14g. Even so, the uncoincident term overestimates convergence by about a factor of 2 nearly everywhere in the domain. Even if we spatially smooth the noisy uncoincident field, the convergence is still much stronger but spread out over a broader area (not shown).

Since grid cells surrounding rain typically occur in convergent conditions, the uncoincident winds contribute convergence to the AFDS_RF divergence and can account for some of the convergence associated with rain. It does not necessarily provide an accurate magnitude of the convergence, as we showed in the last section, but it does counteract, to an extent, the divergent sampling bias of the DFAS_RF method.

The final question is whether the uncoincident component in the AFDS_RF divergence is controlled by the uncoincident mean wind components themselves or by the ratios of the number of points multiplying them. To shed some light on this question, Fig. 14h shows the divergence of the mean uncoincident winds without the factors involving the number of data points [i.e., ]. This field has been divided by 20 to fit the color scale of the other fields in this figure. Although noisy, it contributes mainly very strong convergence. Convergence enhancement near the Gulf Stream is less apparent in this component compared to Fig. 14f, suggesting that spatial gradients in the number of rain-free data points play a role in biasing convergence near the Gulf Stream, where most rain-flagged data points reside (cf. Fig. 7a). Spatial gradients in the number of rain-free observations thus improve representation of the spatial structure of the AFDS_RF divergence compared to the AW fields, although this effect does not have any real physical basis.

6. Satellite estimates of all-weather derivative wind fields

The comparison of the AFDS_RF and DFAS_AW methods is extended to other regions using all-weather scatterometer wind estimates with two goals. First, we wish to determine whether the differences between the AFDS_RF and AW fields over the northwest Atlantic shown in the COAMPS simulations are consistent over the rest of the world’s oceans. Second, as we show, missing data due to rain are not the only factor contributing to the uncoincident component of the AFDS_RF divergence and vorticity computed from scatterometer winds; swath edges also contribute to the uncoincident component that is not present in the COAMPS simulation.

Estimates of AW satellite vector winds from the QuikSCAT, Advanced Scatterometer on MetOp-A (ASCAT-A), and WindSat instruments are used to further evaluate the AFDS_RF divergence and vorticity against the all-weather estimates. Each of these datasets is described below. The period of the analysis in this section is the common 2-yr period August 2007–July 2009 where all three instruments were operational.

a. Datasets

Besides the JPL QuikSCAT dataset discussed earlier, we also show the QuikSCAT dataset produced by RSS. It uses the same GMF as the JPL dataset but is processed differently. As we show, the RSS QuikSCAT winds are modestly less accurate in rain than the JPL winds but have nearly identical performance in rain-free conditions.

The C-band ASCAT-A provides the second dataset (Figa-Saldaña et al. 2002). The dataset shown here was processed by RSS using their C-2013 GMF. We used the RSS ASCAT dataset rather than the Royal Netherlands Meteorological Institute (KNMI) version, since it provides a rain flag similar to the one produced for QuikSCAT. C-band scatterometers are less affected by rain than Ku-band scatterometers, such as QuikSCAT (e.g., Portabella et al. 2012), and are thus able to provide more accurate estimates of winds in rain. ASCAT-A obtains surface wind estimates by using a C-band fan-beam scatterometer design. ASCAT-A collects wind observations along two off-nadir swaths, each 500 km wide and separated by a 700-km nadir gap. Because of the nadir gap, ASCAT-A collects approximately 40% fewer observations than QuikSCAT. ASCAT wind observations have been successful in surveys of surface convergence associated with mesoscale convective downdrafts in the tropics in light to moderate rain (Mapes et al. 2009; Kilpatrick and Xie 2015). Portabella et al. (2012) have also shown surface wind signatures associated with mesoscale circulation anomalies are resolved well in ASCAT, while they are not resolved in ECMWF winds.

The third dataset is from the WindSat passive polarimetric radiometer on the Coriolis satellite (Gaiser et al. 2004). The wind data were processed by RSS, dataset version 7.0.1. It estimates surface vector winds using brightness temperature measurements in C- and X-band microwave frequencies. The performance of WindSat measurements in rain is somewhat complicated, although they do provide reasonably more accurate winds in rain than the RSS QuikSCAT dataset (Meissner and Wentz 2009). Initial assessments of the WindSat vector wind retrievals, while encouraging, indicated a lower overall accuracy compared with those from QuikSCAT (Freilich and Vanhoff 2006; Monaldo 2006). Further refinements in sensor calibration and retrieval algorithm development, however, have significantly improved the accuracy of the WindSat vector wind retrievals, particularly in rain. WindSat also has channels sensitive to rain rate.

b. Evaluation of satellite vector winds using buoy wind measurements

Since satellite wind retrievals are referenced to the 10-m ENW, we use buoy winds converted to the 10-m ENW according to standard procedure (see  appendix A). Our comparison uses the buoys shown in the map in Fig. 6b for the 2-yr period August 2007–July 2009. Statistics of the buoy–satellite wind speed and direction comparisons are shown in Table 1, separated by instrument and for rain-free, rain-only, and all-weather conditions. In rain-free conditions, all satellite datasets perform similarly, with wind speed RMSD between 1.0 and 1.1 m s−1, mean wind speed differences between 0.0 and 0.2 m s−1, and direction RMSD between 17° and 19°. Performance in raining conditions clearly separates the four satellite datasets. Compared to the buoy winds, ASCAT-A performs most favorably in rain using all wind speed and direction metrics, with wind speed and direction RMSD of 1.4 m s−1 and 21°, respectively, and a mean wind speed difference of −0.1 m s−1. WindSat is slightly less accurate (relative to the buoys) compared to the other three satellites. The RSS QuikSCAT provides the least favorable estimates of winds in rain using the metrics of RMSD and mean wind speed differences, although it performs very well in rain-free conditions. Finally, there is a marked difference between the JPL and RSS QuikSCAT datasets, which shows the significant improvement of wind measurements in rain in the JPL dataset.

Table 1.

Comparison statistics between buoy and satellite wind speed and direction for the 2-yr period August 2007–July 2009. The statistics are separated for AW, RF, and R conditions. The rows correspond to satellite dataset with RQ for RSS QuikSCAT, JQ for JPL QuikSCAT, RA for RSS ASCAT-A, and RW for RSS WindSat. Statistics were computed for all buoy wind speeds but excluding absolute wind direction differences between buoy and satellite >100°. Differences are defined as buoy minus scatterometer.

Comparison statistics between buoy and satellite wind speed and direction for the 2-yr period August 2007–July 2009. The statistics are separated for AW, RF, and R conditions. The rows correspond to satellite dataset with RQ for RSS QuikSCAT, JQ for JPL QuikSCAT, RA for RSS ASCAT-A, and RW for RSS WindSat. Statistics were computed for all buoy wind speeds but excluding absolute wind direction differences between buoy and satellite >100°. Differences are defined as buoy minus scatterometer.
Comparison statistics between buoy and satellite wind speed and direction for the 2-yr period August 2007–July 2009. The statistics are separated for AW, RF, and R conditions. The rows correspond to satellite dataset with RQ for RSS QuikSCAT, JQ for JPL QuikSCAT, RA for RSS ASCAT-A, and RW for RSS WindSat. Statistics were computed for all buoy wind speeds but excluding absolute wind direction differences between buoy and satellite >100°. Differences are defined as buoy minus scatterometer.

c. Comparison of DFAS_AW and AFDS_RF divergence and vorticity fields

Comparison statistics were computed for four regions demarcated in the map in Fig. 15, which include portions of the North Pacific, northwest Atlantic, Pacific ITCZ, and Agulhas Return Current in the Southern Ocean. These regions were chosen since they either contain regions with relatively frequent precipitation or are regions of relatively active synoptic weather variability. Time series of the RMSD, mean difference, and correlation coefficient between the monthly averaged DFAS_AW and AFDS_RF fields are shown for each region. The differences are defined as DFAS_AW minus AFDS_RF. These statistics were computed for the four datasets discussed above, and the divergence is shown by solid curves and vorticity by the dashed curves.

Fig. 15.

Comparison statistics of satellite DFAS_AW minus AFDS_RF divergence (solid curves) and vorticity (dashed curves) computed at each grid point within the four regions as shown in the map at the top of the figure. For each region, (top) the RMS difference, (middle) the mean difference, and (bottom) the correlation coefficient are shown. The four satellite datasets used are color coded as indicated in the top North Pacific panel. The mean difference was defined as DFAS_AW minus AFDS_RF.

Fig. 15.

Comparison statistics of satellite DFAS_AW minus AFDS_RF divergence (solid curves) and vorticity (dashed curves) computed at each grid point within the four regions as shown in the map at the top of the figure. For each region, (top) the RMS difference, (middle) the mean difference, and (bottom) the correlation coefficient are shown. The four satellite datasets used are color coded as indicated in the top North Pacific panel. The mean difference was defined as DFAS_AW minus AFDS_RF.

The RMSDs of the monthly averages are quite large, varying from about 0.5 × 10−5 s−1 during April over the eastern tropical Pacific to nearly 3 × 10−5 s−1 during February 2009 over the northwest Atlantic. Other than the Agulhas Return Current, the RMSD varies seasonally by roughly a factor of 2 for all datasets. Over the northwest Atlantic, the RMSDs vary by greater than a factor of 2 between March and July.

The RMSDs are generally largest for ASCAT-A and WindSat and smallest for the two QuikSCAT datasets. It is also generally smallest for the tropical Pacific and largest in the midlatitude regions. Part of the larger difference in ASCAT-A is due to its nadir gap, which introduces two additional swath edges per orbit and thus exacerbates swath edge errors in the AFDS method. The long 29-day repeat cycle of the ASCAT-A ground track distributes the swath edge artifacts over many longitudes, which contaminates many more grid points than with QuikSCAT or WindSat. Additionally, wider swaths have a smaller proportion of measurements at a swath edge than otherwise, so the narrower ASCAT-A and WindSat swaths will see larger errors in the AFDS method as a result of swath edge artifacts. DFAS divergence and vorticity do not suffer from this sampling error since spatial derivatives are never taken across swath edges.

The mean differences between the two methods are between about 0.1 and 0.2 × 10−5 s−1, which is fairly significant given the large geographical extent of the regions considered here. The cross-correlation coefficients are less than about 0.6 in the three midlatitude regions for all months, while it varies between 0.4 and 0.8 over the eastern tropical Pacific.

In summary, analysis of the satellite DFAS_AW divergence and vorticity fields shows that the differences between the AFDS_RF and DFAS_AW methods from the regional COAMPS analysis over the northwest Atlantic occur throughout the midlatitude World Ocean. The differences appear to be somewhat less significant in the tropical oceans, however.

d. Swath edge artifacts in AFDS divergence and vorticity

Another consideration of the AFDS method unique to satellite measurements is swath edges. Since these satellites all have repeating orbits (4 days in the case of QuikSCAT), swath edge artifacts often persist in time-averaged u and υ fields; spatial differentiation amplifies these artifacts in the AFDS method. Fortunately, since winds along swath edges are dynamically similar to anywhere else, these artifacts do not produce the same degree of error in the AFDS method as does rain. This is not an issue in the DFAS method because spatial derivatives are only computed within the swath and never across swath edges.

Effects from computing spatial derivatives across swath edges can be isolated independently from rain by computing the divergence and vorticity differences between the DFAS_AW and the AFDS_AW methods, as shown in Fig. 16. Swath edge artifacts appear as stripes in these figures. Swath edge artifacts in the AFDS_AW divergence and vorticity are largest poleward of roughly 20° latitude in all four datasets, although there are residual artifacts in the tropical convergence zones. ASCAT-A also has rather large, localized differences at about 40° latitude that appear to alternate sign. ASCAT-A and WindSat are the most affected by these artifacts, while the JPL QuikSCAT appears least affected. It is unclear why the AFDS derivative wind fields from the JPL QuikSCAT dataset are affected less than the RSS QuikSCAT dataset, although it may be related to the cross-track bias removal procedure employed in the JPL processing (Fore et al. 2014).

Fig. 16.

Maps of DFAS_AW minus AFDS_AW of (left) divergence and (right) vorticity for the 2-yr period August 2007–July 2009 from the four satellite vector wind datasets: (a),(e) RSS QuikSCAT; (b),(f) JPL QuikSCAT; (c),(g) ASCAT-A; and (d),(h) WindSat.

Fig. 16.

Maps of DFAS_AW minus AFDS_AW of (left) divergence and (right) vorticity for the 2-yr period August 2007–July 2009 from the four satellite vector wind datasets: (a),(e) RSS QuikSCAT; (b),(f) JPL QuikSCAT; (c),(g) ASCAT-A; and (d),(h) WindSat.

Swath edge artifacts are yet another drawback to using the AFDS method to compute the time-averaged divergence and vorticity from satellite vector winds. Note that the AFDS_RF and AFDS_AW methods are equally affected by these swath edge artifacts. To avoid these artifacts, the DFAS method should be used to compute the time-averaged divergence and vorticity.

e. Sverdrup transport estimates from the AFDS_RF and DFAS_AW wind stress curl

We demonstrate the differences in Sverdrup (Sv; 1 Sv ≡ 106 m3 s−1) transport streamfunction if the AFDS_RF wind stress curl is used to approximate the DFAS_AW wind stress curl. Figures 17a and 17b show the Sverdrup streamfunction computed from 2 yr of JPL QuikSCAT AW wind stress fields using the DFAS_AW and AFDS_RF methods, respectively. Figure 17c shows the difference in streamfunctions, defined as AFDS_RF minus DFAS_AW. Differences of 10–20 Sv are apparent in the Northern Hemisphere western boundary current regions, and larger, more widespread differences are also apparent in the Southern Ocean south of 40°S. There are also significant differences of 5–10 Sv in the western equatorial Pacific current system. This calculation shows a potential strong sensitivity of the wind-driven ocean circulation to the details of how the time-averaged wind stress curl is computed: specifically, the order in which the time-averaging and spatial differentiation operations are applied.

Fig. 17.

Sverdrup transport streamfunction computed from the JPL QuikSCAT v3 vector wind fields for the 2-yr period August 2007–July 2009 using the (a) DFAS_AW and (b) AFDS_RF methods. The contour interval is 10 Sv and the color scale runs from ±60 Sv. (c) The difference between (a) and (b) is defined as AFDS_RF minus DFAS_AW. The contour interval is 5 Sv and the color scale runs ±20 Sv. Dashed contours in all three panels indicate negative values of the streamfunction.

Fig. 17.

Sverdrup transport streamfunction computed from the JPL QuikSCAT v3 vector wind fields for the 2-yr period August 2007–July 2009 using the (a) DFAS_AW and (b) AFDS_RF methods. The contour interval is 10 Sv and the color scale runs from ±60 Sv. (c) The difference between (a) and (b) is defined as AFDS_RF minus DFAS_AW. The contour interval is 5 Sv and the color scale runs ±20 Sv. Dashed contours in all three panels indicate negative values of the streamfunction.

7. Summary and conclusions

Two methods were considered to compute the time-averaged divergence and vorticity from rain-free scatterometer winds, differing only in the order in which the time-averaging and spatial differentiation operations are applied. Figure 1 showed radically different time-averaged divergence and vorticity fields using rain-free winds depending on whether spatial derivatives were computed before time averaging (the DFAS_RF method) or vice versa (the AFDS_RF method). Because of missing data (either as a result of rain or swath edges), the time-mean divergence and vorticity from these two methods are not the same. The AFDS_RF method is appealing because it superficially appears to provide a good approximation of the all-weather divergence and vorticity fields. The goals of this analysis were to determine whether the AFDS_RF method really provides a plausible method to estimate the time-mean all-weather divergence and vorticity fields from rain-free winds and to resolve the reasons for the differences between the AFDS_RF and DFAS_RF methods. We showed an example using the surface divergence in Fig. 3 where the differences between the two methods led to two contradictory conclusions about the link between surface convergence and rainfall.

From one year of mesoscale weather simulations over the northwest Atlantic, we found that the AFDS_RF method produces time-averaged divergence and vorticity fields that have normalized RMS errors of 60%–120% for monthly averages and 25%–40% for annual averages. The AFDS_RF method also overestimates convergence along the mean Gulf Stream convergence zone by 50%–100%. The accuracy of the AFDS_RF method also varies by a factor of 2–3 between winter and summer, with less accurate estimates during rainy winter months. The approximation of the AW divergence and vorticity fields by the AFDS_RF method improves with temporal and spatial smoothing, although the gains in accuracy are limited beyond averaging periods of roughly 3 months and spatial smoothing beyond 125 km. Even so, the AFDS_RF method only provides marginally accurate estimates of the all-weather divergence and vorticity fields.

Besides showing the differences that occurred when changing the order in which the operations were applied, we also isolated why the AFDS_RF and DFAS_RF methods produced such different time-mean divergence and vorticity fields. The main difference between the AFDS_RF and DFAS_RF methods centers on the rain-free winds in grid cells bordering rain, which are included in the AFDS_RF method but not in the DFAS_RF method. The AFDS_RF method thus incorporates some information on the convergent, cyclonic winds bordering raining systems which is missing from the DFAS_RF method. We showed that the time-averaged convergence along the edges of raining systems can provide a rough estimate of the time-averaged convergence within the interiors of raining systems given a sufficiently long time-averaging window. The improvement in the AFDS_RF method stems mainly from improved representation of the spatial structure of convergence but not necessarily its absolute magnitude.

Our results show that the method of computing the time-mean divergence and vorticity from scatterometer winds is more sensitive to the calculation details than previously thought and that it is important for researchers to be clear about the calculation details. These details include the order in which the time-averaging and spatial differentiation operations are applied and whether any spatial smoothing has been applied to the time-averaged fields.

Acknowledgments

This work was supported by NASA Grants NNX11AF31G and NNX10AO93G for funding of NASA’s Ocean Vector Winds Science Team activities. We wish to thank Thomas Kilpatrick, Gregory King, and Ad Stoffelen for their thoughtful reviews of the manuscript. AMSR-E data are sponsored by the NASA Earth Science MEaSUREs DISCOVER Project and the AMSR-E Science Team. QuikSCAT data are sponsored by the NASA Ocean Vector Winds Science Team. Both the QuikSCAT and AMSR-E data used here were produced by Remote Sensing Systems. C-2013 ASCAT data are produced by Remote Sensing Systems and sponsored by the NASA Ocean Vector Winds Science Team. The JPL version 3 QuikSCAT dataset (https://podaac.jpl.nasa.gov/dataset/QSCAT_LEVEL_2B_OWV_COMP_12) was obtained via the PO.DAAC web portal and was processed by the SeaWinds Processing and Analysis Center (SeaPAC).

APPENDIX A

Procedure for Evaluation of COAMPS and Satellite Winds with Buoys

Vector winds from COAMPS and satellites are evaluated with moored buoys in the North Atlantic (for the case of COAMPS) and throughout the World Ocean (in the case of the satellites). This appendix describes details in the evaluation.

a. COAMPS

We evaluated the COAMPS 10-m wind forecasts with 11 buoys in the North Atlantic for the calendar year 2009. Locations of these buoys, as part of the NDBC and CDFO observation systems, are shown in Fig. 6a. They provided measurements at typically between 4- and 5-m height. The in situ buoy wind speeds were adjusted to 10-m height using the Coupled Ocean–Atmosphere Response Experiment (COARE) bulk flux algorithm version 3.0 (Fairall et al. 2003) with inputs of buoy air temperature, SST, air humidity, and air pressure. The height adjustment was not sensitive to the specific value of humidity and pressure, so for buoys missing either of these measurements, we used constant values of 75% and 1013 hPa, respectively. No 10-m wind speeds were computed if any of the buoy wind, air temperature, or SST observations were missing.

b. Satellite

Satellite wind retrievals are referenced to the 10-m ENW, which is the wind speed adjusted to 10-m height consistent with the observed surface stress referenced to neutral stability (e.g., Ross et al. 1985; Liu and Tang 1996). The methodology used to convert in situ buoy winds to the 10-m ENW follows previous satellite–buoy wind comparisons (e.g., Freilich and Dunbar 1999; Ebuchi et al. 2002; Chelton and Freilich 2005). Similar to the buoy winds speed adjustment above, we use the COARE bulk flux algorithm along with inputs of ancillary meteorological observations to convert the in situ buoy wind speeds to the 10-m ENW speeds. Comparisons were made only when a buoy observation was within ±30 minutes and a 25-km radius of a satellite observation.

The performance of the 10-m ENWs from the four satellite datasets are evaluated using buoys from the NDBC, Tropical Atmosphere Ocean/Triangle Trans-Ocean Buoy Network (TAO/TRITON), Prediction and Research Moored Array in the Tropical Atlantic project (PIRATA), Research Moored Array for African–Asian–Australian Monsoon Analysis and Prediction (RAMA), and CDFO arrays for the 2-yr period August 2007–July 2009. Buoy locations are shown in Fig. 6b, with the QuikSCAT rain frequency shown in grayscale. Because our emphasis is on winds in rain, as many buoys in relatively rainy areas were chosen as possible.

APPENDIX B

Statistical Significance of Differences between AFDS_RF and DFAS_RF Distributions

The statistical significance of the differences in distributions between the RF methods and the “true” AW monthly averages are assessed using the two-sample Kolmogorov–Smirnov (K–S) test (Kolmogorov 1933; Smirnov 1948), which is applied here to test the equivalence of the RF and AW distributions. The two-sample K–S test is a general and useful nonparametric method for comparing sample distributions and is sensitive to differences in both shape and location of the empirical distribution functions of the two samples. To allow for possible geographic variations of significance, we computed the test statistic in geographic squares encompassing 16 total grid points. Figure B1 shows maps of the results of the K–S test; black areas indicate regions where the RF and AW distributions have less than 5% probability of being equivalent. Red areas indicate areas where the RF and AW distributions have greater than 5% probability of being equivalent, and the hypothesis that the two distributions are equivalent cannot be rejected at this significance level. Few red points in Figs. B1a,b,d,e indicates that the AFDS_RF and DFAS_RF distributions are significantly different from the AW distributions. We can therefore reject the hypothesis that they are all drawn from the same distribution. The same cannot be said for the SM_AFDS_RF distributions, however, since roughly 40% of the grid squares pass the K–S test (Figs. B1c,f). Red points are scattered fairly randomly throughout the domain, showing that the rejection of equivalence between the SM_AFDS_RF and AW distributions is not uniform spatially. This result, however, indicates that the SM_AFDS_RF method provides better statistical approximation of the AW fields than either the AFDS_RF or DFAS_RF methods.

Fig. B1.

Maps of the results of the two-sample Kolmogorov–Smirnov test for the statistical equivalence of the COAMPS monthly averaged RF and AW wind derivative field distributions: (a)–(c) divergence and (d)–(f) vorticity. Red indicates regions where the hypothesis that the RF and AW wind derivative distributions are equivalent cannot be rejected at the 5% significance level. Black indicates regions where it can be rejected; meaning that the monthly average RF and AW wind derivative distributions have <5% probability of being drawn from the same distribution. The methods shown are (a),(d) the AFDS_RF, (b),(e) the DFAS_RF, and (c),(f) the SM_AFDS_RF. The test was performed in geographic squares consisting of 4 × 4 grid points thus yielding 16 × 12 = 192 samples in each square from the 12-month COAMPS simulation.

Fig. B1.

Maps of the results of the two-sample Kolmogorov–Smirnov test for the statistical equivalence of the COAMPS monthly averaged RF and AW wind derivative field distributions: (a)–(c) divergence and (d)–(f) vorticity. Red indicates regions where the hypothesis that the RF and AW wind derivative distributions are equivalent cannot be rejected at the 5% significance level. Black indicates regions where it can be rejected; meaning that the monthly average RF and AW wind derivative distributions have <5% probability of being drawn from the same distribution. The methods shown are (a),(d) the AFDS_RF, (b),(e) the DFAS_RF, and (c),(f) the SM_AFDS_RF. The test was performed in geographic squares consisting of 4 × 4 grid points thus yielding 16 × 12 = 192 samples in each square from the 12-month COAMPS simulation.

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Footnotes

*

Current Affiliation: Marine Meteorology Division, Naval Research Laboratory, Monterey, California.

This article is included in the Climate Implications of Frontal Scale Air–Sea Interaction Special Collection.

1

We refer to all-weather throughout this analysis as composed of both rain-free and raining conditions.

2

COAMPS is a registered trademark of the U.S. Naval Research Laboratory.

3

Skamarock (2004) (section 3a) suggests that NWP models have effective horizontal resolutions of about 7Δx, which is 63 km in these COAMPS simulations. Surface wind spectra from these simulations (not shown) suggest very similar behavior and a similar conclusion to Skamarock (2004). The COAMPS effective horizontal resolution is thus slightly longer than the resolution capabilities of the gridded scatterometer winds of ~40 km.