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version (appendices not included) updated 06/2005
Radar Data and Climatological Statistics Associated with Warm
Season Precipitation Episodes over the Continental U. S.
NCAR Technical Note  448+STR
D. A. Ahijevych, R. E. Carbone, J. D.
Tuttle, S. B. Trier
Table of Contents
Preface
Acknowledgements
1. Introduction
2. NOWrad™ MASTER15
Product
3. Construction of
Hovmöller Diagrams
4. Rain Streak Statistics
4.1 Zonal phase speed
of largescale forcing
4.2 Periodicity of rainfall streaks
in Hovmöller space
4.3 Zonal progression of rainfall
over the diurnal cycle
Tables
Appendix A. Rainfall
rate Hovmöller diagrams, 1996 through 2000
Appendix B. Power spectra
of rainfall rate Hovmöller diagrams
Appendix C. Daily histograms
of rainfall occurrence
Appendix D. Power spectra
of daily histograms of rainfall occurrence
References
This NCAR Technical Note supplements the paper "Inferences
of Predictability Associated with Warm Season Precipitation Episodes"
by Carbone et al. (2001). It includes a superset of the meridionallyaveraged
and zonallyaveraged rainfall rate diagrams that underlie the Carbone
et al. study and explains the construction of these Hovmöller diagrams.
Additional monthly and seasonal statistics related to the precipitation
structures found within the Hovmöller diagrams are computed.
The analysis period spans the warm season months of May through
August, 1996 through 2000.
This research was sponsored by National Science Foundation
support to the U.S. Weather Research Program. The authors
are deeply appreciative for assistance received from the National
Climatic Data Center, the website of NOAA/CIRES Climate Diagnostics
Center at U. of Colorado, S. Goodman of NASA's Marshall Spaceflight
Center, NOAA/CIRA at Colorado State University, NOAA/NESDIS, and
our sister organizations at NCAR/UCAR, namely the Research Applications
Program and COMET. These organizations provided critical access
to datasets, without which this study would have been impossible.
Comments from L. J. Miller of MMM/NCAR have also greatly improved
this Technical Note.
One of the biggest problems facing atmospheric scientists
today is the accurate representation and prediction of summertime
precipitation. While faster computers and better analytical
models have led to significant improvement in wintertime forecasts,
our inadequate representation of dynamical and microphysical processes
important to summertime convection has meant slower progress in
the arena of warm season quantitative precipitation forecasts.
This Technical Note supplements the Carbone et al.
(2001) study, for which meridionallyaveraged composites of national
radar reflectivity were used to compile seasonal statistics on the
zonal span, phase speed, duration and recurrence frequency of temporally
and spatially coherent rainfall episodes. At the upper end
of this episode spectrum are precipitation events that persist longer
than 24 h or have zonal span exceeding 1250 km. Carbone et
al. found that these longlived events are not exceedingly rare;
their recurrence interval is only two days. Their characteristic
phase speed cannot be explained either by transient synoptic disturbances
or steering winds, and their duration exceeds the lifespan of a
single mesoscale convective complex (e.g. Laing and Fritsch 1997).
Inspection of radar loops usually reveals a chain of mesoscale convective
systems (MCSs) responsible for the longer episodes (>36 h), but
further research is necessary to fully explain their longevity.
Twelve to 36h forecasts of summertime precipitation stand to benefit
from a better physical understanding of the linkage between these
successive MCSs. Interested readers should refer to Carbone
et al. (2001) for further motivation and background material.
The remainder of the document consists of three text
sections, a table section, plus four Appendices. The following
section describes the principal data set used in the Carbone et
al. (2001) study. The next section details the process of
constructing longitudetime and latitudetime sections from the
radar reflectivity data. The steps taken to analyze the longitudetime
sections (Hovmöller diagrams) are found in the final section.
The Appendices contain Hovmöller diagrams produced for the years
1996 through 2000 as well as related analyses for seasonal, monthly,
and daily periods.
Hovmoller data archive
This study relies heavily on a national radar reflectivity
composite produced by Weather Services International Corporation
(WSI). Known as the NOWrad™ MASTER15, this product is
synthesized every 15 minutes from all operational National Weather
Service (NWS) radars (the WSR88D network). The data undergoes
three levels of quality control at WSI: two at the singlesite level
and one at the national composite level. The first two levels
are completed via computer algorithms, while the third and final
stage involves a radar technician who attempts to manually remove
anomalous propagation echoes, ground clutter, and other miscellaneous
errors. WSI also produces a product every five minutes with
no manual corrections. However, the positive impact of the
third level of quality control is too great to justify using the
5minute product exclusively in our study. The 5minute resolution
product was used primarily to fill data gaps in our 15minute resolution
NOWrad™ product archive. These data gaps accounted for
only 0.4% of the total time reported here.
Using all available tilt angles, WSI merges WSR88D
radar reflectivity volumes into a twodimensional mosaic over the
U.S. with sixteen 5dBZ_{e} reflectivity levels, beginning
with level 0 (05 dBZ_{e}). Although the exact algorithm
by which WSI integrates singlesite radar data into a national composite
is proprietary information, it is generally understood that the
mosaic contains the highest reflectivity found within the vertical
column centered on each grid point. The reflectivity data
is mapped to a cylindrical equidistant projection with 1837 rows
and 3661 columns, and a grid spacing of approximately 2 km.
See Table 1 for additional NOWrad™ MASTER15 specifications.
Our NOWrad™ MASTER15 national radar composite
archive was obtained from the Global Hydrology Resource Center (GHRC)
at the Global Hydrology and Climate Center, Huntsville, Alabama.
The GHRC maintains a 5year archive of NOWrad™ MASTER15 products
dating back to 10 October 1995. At the time of this writing,
data may be accessed through the website: http://ghrc.msfc.nasa.gov/hydro.html
or by mail:
GHRC User Services Office
Global Hydrology and Climate Center
320 Sparkman Dr
Huntsville, AL 35805
Email:ghrc@eos.nasa.gov
In order to conserve disk space, GHRC reformatted
and stored the NOWrad™ data in Hierarchical Data Format (HDF)
format as run length encoded raster 8 images. Information
on decoding the HDF data structure is available on the GHRC web
site, and the HDF software library is at http://hdf.ncsa.uiuc.edu.
As stipulated by the contractual agreement between
the GHRC and WSI, use of this data set is restricted to educational
(including K12) or research, noncommercial purposes. Researchers
are prohibited from redistributing it. Therefore, interested
parties must contact the GHRC to access the national radar composites
used in this study.
The NOWrad™ gridded
radar data were used to create Hovmöller diagrams of estimated rainfall
rate over the contiguous U.S. for the warm seasons of 1996 to 2000.
In general, a Hovmöller diagram maps a scalar quantity to distancetime
space. The scalar variable is averaged along the spatial dimension
(or dimensions) orthogonal to the spatial dimension plotted in the
Hovmöller diagram. Among other uses, these diagrams have been
used in the past to diagnose patterns and isolate signals in equatorial
convection (e.g. Lau and Peng 1987; Hayashi and Nakazawa 1989).
Figure 1. This
flowchart illustrates the method used to compute longitudetime
sections of radarestimated rainfall rate (i.e. the zonal Hovmöller
diagrams.) See text for details.
In our case, we begin with a time series of radar
reflectivity mapped to a latitude/longitude grid (the NOWrad™
MASTER15 product). The two spatial dimensions are reduced
to one dimension using a procedure illustrated in Fig. 1.
First, the region of interest is divided into narrow striplike
subdivisions. After converting to rainfall rate, the arithmetic
average is taken of the rainfall rate values within each strip.
This average rainfall rate, in turn, becomes one data point in the
final Hovmöller diagram. In Fig. 1, the distance dimension
of the Hovmöller diagram happens to be longitude, making this a
zonal Hovmöller diagram. In the zonal Hovmöller diagrams,
meridional information is lost in the averaging process, effectively
eliminating this spatial dimension. Conversely, in the meridional
Hovmöller diagrams, zonal information is lost. Throughout
this document, distancetime section and Hovmöller diagram
are used interchangeably. The terms longitudetime section
and zonal Hovmöller diagram are also synonymous.
Figure 2. The
computational domain for the Hovmöller diagrams is delineated by
the solid, bold interior rectangle. The domain covers most
of the continental U.S. between the Rocky Mountains and the Appalachians.
Vertical lines within the computational domain represent the subdivisions
used to produce zonal Hovmöller diagrams. For clarity, the
subdivisions are shown with 1° width instead of their actual 0.05degree
width. There are actually 740 strips between 115 and 78
W. The meridional Hovmöller diagrams use the same computational
domain, except with 0.05degree subdivisions orthogonal to those shown
here. Threehundredsixty subdivisions for the meridional
Hovmöller diagrams lay between 30 and 48 N.
Our region of interest (delineated by the solid, bold
rectangle in Fig. 2) lies between 30 and 48 N latitude and between 115 and
78 W longitude. The vertical lines drawn across the domain
in Fig. 2 illustrate the orientation of the subdivisions for zonal
Hovmöller diagrams. For clarity, subdivisions are shown with
1°wide vertical strips whereas, in actuality, there are 740 strips
of width 0.05degrees between 115 and 78 W. Due to the convergence
of meridians, the zonal width varies from 4.8 km at 30 N to 3.7
km at 48 N. For the meridional Hovmöller diagrams, the domain
of interest is divided into subdivisions that stretch from 115
to 78 W and are 0.05° (5.6 km) in northsouth extent. Threehundredsixty
meridional subdivisions lay between 30 and 48 N.
In order to transform radar reflectivity factor in
dBZ_{e} to estimated rain rate R in mm h^{1}, the
logarithmic reflectivity was linearized to Z_{e} (mm^{6}m^{3})
and then converted to R using an exponential relationship:
Z_{e} = 300 R^{1.5}.
(1)
The multiplicative coefficient, 300, seems to render
a relatively small net bias in radarestimated national rainfall
when compared to rainfall analyses derived from rain gauges (Klazura
et al. 1999). While using a single Z_{e}R relationship
is fraught with uncertainty, quantitative precipitation estimation
was not central to our study. The radar data was primarily
used to determine the zonal span and duration of rainfall events.
Finally, to produce the actual Hovmöller diagrams,
the estimated rainfall rates were plotted in longitudetime (latitudetime)
space for the zonal (meridional) Hovmöller diagrams, with each rainfall
rate pixel centered between the longitude (latitude) bounds of its
representative subdivision. Zonal and meridional Hovmöller
diagrams for each of the warm seasons (May through August) from
1996 to 2000 are included in Appendix A.
We perform various analyses on the Hovmöller diagram
data to quantify rainfall event coherency, duration, zonal distance
(hereinafter referred to as "span"), and rate of propagation.
To identify events, rectangular autocorrelation functions
were fit to the rainfall rate data in Hovmöller space (illustrated
in Fig. 3). The bounded functions are uniform in one dimension
and cosineweighted in the other:
(2)
Its spatial dimensions and amplitude were chosen
to match that of an archetypal rainfall rate "streak"
segment in Hovmöller space. The two dimensional function was
centered on a given longitude/time coordinate and rotated at 1degree
intervals through all angles (except for slope zero) until the linear
correlation between the function weighting and the underlying rainfall
rate values was maximized. The function was stepped through
all longitude/time coordinates at 0.2°/15 min intervals. Sequences
of contiguous "fits" defined the coherent span, duration,
and propagation characteristics for each event. The minimum
rainfall rate threshold to initiate a "fit" was 0.1 mm
h^{1} and the correlation coefficient had to be 0.3 or
higher. The zonal span and duration of the streaks were determined
by the coordinates of the first and last "fit" in the
contiguous sequence. To convert from degrees longitude to
kilometers, we used the relation
,
(3)
which is valid at the center latitude of our domain
(39 N). Shown in Fig. 3 are several examples of "fits"
during a twoday period in June 1998.
Figure 3. Coherent
rainfall streaks were identified in the longitudetime sections
of rainfall rate by correlating the data with bounded, rectangular
autocorrelation functions. These functions are depicted at
the top of the figure, and their application is explained in the
text. Each function is constant in one dimension and cosineweighted
in the other, trailing off to minus one at its edges. The
straight, black lines in the Hovmöller diagrams are examples of
"fits" where the autocorrelation functions matched well
with the underlying rainfall rate. For clarity, not all fits
are shown. In the longitude dimension, one unit is equivalent
to 0.2°, while in the time dimension one unit equals 15 min.
For the purpose of zonal span and duration statistics
(as summarized in Tables 24), the uniformly weighted dimension
of the autocorrelation function was only 15 grid points, or ~3 degrees,
consistent with the size of an individual mesoscale convective system
and able to exceed the correlation threshold near the beginning
and end of a rainfall streak. Along the longitude axis, 15
grid points cover exactly 3°. For the purpose of propagation
statistics (e.g. Table 5), the uniform dimension was extended to
60 grid points, or ~12°, in order to have a stable measure of sustained
movement for significant rainfall episodes with a zonal span of
order 1000 km. The span of the autocorrelation function in
the cosineweighted dimension in both applications was 12 grid points,
equivalent to ~3 h rainfall duration at a given longitude.
The length, width, and shape of the autocorrelation functions are
somewhat arbitrary; however, adjusting these parameters does not
significantly alter the rainfall event statistics.
Most major episodes in Hovmöller space continuously
produce detectable precipitation throughout their lifetime.
However, some events exhibit intermittency while retaining phase
speed coherence. Consider an eastward propagating, dissipating
squall line whose remnant cold pool/gust front initiates another
MCS 100 km downstream. The two precipitation entities are
causally related and should be viewed as a single longlived precipitation
episode. However, our algorithm may accidentally classify
them as two separate events if the meridionallyaveraged rainfall
rate dips below 0.1 mm h^{1} before the second MCS forms.
This type of error was fixed by studying radar loops and determining
whether the two events were causally linked (perhaps by a cold pool
whose weak radar return was lost in the meridional average).
If the events were indeed linked, then the two rainfall streak segments
in Hovmöller space were reclassified as a single episode.
Subjectively determined corrections such as this, which took the
form of disconnects and reconnects of precipitation entities, occurred
in approximately 2% of all cases.
Figure 4. Rainfall
streak duration and zonal span data. Solid line indicates
the median phase speed (14.3 m s^{1}) for rainfall streaks
> 1000 km and 20 h. Dashed lines locate the 30 m
s^{1} and 7 ms^{1} constant phase speed lines.
Most "long" events fall between these two speeds.
The coordinates of each point in the scatter plot
in Fig. 4 represent the zonal span and duration of an individual
rainfall streak during the period May through August, 1997 to 2000.
The median zonal span/duration ratio for the population of rainfall
streaks with span >1000 km and duration >20 h was
14.3 m s^{1}, represented by the solid line in Fig. 4.
In Table 2, we employ recurrence frequency as the
means to express rainfall streak zonal span and duration for the
"longest" 10% of all rainfall episodes. Table 2
lists the rainfall streak zonal span (km) and duration (h) corresponding
to several recurrence frequencies.
Figure 5. Cumulative
probability histograms for zonal span and duration of rainfall streaks.
The period of record is May through August for the years 1997 through
2000. The cumulative distributions can be approximated by
the power laws described in the histograms.
The cumulative probability histogram of rainfall streak
zonal span, as defined earlier, is shown in Fig. 5 for the years
1997 through 2000, May through August. The histogram exhibits
a continuum of events with decreasing frequency out to 2600 km,
the approximate distance from the western Great Plains to the eastern
edge of the domain. For the purposes of parameterization,
the cumulative distribution may be approximated by a power law of
the form
(4)
where N(S) is the number of rainfall streaks
of zonal span > S (km), N_{0} is the total
number of streaks, and L
= 0.001 km^{1}.
Figure 5 also exhibits the cumulative probability
histogram of rainfall streak duration. As with rainfall streak
zonal span, the cumulative distribution for streak duration may
be approximated by a power law
(5)
where D is duration (h) and G = 0.05 h^{1}. Equations 4 and
5 can be combined to produce a characteristic phase speed
(6)
assuming N(S) = N(D).
Table 3 lists a pair of zonal phase speeds beneath
each year. Both speeds were derived by dividing a median zonal
span by a median duration for that particular year. The median
values for the first row were taken from all eastward moving rainfall
streaks (for a particular year), while the median values for the
second set of numbers were drawn from the population of rainfall
streaks with zonal span > 1000 km and duration >
20 h.
Table 4 summarizes three
additional sets of phase speed estimates, stratified according to
recurrence frequency. The top row within each recurrence frequency
is simply the span/duration ratio from the span, duration couplets
in Table 2. The middle row within each recurrence frequency
is the exceedance speed, defined as the span/duration ratio that
is equaled or exceeded with a particular frequency. For example,
during the 123day period of record in 1997, the span/duration ratio
for a rainfall streak exceeded 23.7 ms^{1} about 123 times,
or once per day.
The bottom row within each
recurrence frequency arose from scatter plots of rainfall streak
duration and zonal span for individual years. This process
is illustrated for 1997 in Fig. 6. First, the locations in
the scatter plot whose coordinates share a particular recurrence
frequency were established. As seen in Fig. 6, these anchor
points were marked with bold dots. Next, the constant phase
speed line was drawn whose slope equaled the median span/duration
ratio for long rainfall events (span > 1000 km and duration
> 20 h) for that year. On either side of each anchor
point, a pair of lines was drawn. Both lines were perpendicular
to the constant phase speed line and equidistant from their anchor
point. Note that all events along one of these lines share
a characteristic length, defined as a linear combination of duration
and zonal span. The lightly shaded bands in Fig. 6 that straddle
each anchor point do not overlap and each band encompasses up to
25 data points (depending on the density of the data points).
Once the bands were established, the median span/duration ratio
for each band was entered into the bottom row of phase speed estimates
for each recurrence frequency in Table 4. The "one per
month" category is not shown since there were almost no data
points at this recurrence frequency. The median span/duration
ratio within each recurrence frequency band was comparable to the
phase speed derived from the span, duration couplets in Table 2
because the sample of rainfall streaks within each particular recurrence
frequency band was not skewed towards fast or slow phase speeds.
Figure 6. Scatter
plot of rainfall streak duration and zonal span for May through
August 1997. Bold dots mark locations with coordinates that
share a particular recurrence frequency. The lightly shaded
bands are perpendicular to the phase speed line whose slope is equal
to the median zonal span/duration ratio for rainfall events >
1000km and 20 h. Each band encompasses data points within
a range of characteristic lengths—characteristic length being
a linear combination of duration and zonal span. The median
duration/zonal span ratio for data points within each band corresponds
to the bottom set of phase speed estimates in Table 4.
For the years 1997 through 2000, we examined both
zonal span and duration properties from 5406 rain streaks over 492
warm season days, or approximately 11 per day. In addition,
we measured phase speeds and compared them to the phase speed of
upper tropospheric anomalies and zonal "steeringlevel"
winds. This process is described in the next section.
Using images provided by the NOAACIRES Climate Diagnostics
Center, we examined 1998 and
1999 NCEP 2.5° gridded analyses in latitude bands corresponding
to the largest convective events. We compiled statistics on
the zonal speed of 30 kPa anomalies and "steering" winds
at 70, 50, 40, 30, and 25 kPa (Table 5).
Figure 7. Longitudetime
section depicting anomalies of meridional wind for the period 1
to 9 May 1999. Vwind anomalies have been averaged over the
latitude range 30 to 40 N. The zonal phase speed of the
meridional wind anomaly is about 4 m s^{1} in this case.
The zonal speed of 30kPa meridional wind anomalies
served as a proxy for zonal speed of synopticscale forcing.
We examined longitudetime sections of meridional wind anomalies
at 30 kPa, such as Fig. 7. This field usually revealed a phase
speed signal, though its amplitude and clarity varied. On
those occasions when meridional wind anomalies at 30 kPa failed
to reveal a clear phase speed signal, the examination of other upper
level anomalies (geopotential height, temperature, etc.) sometimes
helped define the phase speed.
The median zonal phase speed of synopticscale forcing
was drawn from 45 estimates. Each of the 45 phase speed estimates
represented the average zonal phase speed of a 30kPa anomaly over
a 3 to 7 day interval between 35 and 42.5 N latitude. The
1998 and 1999 period of record spanned 34 weeks from May through
August. An effort was made to divide the period into 3 to
7 day intervals such that the zonal phase speed of synopticscale
forcing within each interval was somewhat constant.
The combined 19981999 statistics were quite stable
and exhibited small differences. The mean phase speeds for
the 1998 and 1999 periods were 2.6 and 3.3 m s^{1}, respectively.
The standard deviation was 3 m s^{1} for both years, with
speeds ranging from 2 to 10 m s^{1}. As shown in
Table 5, the median zonal phase speed of synopticscale forcing
(30 kPa anomalies) was ~3 m s^{1}.
Median zonal wind speed at standard pressure levels
was found for 1week and 5° longitude intervals in the 35° to 42.5°
N latitude band. Most major convective events occurred within
this latitude range. Since organized convection is known to
occur with greater frequency just southward of stronger winds in
the upper troposphere (Laing and Fritsch 2000), the calculations
include departures to the 3035° N or 42.545° N latitude bands
as appropriate after inspection. These speeds (u) are in column
two of Table 5. For our study, "steering level"
is defined as the lowest standard pressure altitude where median
zonal wind speed, u, equals or exceeds the locally averaged rainfall
streak phase speed (U) within a 7day x 5degree longitude domain.
Remember that the locally averaged rainfall streak phase speed (U)
was obtained by applying the longer autocorrelation function (60
grid points or ~12 degrees) to the rainfall rate data. This eliminated
minor rainfall events from these phase speed statistics. A
high correlation between a particular rainfall streak and the long
autocorrelation function was only possible when the rainfall streak
span was at least 1000 km. In any given 7day x 5degree calculation
bin, there existed a spectrum of winds and rain streak phase speeds;
however, the aggregation of data was sufficient to capture gradients
associated with major changes while retaining at least a moderate
degree of stability within each bin.
Fourier analysis allows us to analyze the various
frequency components of a time series. For the theory behind
Fourier analysis and a more indepth treatment of the discrete Fourier
transform (DFT), the reader is referred to the extensive literature
(e.g. Morrison, 1994). In essence, the Fourier transform decomposes
a function f(t) into sinusoids of different frequencies w which integrate to the original function
,
(7)
where F(w)
is defined by
.
(8)
The complex function F(w) is called the Fourier transform of f(t).
It contains the complex amplitudes of the different frequency sinusoids
that compose the original function.
In the above relationships, the function f(t)
is assumed to be defined at all times. However, in reality,
we must work with time series constructed from measurements taken
at discrete times. Therefore, we turn to the discrete Fourier
transform or DFT (Morrison 1994, 323347). In general, the
k^{th} element f_{k} of an Nelement
time series f can be written as a linear combination of sinusoids.
(9)
where each coefficient F_{s} is equal
to
.
(10)
Together, the N coefficients define the DFT of f.
Each coefficient F_{s} represents the complex amplitude
of the s^{th} sinusoid component of f_{k},
while its magnitude equals the power associated with that component.
In frequency space, the difference between adjacent elements in
the DFT is equal to 1/(NT) where T is the time between
two consecutive elements in the time series. In other words,
the frequency resolution of the DFT is proportional to the number
of elements in the time series, N. Furthermore, the
highest resolvable frequency in the time series is 1/2T.
For our T=15 min data set, this corresponds to 1/30 min^{1}
or 5.556 x 10^{4} s^{1}.
Power spectral analyses permit the objective identification
of periodicity in the observational record over timescales from
less than one hour up to seasonal. DFTs were calculated for
the longitudetime sections of radar rainfall rate (the zonal Hovmöller
diagrams in Appendix A) at 0.2° longitude intervals. The daily
fluctuation of rainfall rate overwhelms any systematic change in
rainfall rate over the course of the warm season, so it was unnecessary
to apply a linear fit to the rainfall rate data and remove the trend.
From the DFTs, we obtained power spectra and used a sevenelement
boxcar average to smooth the spectra. The power spectra were
normalized such that they take the form of probability histograms.
Probability histograms for various periods (monthly and seasonal)
are found in Appendix B. Note that in Appendix B, the probabilities
have been scaled by 1000.
In addition to examining the periodicity of rainfall
rate for various longitudes, we produced daily histograms of rainfall
occurrence. This permits us to study the zonal progression
of rainfall coupled with the diurnal cycle. The regular occurrence
of rainfall at a particular longitude at the same time of day manifests
itself as a relative maximum in the daily histogram. The statistics
are mainly indicative of precipitation residence time at a specific
longitude and phase of the diurnal cycle. Coherent patterns
of rainfall occurrence in this coordinate system represent "phaselocked"
rainfall episodes. To construct the daily histograms, we counted
the number of times that the rainfall rate exceeded 0.1 mm h^{1}
(equivalent to a radar reflectivity of ~10 dBZ_{e}) at each
longitudetime coordinate with 0.2° and onehour resolution.
Histograms of rainfall occurrence have been compiled for monthly
and seasonal periods, in addition to multiyear periods. The
complete set resides in Appendix C.
Lastly, DFTs were applied to the time dimension of
the daily rainfall histograms in Appendix C. Since the histogram
period is 24 hours, the smallest nonzero frequency in the DFT is
1/day. This allows for greater sensitivity to spectral maxima
at frequencies higher than 1/day. As in Appendix B, the power
spectra in Appendix D were normalized such that they take the form
of probability histograms. The sum of probabilities across
all frequency bins for a particular longitude is equal to one.
A rectangular autocorrelation function, designed to
match the archetypal rainfall streak in Hovmöller space, was stepped
through the rainfall rate longitudetime sections at 0.2°, 15 min
intervals. Centered on each point having rainfall rate >
0.1 mm h^{1}, the function was rotated at 1° intervals
until the linear correlation between it and the rainfall rate underneath
was maximized. In order to count as a "fit," the
correlation coefficient had to exceed 0.3. Continuous sequences
of fits were designated as coherent rainfall events, subject to
manual verification. The endpoints determined the rainfall
event zonal span and duration. For the statistics below, the
autocorrelation function was 15 grid points (3°) long in its uniform
dimension and 12 grid points (3 h) in its cosineweighted dimension.
See Fig. 3 and related text for further details on the autocorrelation
functions and their application.
The total number of rainfall streaks for 19972000
is listed below, along with span and duration statistics stratified
according to recurrence frequency. For example, in 1997, for
the 123 days on record, there were approximately 123 rainfall streaks
with a zonal span > 850 km. This was a frequency
of one per day. In 1997, rainfall streaks lasting >
20 h also had a recurrence frequency of one per day. The final
column is simply the arithmetic mean of the yearly values.
Table 3 summarizes zonal phase speeds derived from
the rainfall streak span and duration statistics. The first
row of phase speeds consists of the median zonal span for a particular
year divided by the median duration (for all eastward moving rainfall
streaks). The second row shows similar ratios when the sample
is restricted to rainfall streaks with zonal span > 1000km
and duration > 20 h (i.e. large rainfall streaks).
Three additional sets of phase speed estimates are
listed in Table 4 for each year and recurrence frequency.
The first set, the span / duration ratio, is simply the ratio of
the span, duration pairs in Table 2. The middle set, the exceedance
speed, consists of span / duration ratios that are exceeded
with a particular frequency. For example, in 1997, the span/duration
ratio for a rainfall streak exceeded 23.7 ms^{1} about
once per day. The final set, the median span/duration ratio,
was determined by examining the duration, zonal span scatter plots
for each year. This procedure is explained in the text.
The final column is the arithmetic average of the numbers in the
yearly columns.
This table indicates propagation speed, or zonal phase
speed relative to background flow. The zonal phase speed of
rainfall streaks is compared to that of upperlevel synoptic features
and to wind speed at standard pressure levels. Once again,
all speeds are in m s^{1}.
The median zonal phase speed of synopticscale features
at 30 kPa is shown at the top of column two (u). Longitudetime
sections of NCEP 2.5° data from 1998 and 1999 were used to track
wind and height anomalies at 30 kPa for 3 to 7 day segments during
the months of May through August. The 34week period of record
was divided into 3 to 7 day periods for which the zonal phase speed
of upperlevel anomalies was somewhat steady. This is explained
in the text. As seen in the table, the median zonal phase
speed from the 45element sample was 3 m s^{1}.
The remainder of the second column lists median zonal
wind speed (u) retrieved from 1998 and 1999 NCEP 2.5° data.
Average zonal wind speeds were found for 5° longitude x 7day intervals
with the majority of wind speed measurements averaged over the 35°42.5°
N latitude band. The northsouth boundaries were occasionally
adjusted to follow the major convective events. Median phase
speeds of rainfall events (U) for matching 5° longitude x
7day intervals are listed in column three. As explained in
the text, these rainfall streak phase speeds were calculated using
the longer autocorrelation function (60 grid points or ~12°).
The fourth column is simply the difference between U and
u.
The final column is the percentage of 5° longitude
x 7day sections for which u exceeds U. For
example, the 70kPa zonal wind u was greater than the rainfall
streak phase speed U in 10% of the cases. In other
words, the "steering level" was at or below 70 kPa 10%
of the time. By the time one reaches 25 kPa, the likelihood
of finding a steering level at lower altitudes is 94%.
This appendix contains distancetime sections of radarestimated
rainfall rate over the U.S. during the warm season from May through
most of September. Dates are listed on the time axis (the
yaxis) along with UTC time in hours. A bold vertical line
separates the longitudetime section (zonal Hovmöller diagram) from
the latitudetime section (meridional Hovmöller diagram).
For the zonal Hovmöller diagram, rainfall rate was averaged over
the latitudes 30 N to 48 N, while an averaging range of 115 W
to 78 W was used for the meridional Hovmöller diagram.
The rainfall rate Hovmöller diagrams in Appendix A
were analyzed with a discrete Fourier transform (DFT) performed
on the time dimension at 0.2° intervals. The resultant power
spectra were normalized such that they take the form of probability
histograms. In other words, the probabilities sum to unity
over all frequency bins for a particular longitude. Since
the number of frequency bins is proportional to the number of elements
in the time series, the MayAugust histograms have four times the
frequency resolution of the singlemonth histograms. This
explains the lower probabilities in the MayAugust histograms; the
width of the frequency bins is only onefourth that of the singlemonth
histograms. Note that the probabilities have also been scaled
by 1000. Probability histograms for various periods (monthly
and seasonal) are found in Appendix B. They are plotted as
functions of longitude and frequency. Semidiurnal and diurnal
frequencies are marked on the frequency axis with letters "S"and
"D,"respectively.
Rainfall rate estimates from 1996 through 2000 were
used to create histograms of rainfall occurrence for various warm
season periods from May through August. Rainfall occurrence
is shown as a function of longitude and UTC time with 0.2° x 1h
resolution. The histograms indicate the number of times the
estimated rainfall rate exceeded 0.1 mm h^{1} in a particular
(0.2° x 1h) longitudeUTC time section. As mentioned in the
text, the rainfall rate data have been averaged in the meridional
dimension (30 to 48 N). There are histograms for single
month and multimonth periods, covering both specific years and
fiveyear periods. The histogram totals for fiveyear periods
have been scaled by one fifth to make them comparable to the singleyear
histograms.
For the MayAugust histograms, a value of 295 means
that rainfall was present at that (longitude, UTC time) coordinate
on ~60% of all days, while for the singlemonth histograms, a value
of 74 represents ~60% of all days. Lastly, a value of 110
in the 1 July15 August histograms represents ~60% of all days.
Ultimately, we are concerned with the distribution and variability
of rainfall occurrence in the histograms, not the absolute values.
See diurnal cycle composites.
Most of the rainfall occurrence histograms in Appendix
C were analyzed with a discrete Fourier transform performed on the
time dimension. The resultant power spectra as functions of
longitude and frequency are found in Appendix D. The power
spectra were converted to probability distributions such that the
sum of probabilities across all frequency bins equals unity for
any particular longitude. As in Appendix B, the probabilities
have been scaled by 1000. In addition, semidiurnal and diurnal
frequencies are marked on the frequency axis with letters "S"
and "D," respectively.
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