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Technical Reports of the MRI，No．392000
4．Algorithms for precipitation nowcasting focused on detailed anaIysis
using radar and raingauge data
4、1 1ntroduction
The Japan Meteorological Agency（JMA）began operating of a precipitationnowcastingsystem inApri11988．
One of the purposes of this study is to improve products this system provides．Since the algorithms an（1
accuracy of the pro（1ucts depend highly on the characteristics of observation instmments and the configuration
of the nowcasting system，we first outline the JMA nowcasting system。We then describe in detail the
algorithms and techniques developed in this study for detailed precipitation analysis．
The subjects included in Section4are：
（1）Newly determine（1representative values of digitized radar echo intensity levels suitable for precipitation
nowcasting，
（2）A method for improving radar estimates of precipitation by comparing data from multiple radars and
raingauges，
（3）Radar－estimate calibration by raingauge in view of Z－R relationship modification and appropriate
correspondence between calibration targets，
（4）Evaluation of the product for detailed precipitation analysis，called Radar－AMeDAS precipitation．
4．20utline of the JMA nowcasting system
Japan Meteorological Agency（JMA）precipitation nowcasting system went into operation in Apri11988，as
one of the components of the National Weather Watch（NWW）system，which was programmed to mitigate
disasters caused by heavy rainfa11，such as landslides，flash floods，and debris flows．The JMA nowcasting
system provides hourly precipitation charts on a5km grid，namely Radar－AMeDAS precipitation，hourly
precipitation forecast charts up to3hours，radar echo intensity composite，and radar echo top－height composite，
using data from conventional weather radars and a network of automated weather stations，called AMeDAS
（Forecast Division，1991；Makihara et a1．，1995）．These products are disseminated in digital form to Iocal
meteorological offices，TV stations，and meteorological consulting corporations about20minutes after each
hourly observation．
Features of the JMA nowcasting system are：
（1）A network of radars with a Moving Target Indicator（MTI）filter for rejecting ground clutterl
（2）A dense raingauge network with an average spacing of nearly one station per280km2employed for
calibrating precipitation estimates by ra（1arl
（3）Process domain of about1，000km by3，000kml
（4）Utilization of Numerical Weather Prediction（NWP）values for forecasts up to3hours aheadl
（5）Prediction in view of orographic effects often found in heavy rainfall eventsl
（6）A format of products identical to that of the digital topography or the digital river information issued by
the Geographical Survey Institute of Japan．
The domain for the nowcasting system is shown in Fig．4。2．1．A projection with a standard line slanting about
450against parallels and meridians，the so－called Biased－Lambert conical projection，is employed for the least
一63一
Technical Reports of the MRI，No．392000
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Fig．4．2．1DomainoftheJMAnowcastingsystemanddistortionrateondistance．Thethickrectangledenotesthedomain． Numbers along the thin arc lines are the distortion rates calcuIated by
（distance on map）／｛（actual distance）／（representative fraction of map）｝
distortion of direction and distance．By use of this projection，the maximum rate of distortion in the domain
is about1％（Makihara et a1．，1995）．
An example of a Radar－AMeDAS precipitation chart is shown in Fig．4．2．2，and a flowchart for on－1ine data
processing of the nowcasting system is shown in Fig．4．2。3．
4．3Data
The following data are used in the JMA precipitation nowcasting system：
（1） Radar data
Echo intensity
Echo－top height
Hourly precipitation estimate
（2） Raingauge data：
Hourly precipitation
（3）NWP data
Wind forecast at700hPa
Wind and temperature forecast at900hPa
Radar data are provided by the JMA radar network，and raingauge data by a network of automated surface
weather stations called Automated Meteorological Data Acquisition System（AMeDAS）．With the exception
of some radar data，all data are usually obtained hourly．
NWP data are used only in the forecasting process，so the detailed description is not included in this study．
一64一
Technical Reports of the MRI，No．392000
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of each pixel is3minutes of the latitude and3．75minutes of the longitude，which is equivalent to about5．5km by5。
5km in this area．
一65一
Technical Reports of the MRI，No。392000
AMeDAS1，300s詫es
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Fig．4．2．3A flowchart for on－1ine data processing ofthe JMA nowcasting system．
1ine，and single thin line denote outputs，inputs，and processes，respectively．
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2km3km4km5km
5km～ 8 － 1 一
Fig．4．3．1JMA weather radar network．Calculation in this study is made over the unhatched area，which is the entire
detection range of the radars．The observation heights of the radar beam are also indicated．The lowest value is
chosen as the observation height at a pixel where more than one radar observes precipitation．
一66一
Technical Reports of the MRI，No．392000
Table4．3。1．Specifications of the radars in JMA（March1999）
Ground－based radar 長》r generalSpectEcation
u oses VbssehadarStandard weather Mt．FujiRadarRadar JMA・MR JMA・MR－78
Number ofstations 18 1 1Frequency 5300MHz 2880MH：z 5300MHz（wavelength） （5．7cm） （10．4cm） （5．7cm）
Peakpower 250kW 1500kW 250kWPulse precipitation丘eq． 260Rz 160Hz 260：Hz
Antenna diameter 3．O m 5．O m 2．4mAntenna scan speed 4rpm 2rpm 4rpmEf飴ctiverange 400km 800km 400kmNumber ofREDIS・radars 18 1 1
4．3．1Radar data
The JMA operates19gromd－based weather radars as of March1999，and their entire detection range covers
almost all of the Japanese Islands（Fig4。3ユ）。Of these，18are the standard JMA type，and the other，Mt Fuji
Radar，is specialized（Table4．3．1）．
An automatic data processing system，the Radar Echo Digitizing an（1Disseminating System（REDIS〉，was
installed at all of these radars（Sakota，1990）．Radars with REDIS provide three types of information on
precipitation，1isted at the beginning of Section4．3，for the nowcasting system with a spatial resolution of5km
by5km．These data are transmitted to the Computer System for Meteorological Services（COSMETS）at
JMA Headquarters through dedicated lines．
Radars with REDIS are usually operated in two observation modes：continuous and3－hourly selected when
there is little precipitation within the coverage．In continuous mode，all types of data are observed every hour．
In the3－hourly mode，only echo intensity and echo top－height are measured at3－hour intervals．Either mode
is selected manually by the operator according to the forecaster’s judgment．
The JMA standard radar has the following features：
（1）5－cm conventional radar with no Doppler processing unit
（2）Beam width of about1．4。
（3）MTI filter installed for gromd clutter rejection（Tatehira and Shimizu，1987；Aoyagi，1983）
（4〉Height of up to2km for echo intensity observation with510w elevation angles
（5）Echo top－height estimated from observations with14elevations
（6）Hourly precipitation estimates accumulate（1at7．5－or10－minute intervals
The field of precipitation provided by the radar comprises a square domain on a5－km grid with sides of500
km（600km for Mt．Fuji Radar）．Echo intensity data from the lower five elevation angles are processedto give
an echo intensity field by using an allocation chart that indicates which elevation should be selected for the value
of every grid of the field．The allocation chart is made beforehand for each radar so that the altitude may be
the lowest，under the condition that the radar sampling volume should not have interference by mountains，and
it should not be contaminated by sea clutter．
Range correction on wave propagation and compensationfor the attenuation of wave intensityby air are made
一67一
Technical Reports of the MRI，No．392000
beforehand．Attenuation due to precipitation in the path of the radar，and due to a film of precipitation over
the radome，is not corrected．
Echo intensity Z in the mit of dBZ is converted to a precipitation rate in the unit of mm／h by the typical
relationship：Z二200ノ～L6（Marshall and Palmer，1948）．
An estimate of l－hour precipitation by radar，a radar－precipitation amount hereafter，is produced by averag－
ing echo intensities in mm／h，radar－precipitation rates hereafter，observed six or eight times during l hour．
Radar－precipitation rate，echo top－height，and radar－precipitation amount are sliced into16，9，and641evels，
respectively，The original grid size of these data is2．5km，but it is changed into5km before the data are
transmitted to the JMA forecast center，by choosing the maximum value among four pixels of a2．5km－square．
4．3．2AMeDAS
AMeDAS includes about1β20automatic surface weather stations．About8400f these，called four－parameter
stations，observe four meteorological parameters：1－hour precipitation，wind direction and speed，temperature，
and smshine duration per hour．The remaining480have only raingauges．The density of the raingauge
network is approximately one station per17km by17km．Data from these stations are collecte（1every hour
automatically and sent via public telephone lines to the AMeDAS computer center in Tokyo，where they are
edited and sent to COSMETS to be introduced into the nowcasting system．
4．4Newly determined representative values of digitized radar echo intensity levels suitable for precipitation
nowcastin9
4．4．1 1ntroduction
Continuous radar echo intensity acquired by JMA standard radars is converted into digital values which are
categorized in restricted numbers of levels．The representative value for each level should not be a unique one，
but should be determined according to the purpose for which the radar data are used．For example，in severe
rainfall watching，the maximum intherange ofa level issuitableforthepurpose．Incontrast，Radar－AMeDAS
precipitation and very－short－range forecasts up to3hours are used quantitatively in the hydro－meteorological
field．What values wouldbe appropriate，for example，whenwe derive rainfall total ortotal water contentfor
a drainage basinP Suppose the criteria be l mm／h，10mm／h，50mm／h．In genera1，events near l mm／h occur
more frequently than near10mm／h or50mm／h．Hence，rainfall total would be overestimated，if the represen－
tative value was fixed to the average of l mm／h and10mm／h，that is5．5mm／h，and summed in a large area
or for a long Period．
In this stu（1y，a set of representative values are determined in order that the statistical values of Radar－
AMeDAS precipitation and very－short－range forecast may be quantitatively equal to those derived from
continuous data．
4．4．2Data
Hourly radar precipitation amounts and radar precipitation rates with a5－km grid are used in this study．
Criteria for digitizing these continuous data are listed in Table4．4．1．Tanegashima Radar and fifteenradars to
the north of Tanegashima Radar from January1990to October1990are used in this stu（1y．Data of January
1991are used for Kushiro and Sendai Radar instead of those of January1990because their digitization was not
completed until Apri11990．
一68一
Technical Reports of the MRI，No．392000
4．4．3Method
Let the minimum and the maximum of a specific leve1，L，beαandδ，and the frequency of a specified value
劣betweenσand6for a certain period be∫（x）．Then，the total number of radar－precipitation amounts of L
in the period，F、δ，is described as follows：
孔イ∫（X）ぬ （4．4．・）
We detemine representative values of五and〃、δ，so that the total precipitation amomt estimated with〃、δ
and∫（x）will be the same as that derived from the sum of the continuous radar－precipitation amomts，as
follows：
M．δゴ∫（X）吋が（X）ぬ （4．4．2）
The data are accumulated for one month in this study．
4．4。3．1Determination using gamma distribution
In order to determine the representative values，we have to know details of∫（x）．’In this study，we apply a
gamma distribution，which is often used to represent rainfall distribution．
A gamma distribution is described as follows：
∫（刃）目灘たexp（一航）
where h，吻，andηare positive parameters．
In logarithmic form，the equation is described as：
ln（ノ（x））詔1n（規）＋え1n（x）一胤 （4。4．3）
Frequencies of hourly radar precipitation amounts for different levels are shown in Fig．4．4．1．Approxima－
tions to these frequency distributions are made with the gamma distribution function and are also shown in Fig．
4．4．1．
As Fig．4．4．1indicates，frequencies for ra（1ar－precipitation amounts are we11（1escribed by the gamma distribu－
tion．Those of the radar precipitation rate are described with only small errors（figures not shown）．Further－
more，as three continual levels are concemed，the distributions can be well described as a linear function，and
海1n（％）can be treated as constant．Under these assumptions，∫（x）is approximated as follows：
ノ（x）謂’πexp（一nx）
The representative value of a specific level between the valuesσandα十x，〃’砿，is then（1etermined as：
㌃∬＋x∫（∫）4f＋醒exp（一nf）／nll＋x （4・4・4）
∬＋xゲ（‘）4ご［’exp（一nご）】1－1
〃’ 呂 ＝ ＋一
㍑＋x∫（∫）…exp（一n∫）ll＋xn （4・4・5）
Likewise，the frequency of the data between6and6十ッ，Fめ，is describe（1as＝
罵ヅ＋醒exp（一nご）／n】1＋y （4・4・6）
一69一
Technical Reports of the MRI，No．392000
む器魯①占
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15
14
13
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109876543
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●一● Fujisan
▽一一▽Nigata
o＿■Fukui
の一・e Nagoya
く〉ゆOsaka
田一田Matsue
o・一〇（Fukui）
2 3 4 5 6 7 8 9 1 Leve10
（b）
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1n
15
14
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2109876543
111
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Radar
O一●Hiroshima▽…・▽ 八一uroto
巳一臨Fukuoka
＄一モ9Tanegashima
◇・一つ（Iliroshim a）
2 3 4 5 6 7 8 9 eaL10（c）
Fig．4．4．1Frequency distribution of hourly radar－precipitation amounts．Frequencies of each level are indicated for
different radars。Data were observed in January，Apri1，July，and October1990．Data observed with observation
height of2km and lower were used in the statistics．The vertical axis shows total numbers on a logarithmic scale，
and the horizontal axis shows levels on a linear precipitation intensity scale．
Fukui Radar and Hiroshima Radar in parenthesis are gamma distributions to approximate the actual distribution．
Parameters窺，ゐ，and％were detemined with the least squares method so that difference between the actual
distribution and the approximation may be the smallest。Specific values for彫，h，and％for Fukui Radar and
Hiroshima Radar are14．2，13．9，一4．7，一2．1，0，and O．1，respectively．
一70一
Technical Reports of the MRI，No．392000
From Eqs．（4．4．4）and（4．4．6），the following equation for錫is derived：
罵ツ［一exp（一脳）〆n】1＋㌔殊［一exp（一航）／nll＋y （4・4・7）
We detemine the parameter％by Newtonフs approximation，then，π砿，a function ofηin Eq．（4．4．5）．
4．4．3，2Determination of the value for the lowest IeveI
In Eq．（4．4．2），the minimum and the maximum of a specific level must be known．：However，the minimum of
the lowest level cannot be fixed because the value depends on the minimum detectable power of radar signals．
For that reason，we use raingauge data from AMeDAS．Actually，the right hand side of Eq．（4．4．2）is replaced
with the rainfall total measured from AMeDAS raingauges．
In this method，however，radar precipitation does not always correspond to the raingauge measurement on the
same grid for the following reasons：
（1）Raingauges represent values on a spot，while radar estimates spatial average values，so they sometimes
differ in a severe local storm event，
（2）Raindrops aloft observed with radar sometimes drift before reaching the ground，which causes the
difference of the corresponding grids between radar estimates and raingauge measurements．
To avoid this problem，the data are used only if the eight grids surrounding the center gridhave the same level
as the center gri（1．
This method is considered to be effective only when precipitation is from stratus clouds，so the method is used
only for the lowest level．The difference between this method and that in4．4．3．1for the second lowest level is
less than O．2mm for each radar station．
Another problem is that radar estimates are not equal to raingauge measurements．For this problem，we
assume that radar estimates of the lowest an（i the second lowest level can be calibrated with the corresponding
raingauge measurements with a parameter g as follows：
躍1Fl＝Rl／9 （4．4．8）
M2F2－R218 （4・4・9）
Where
〃1：representative value for leve11（lowest leve1）
F1：total mmber of level l
R1：total precipitation measured by AMeDAS raingauge
9：parameter．
The parameter g is detemined by substituting〃1、that is derived from Eqs．（4．4．5）and（4．4．7）into Eq．（4．4．
9）．
4．4．3．3Modification due to the change of grid size
Representative values for digitized levels of both radar precipitation rate and radar precipitation amount can
be determined individua11y from the algorithm in the previous sections．It is considered that although fluctua－
tions during one hour camot be described in the rainfall rate，the total sums for the two data for》a long time
or for a large area shouldbe equaL However，precipitationtotal from radar－precipitationrates is always larger
一71一
Technical Reports of the MRI，No．392000
than that from radar－precipitation amomts if those representative values are determined individually，as Fig．
4．4．3shows．In a very－short－range forecast，precipitation rate is used as the initial field．For using the
precipitation rate in the forecast，the two sets of representative values should produce statistically the sametotal
precipitation．In this section，the reason for this difference is discussed and a method for modification is
proposed．
The primary reason for this difference is considered to be the change of the radar grid size、In genera1，the
maximum of averages of radar precipitation rates in one－hour，that is radar－precipitation amount，is smaller
than the maximum of radar－precipitatiqn rate at any given time．For example，1et an isolated echo of10mm／
h with a size of2．5km square move eastward at a speed of5km／h，and the echo be on the gridαfrom time
渉toオ十30minutes．During the following30mimtes，the echo might be on the grid2．5km east of4．Then，we
consider the precipitation values for the period from渉to渉十60minutes in an area includingαand the grid2．5
km east ofα．The precipitation amount is calculated as5mm／h for two2．5－km grids，and the precipitationrate
indicates10mm／h at a2．5－km grid．Although the averages of four2．5－km grids are equa1（2．5mm／h），the
maximum of precipitation rate for a5－km grid is10mm／h and that of precipitation amount is5mm．
To make the difference clearer，the ratio of the frequency of radar observations with a10－km grid to that with
a5－km grid for different levels is presented for ra（1ar－precipitationrates andradar－precipitation amounts in Fig．
4．4．2．
In view of the above idea，representative values of precipitation rate are modified from the upper leve1，in
order，so that the precipitation totals from the highest to the specific level may be the same．
4．4．4Results
4．4．4．1Difference between central values and representative values
Table4．4．1shows the resulting representative values．For reference，central values are also indicated．
Representative values for radar－precipitation amomts，which are determine（i in Sections4．4．3．1and4．4．3．2are
smaller than central values by l to4％，and those for radar－precipitation rates by2to9％．Representative
values for radar－precipitation rates to be used in forecast are smaller by15to25％．
4．4，4。2Difference between precipitation total from radar precipitation rate and that from radar－precipitation
amount
Figure4．4．3shows the ratio of precipitation total estimated from radar－precipitation rates to that from radar
－precipitation amounts for each radar station．About20％overestimation is improve（1to3．5％by the process
in Section4．4．3．3．For several radar sites such as Hakodate，Akita，and Murotomisaki，a change of the
representative value of the lowest level in view of different minimum（ietectable powers of radar signals further
improves the overestimation．
4．4．5 Concluslons
Contimous values of1－hour precipitation amounts and precipitation rates that JMA radar provides are
digitized into64and161evels．For using these digital data quantitatively with JMA nowcasting system，
apPropriate sets of representative values for（iigital levels have been proposed．
The resulting representative values have the following features．
（1）Central values for radar－precipitation amount overestimate the actual precipitation by l to4％，while the
determined values are statistically almost the same as the actual ones．
一72一
Technical Reports of the MRI，No．392000
10km／5km2．2
2．0
1．8
1．6
1．4
1．2
L O
9．一“・・▽0●
．．．▽9●●
．．9▽
．．．▽9’
1 1 1 1 1 ㎞ve1
0 1 2 3 4
●一●Amount▽＿マRate
123456789Fig．4。4．2Ratio of frequency of radar observations with10－km grid to that with5－km grid for different levels．Numbers
of10－km grid are calibrated with the factor of4since the total number of grids decreased to1／4．The horizontal
axis shows levels for1－hour radar－precipitation amount．
Table4．4．1．Radar observation levels and representative values
One・hour radar一 reCl itationamount Radar． reCl itation rate
Level Range Central New Leve1 Range Central New Newmm出 value Value1 mm伍 value value1 Value2
1 ＜0．5 0．25 0．30 1 ＜ 1．0 0．75 0．67 0．67
2 0．5一 0．75 0．72
3 1．0・ 1．25 1．22 2 1．0一 1．50 1．42 1．28
4 1．5・ 1．75 1．73
5 2．0。 2．25 2．23 3 2．0・ 3．00 2．82 2．45
6 2．5・ 2．75 2．73
7 3．0一 3．50 3．45
8 4．0一 4．50 4．46 4 40・ 6．00 5．46 4．40
9 5．0・ 5．50 5．4610 6．0・ 6．50 6．4711 7．0・ 7．50 7．4712 8．0・ 8．50 8．47 5 8．0・ 10．00 9．67 8．4013 9．0一 9．50 9．4714 10．0・ 11．00 10．91
15 12．0一 13．00 12．92 6 12．0・ 14．00 13．79 12．40
16 14．0・ 15．00 14．93
17 16．0一 17．00 16．94 7 16．0・ 20．00 19．41 16．80
18 18．0一 19．00 18．94
19 20．0・ 21．00 20．95
20 22。0・ 23．00 22．95
21 24．0。 25．00 24．96 8 24．0・ 28．00 24．80
9 32．0一 36．00 32．80
40 62．0・ 10 40。0。 44．00 40．80
41 64．0。 11 48．0一 52．00 48．80
42 68．0。 12 56．0一 60．00 56．80
13 64．0・ 72．00 65．60
63 152．0・ 14 80．0・ 88．00 81．60
15 96．0一 126．00 99．20
Note：New valuel indicates results by the algorithm in Sections4。4．3．1．and4．4．3。2，
and New value2denotes those in Section4．4．3．3．
一73一
Technical Reports of the MRI，No。392000
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Fig．4．4．3Ratio of precipitation total calculated with radar－precipitation rates to that with1－hour radar－precipitation
amounts．The solid line，the dotted line，and the broken line show the ratio that used representative values derived
in Sections4．4。3．1and4．4．3。2，that used values calculated in Section4．4．3．3，and the ratio that used values in Section
4．4．3．3with the lowest level individually determined for every radar，respectively．
（2）Central values for radar－precipitation rate overestimate the actual by15to25％，and new values about
3．5％．The main reason for excessive central values is considered to be due to the change of the grid size
and the adoption of the maximum as the representative of four2．5－km pixels。
Individual values in actual situations vary in the extent that the same level allows，which means newly
detemined representative values are not always appropriate．However，in estimating areal precipitation total
and precipitation for a long time or for a large area，the representative values that this study proposes might
reduce statistical errors of those data．
4．5A Method for lmproving radar estimates of precipitation by comparing data from multiple radars and
raingauges
4．5．1 1ntroduction
Withthecapability ofcontinuous observation ofprecipitationover awide area，aswell asthefacilityforrea1
－time processing，weather radars are being utilized not only in the meteorological field，but also in the
hydrological field．To meet hydrological needs，however，the quantitative accuracy of radars，especially those
under constant operation，requires some improvement when compare（1with that of raingauges．
Studies have been made intensively for improvement in the accuracy of precipitation estimates byradar．One
proposal is to increase information by additional har（iware．It is well known that measurement of the size
distribution of the rain drops in a radar sampling volume provides a more accurate estimation of the precipita・
tion amomt in the volume．For example，Seliga and Bringi（1976）utilized polarized waves，while Doviak and
Zmic（1984）showed that the use of two different wavelengths can also determine the distribution．
Another approach is to obtain other kinds of information different from that of rain drops in the air．Brandes
一74一
Technical Reports of the MRI，No．392000
（1975〉calibrated a field of precipitation estimatedbyradar with data from denselydeployedraingauges．Colier
et a1。（1975）investigated the effectiveness of the density of raingauges to the calibration of radar estimates．
Austin（1987）stressed the importance of the change of the Z－R relationship according to the type of storm，On
the other hand，Joss（1990）proposed some correction methods based on the average vertical profiles of radar
echoes to obtain precipitation on the ground from the radar echo observed above the ground．Using average
seasonal correction fields，Takemura et a1．（1984）corrected the radar estimates of hourly precipitation over the
seatoproduceradar－AMeDASprecipitationcharts，whichwereproducedoperationallyusingJapanMeteorolog－
ical Agency（JMA）radars．
However，in compositing the estimates by the JMA radars，differences in intensity often arise along the
borders of the domains of the respective radars，especially over the sea，even if the estimate（1precipitation maps
are modified by the methods mentioned above．
Some of the reasons for the differences are：
1）Differences in height and volume of the radar beams which observe the precipitation having a vertical
profile in which the strength of precipitation changes with heightl
2）Differences among the sensitivities of radar receiversl
3）Modification made for a radar with no relation to another nearby radarl
4）Difficulty of modification by raingauges over the sea．
JMA operates19weather radars over an area of370，000km2withspacing about three times denser thanthat
of NEXRAD（Golden et al．，1986）．However，mountains often obstruct observations in Japan．As a result，there
are some areas where the distance between an observed point an（1a radar site is more than150km，even with
this dense radar network．The distance over which precipitation can be estimated accurately from the radar
equation without any problems，such as vertical difference of distribution of rain drops or a radar sampling
volume not filled up with rain（1rops，is within about100km．For larger distances，the errors（1escribed in1）
become dominant．
Regarding2），Takase et aL（1988）pointed out that ratios of radar estimates to the corresponding raingauge
measurements are often different from those by another radar even if specifications for those radars are the
same．Joss and Pittini（1991）also stated a similar conclusion．
This section proposes a method for calibrating radar estimates by comparing them not only with raingauge
measurements but also with radar estimates from other radars．This method eliminates the discontinuity in
ra（1ar echo compositing and improves the estimates over a wide ra（1ar detection area．This method is effective
for an area where errors base（i on the vertical profile of precipitation are dominant，especially over the sea，and
for a radar echo composite which needs smooth compositing of data from different radars．In this method，a
calibration factor is first described with two parameters，and the parameters are then（ietermined by the least
－squares method using hourly radar and raingauge data、
4．5．2．Data
Data of the digital radar network of the JMA from January1990to Febmary1993are used in this study．
Throughout the period，16digital radars（three ra（iars in the westem part of Japan were excluded）were
operated，and Okinawa Radar and Naze Radar were equipped with a digital processing unit in April1991and
April1990，respectively（Fig．4．3・1）。
一75一
Technical Reports of the MRI，No．392000
Elevations for the field of precipitation are selected so that the altitude is the lowest，under the condition that
the radar sampling volume should not have interference by momtains，and it should not be contaminate（i by sea
clutter．For the Z－R relationship，the values proposed by Marsha11－Palmer（1948），that is，200forβand1．6
forβ，are employed（Sakota1990）．
To obtain raingauge measurements for calibration，the raingauge network of AMeDAS is used。
Calculations in this study are conducted in the domain for JMA nowcasting system、
4．5．3．Relationship between radar beam height and the statistical field of precipitation observed by radar
Joss andWaldvoge1（1989）estimated the percentages of the precipitation that canbe observedby a radar at
various ranges for convective rain，widespread rain and snow，and low－1evel rain on the basis of the average
vertical reflectivity profiles．Although the percentages show different distributions，they decrease as the
（1istance from the radar increases，and the rate of decrease is larger for the precipitation with lower echo top
－height．Takemura6渉α1．（1984）also made a similar investigation using ra（1ar and raingauge data during6
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elevations，0．0。，1．0。，and2．0。，are used for the observation．Range circles are drawn at50－km intervals．Borders
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一76一
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．
Fig．4．5．2（a）Average hourly amounts for considerable precipitation observed by Fukui Radar．
Average amounts during September l990．Observations were usually made at1－hour intervals，and the intervals were
changed to3hours when there was little possibility ofprecipitation．The numbers inthe legends indicatethe average
hourly amounts ofprecipitation estimates that were the strongest2％of all the observationrecords duringthe period．
Range circles are drawn at50－km intervals．Over the sea，1arge rates are found along a170－km radius from the site．
Another peak is found around90－km radius．Both areas are below2－km observation height．
months in the warm season．They pointed out that the average field of calibration factors on land showed a
high correlation to the field of the minimum height through which radar beams pass without obstruction．They
applied this relationship to an algorithm，and produced radar－AMeDAS precipitation maps，in which the
calibration factor fie1（1for the radar was given as constant with time，except for areas where raingauges could
be used for the calibration by a correction method．This relationship will be clarified by another investigation
in this stu（iy．
Figure4．5．1indicates the field of height of the beam by which Fukui Radar，one of the JMA radars，observes
precipitation．Fukui Radar was using three elevations for intensity observation with an almost constant
altitude when the data in this study were observed．The heights over any area more than130km from the radar
are of the lowest elevation because sea clutter has no influence beyond the tangent point of the beam path of
一77一
Technical Reports of the MRI，No．392000
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Fig．4。5。2（b）Average hourly amounts for considerable precipitation observed by Fukui Radar．
The same as（a），except during January1990．There are areas of large amounts along140－km and90－km radius
from the radar site．There is another maximum around the center．
Fukui Ra（iar at the earthフs surface．Around110km from the radar，the second highest radar beam is adopted
to avoid sea clutter．
Figure4。5．2a is the field of the averages of hourly precipitation estimates for Fukui Radar which were
collected from the strongest2％of the observations during August1990．In other words，it is the field of the
averages of severe precipitation estimates during the month．Zero was assigned for locations where the
frequency of precipitation echoes did not reach2％of the observations．If all of the radar estimates were
accurate，the pattem would show the average precipitation amomts．However，the most obvious characteristic
is that the intensities are weak in the areas with high observation altitudes．Though it camot be denied that
precipitation happens to be weak in all the areas with high observation altitude，it is natural to regard，as Joss
and Waldvoge1（1989）stated，that the lesser amount is caused by weaker estimates due to smaller occupancy
of the radar sampling volume by rain drops，or to observing different contents of radar echoes along the vertical
direction・Discontinuity around the border of different observation angles seen along a circle with a radius of130
一78一
Technical Reports of the MRI，No。392000
一km from the ra（iar in Fig．4．5．2is considered to occur for this reason．
In the above investigation，only a part of the observations were used．On the grid with the1、igher observation
altitude，not only the intensity but also the frequency of detection decreases． It is not reasonable to include
those grids with very small detection rates in the data for the analysis because they may be observed only for
a particular restricted condition，such as precipitation from tall clouds．
The above relationship changes with season，as shown in Fig．4．5．2b．Because the average echo top is lower
in w三nter，a small increase in observation height often corresponds to small occupancy of the radar sampling
volume，and causes a large decrease in the precipitation estimate by radar。This result is consistent with those
of Joss and Waldovage1（1989）and Takemura et a1．（1984），
Now that we have investigated the nature of errors for the estimates，we consider how to modify the radar
estimates observe（1above the ground．The following assumptions are madel
（1）The vertical profile of an observed radar echo can be described by a functionl
（2）The ratio of a raingauge measurement to the corresponding radar estimate on the ground changes for
various reasons，such as the variability of the Z－R relationship，or the difficulty of ideal maintenance of
the sensitivity of operational radar hardware．
We represent the ratio of a raingauge measurement to the corresponding radar estimate observed，一F，as
F＝！媛 （4．5．1）
where。4，which is the ratio of the actual precipitation to the corresponding radar estimate on the ground，is a
parameter for assumption（2），and∫，which is the ratio ofthe radar estimate on the groundto the radar estimate
observed，is a function for assumption（1）．
We can derive∫in Eq．（4．5．1）from the distribution of the intensity of the radar beam and a function
representing the vertical profile of an observed radar echo．The intensity of a radar beam at a specified angle
is known to be described by an exponential function as follows：
1－2一（θノθ晦）2，
where
θ：the angle from the center of a radar beam，
θh：the angle where the strength decreases to half of that at the center．
The intensity I of the radar beam illustrated in Fig．4．5．3is then described as：
1（z，ン，R，β，θね）一exp［一1n2｛（z一β）2＋y2｝／｛Rtanθ＾｝21， （4．5．2）
where
z：height of observation，
y：distance from the center in the direction normal to both the vertical and beam directions，
1～：（1istance from the radar，
β：altitude of the center of the radar beam，which is calculated from the altitude of the radar site，the angle
・felevati・n，andR，bythepr・paga㌻i・nequati・n・
In the above equation，the path of the radar beam is assume（1to be almost horizonta1．
一79一
Technical Reports of the MRI，No．392000
Further we assume that the vertical profile of precipitation be denoted by：
V（z，T，α）昌｛（T－z）／T｝118（4．5．3）
where
T＝echo top height，
召：parameter for determining the shape of the profile．
The intensity of the profile is l at the ground，and O at the echo top－height．Figure4．5。4shows vertica1
ム〃
zRadarpu蓋8e
β・一一＿臨
R一一一→
Fig．4．5．3Geometry of the radar beam used in Eq．（4．5．2）．
0．96
3
軌
三〇〇一。＝
0．6
0．4
0．2
一■＆30．5
ロ圏a31．0
－a＝2．0
一40 －30 －20 －10 0 dB
Intenslty
Fig．4．5．4Vertical profiles used for calculating the calibration factor．The vertical profile of precipitation with T km of
echo top－height is described by｛（T－z）／T｝1／a，where z is radar beam－height．In the figure，radar beam－height is
described as the rate at the echo top－height．Hence，the vertical axis shows2／T．Lines in the figure are for
different values of召，that is，0．5，1．0，and2。0。Intensity is in dB．
一80一
Technical Reports of the MRI，No．392000
profiles described by the above equation for three different values of召．When召is l，the intensity decreases
linearly in proportion to altitudel this is the assumption Andrieu and Creutin（1991）adopted for the vertical
profile with no bright－band effect．By changing a from O．5to2，most of the profiles investigated by Joss and
Waldvoge1（1989）can be apProximated．
With these equations，∫is represented as：
ノ（R，瓦θ，，T，α）、」趣＿
窟・（z，y，7，β）V（z，T，α）み4z
窟・xpl－ln餐識y2｝み4z （4a4）
㍑exp［一1nl繍許y2｝】｛（T－z）／T｝・！・帥
where
7：size of the radar main beam defined by R tan（θh）．
Here，integrals are calculated for the area where the main beam passes．
In Eq．（4，5．4），the echo top－height T and the parameterσfor the vertical profile are needed，and they are
included as a fraction an（1as an integral format．Because a simple form of∫is desirable for the process
described later，an approximated one is proposed，Considering that∫is almost l near the radar site，and begins
to grow rapidly at a certain distance，∫is approximated by the following function：
∫（X，β）＝（1＋X82） （4．5．5）
Itshouldbenotedthatβisafunctionoflocationonly，andisindependentoftime，whiletheparameterXis
treated as a variable of time only．
In this approximation，θh is not used because it is constant for all radars．Distance1～has some effect on∫，
although the effect is not as large as that of B．We don’t include the effect of R inthe approximationso that
we may get sufficient results with as small a number of parameters as possible when we camot know all of the
detailed vertical profiles of radar echoes．
TheparameterXrepresentstheeffectofradarsamplingheightandanechoprofiledescribedwithTandα
in Eq．（4．5．4）on the calibration factor．By introducing X，radar estimates are well calibrated when the ratio
of the echo intensity observed by radar to that on the ground decreases as the radar beam becomes higher．
The fmction∫depends on B only through its square．The form ofβ2was selected as a result ofthe following
investigation，which was made to evaluate the performance of B to the power of different orders．For this
evaluation，the correction factor described by Eq．（4，5。4）was approximated using the least－squares method．In
Fig．4．5．5，calibration factors for various distances from the radar site are approximated by three lines described
as∫＝（1十X8々），whereゐis1，2，0r3fortherespectivelines。TheparameterXforeachlinewasdetemined
with the least－squares method to minimize total difference between the values from Eq．（4。5．4）and those from
（1＋Xβ々）．
For the various situations shown in Fig．4．5．5，where the parameterσin Eq．（4．5．3）is O．5and2，and echo toP
1一
Technical Reports of the MRI，No。392000
X O．1
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Fig．4．5．5Approximation to the calibration factor variable according to radar beam－height．The calibration factor
calculated by Eq．（4．5．4）for the vertical profile in Fig．4．5．4，is approximated with lines described by1十㎜κy．The
line describing calibration factors for different beam heights is denoted by mmber1，and the approximations are by
2，3，and4，where the value of y is1．0，2．0，and3．0，respectivelyl窺for each approximated line is determined by the
least－squares method so that the Iine may be closest to the Iine1．In this study O．7。is used forθh．The beam is shot
from1－km height with O。elevation angle．The altitude of the beam becomes higher as the effect of the earthys
effective radius becomes apparent．Figure4．5．5a，4．5．5b，4．5．5c，and4．5．5．d are for4km，4km，6km，and6km ofecho
top－height and for O．5，2．0，0．5，and2。O of a，respectively．Comparisons for the approximation are made at10－km
intervals from the radar site for the points where the height of the radar beam is lower than4km．
一heights are4km an（16km，the approximation with the square of B shows the least difference as a whole．
The other parameter∠4is also a variable of time only．JMA radars are mdergoing careful maintenance，as
are many others．There are，however，several difficult problems，even with the best maintenance．For
example，there are large differences in the installation dates of the radars，reaching over more than10years in
some cases，which leads to different types of parts used for the same function of those radars．As a result，it
is almost impossible for the hardware，especially any working operationally，to be in strictly ideal condition for
all the ra（1ars at all times．Consequently，there are errors in the measurement of the intensity reflected by
raindrops．A compensation for those errors can be included in∠4．It should be noted that those errors are
represented as a function of time only，and independent of location，although they are different among the
respective radarS．
Variability of the Z－R relationship also affects且．The variability is considered as a function of locati6n，so
且，which is constant for the entire detection area of the ra（1ar，cannot represent a detailed distribution of the
一82一
Technical Reports of the MRI，No．392000
calibration factor field．However，∠4can be changed for the variability in different storm types which are
almost identical in their detection area and change gradually with time．Thus，this variability is better
described by being included in∠4as the average value at the current time．
Consequently，F in Eq。（4。5。1）is described as a function of∠4，X，andβby combining Eq．（4．5．5）as follows：
F（オ，X，β）一オ（1＋Xθ2） （4．5．6）
4．5．4，Algorithm
4、5．4、1．Necessary conditions for the calibrating parameters
The following conditions are proposedfor parameters∠4s and Xs definedbyEq．（4．5．6），whichdifferent radars
should satisfy．
1）Since radars have several errors in observation，precipitation estimates from different radars for a point
where radar observations overlap are，in genera1，different．Their modified values，however，should be
equa1：
オ、（1＋X、砿）E、た冨4（1＋X澱）E厩 （4．5．7）
where
β説：radar beam height of Radar召at the々一th point，
E励：estimate of1－hour precipitation amount by Radarα，at theゐ一th point．
Because the number of such points is usually not sma11，the following sum of the differences betweenboth si（1es
of Eq．（4．5．7）in the whole overlapping area，described as五（X，且），should be a minimum＝
」1一羽屡｛ln（4（1＋礁）E・た）一ln（オわ（1＋Xみβ姦）Eわた）｝2 （4．5．8）
一写溺｛4一（イ1＋β・みた）｝2（4．5．9）
where
オσ31n（孟、），
β、わ是胃ln【｛（1＋Xみ壕）Eゐた｝／｛（1＋X、β姦）E．た｝
The major reason for adopting the logarithm in Eq．（4．5．8）is to avoid unreasonable solutions．If Eq．（4．5．8）
were described without the logarithm，zero might be another solution for alh4s，and in most cases with some
residue for五，zero might be the only solution，which means all ofthe modifiedradar estimates become zero．The
logarithm might tend to place a high weight on a small precipitation rate in determining the correction factor．
However，a small estimate of a radar is not always small when it is observedby another radar，and this tendency
can be modified，if necessary，by placing a high weight on samplings with a large precipitation rate，to a certain
extent．
2）By transforming Eq．（4．5．7），the following relationship is obtained：
｛（1＋X、βα）E、，｝／｛（1＋X議）Eわ，｝鴫1オ， （4．5．10）
The right－hand side of Eq．（4．5．10）is composed of parameters independent of location，and the number of the
一83一
Technical Reports of the MRI，No．392000
points that should satisfy Eq．（4．5．10）is not smal1．In other wor（is，the value of the left－hand side，representing
the ratio of a modified estimate from Radar a to that from Radar b，is constant for all the overlapping area after
modification by X．It directly represents the condition that should be satisfied by estimates modified for the
vertical differences．Under this condition，the standard deviation of the samples of the left－hand side for the
overlapping points should be the minimum．The following should also be taken into account before the
formulation of this condition：
（1）A small average of the left－hand side can also make this standard deviation sma11．For this reason，the
magnitude of the average should be normalized．
（2）Since the situation is the same for Radar a and Radar b，it is desirable for the constraint for each of them
to be equa1．
Consequently，we adopt the followingゐ（X、，Xδ）as the value which should be minimum：
麦写（α謝2一｛孚（α・走／わゐ陀）／K｝2
」2（X、，Xふ）留 ｛孚（ααた／わゐ是）／κ｝2
1 π孚（わわた／α・た）2一｛孚（わわた／α・た）／κ｝2
十 ｛孚（わわ是／ααた）／κ｝2 （4●5’11〉
写（ααた／わわ髭）2 写（δみた／α盈）2
3κ 一1＋κ 一1 ｛孚（ασた／わわ走）｝2 ｛写（わわた／α・走）｝2
where
α、是冨（1＋X、砿）E、た一E、疋＋h．たX、，
娠胃（1＋Xみ峨）Eδた一Eみた＋hわをXわ，
h、走一峨E、た，
K：total number of points，鳶．
InderivingEq，（4．5．11）from Eq．（4．5．10），the logarithm wasnot employed．This isbased onthe nature of X．
TheparameterXmodifiesmainlytheestimatesthatareobservedwithahighbeam－elevationalthough且
modifies all of the estimates with the same weight．In order to place a large weight on data with large
deviations，which is attributed mainly to sampleswith a highbeam－elevation，wehavenot adoptedthe logarithm，
because it causes a lower evaluation of the data with large deviations．
It should be noted that the objective v31ues for Xs inゐare generally different from those inみ，althoughboth
／i andゐare based upon Eq．（4．5．7）
The constraint onゐdirectly determines Xs，so Xscanmodifythevertical differences ofradar estimates．On
the other hand，五fixes Xs as one ofthe components that modify estimates from different radars into those with
the least differences from each other。For example，let us consider three overlapping points．The value ofゐ
is not，in genera1，zero simply because the mmber of parameters is two，while／1can be zero with four
一84一
Technical Reports of the MRI，No．392000
parameters．When Xs determined byゐare used，孟cannot be fixed to zero becauseβ、δたin Eq．（4．5．9）cannot
be treated as constant owing to the deviation still remaining for modifiedsamples．Therefore，the values of Xs
forゐare different from those ofみ In order to be accurate for the entire radar－detecting area，which includes
points where radar observations（10not overlap，Xs detemined by丑are considered better than those by五．
3）On a gridwhere an AMeDAS station is located，the calibrationfactor shouldequal theratio ofthe AMeDAS
measurement to the ra（1ar estimate：
オ、（1＋X、β孟）一R、，／E、， （4．5．12）
where
R。，：raingauge reading at theズーth point in the detection range of Radarα，
∠4、：∠4forRadarα，
X、：X for Radarσ，
β、，：radar beam height of Radarσat theづ一th point，
E、∫：estimate of1－hour precipitation amount by Radarαat the歪一th point．
Becausethere are manyAMeDAS stations to be comparedwithinthelandarea ofoneradar’s detectionrange，
the following summation，ゐ（且，X），obtained by transformation of Eq．（4．5．12）should be a minimum：
」3一ΣΣ［1n｛4（1＋X・環）｝一1n（R．’／E．f）】2
α 1
一ΣΣ（4一δ．，）2
σ 8
（4．5．13）
（4．5．14）
where
ロオσ冒1n（オ、），
δσ，一1n［R4，1｛（1＋招盈）Eα，｝】．
Since Eq．（4．5．13）directly compares estimates with the actual measurements，Xs and∠4s can be derived only
by this equation．The areas for this comparison，however，are limited to land only，and there are much fewer
points for comparison than for Eq．（4．5．8）or Eq．（4．5．11）．
4，5，4．2．OutIine of the procedure
For the procedure in this study，Xs are determined firstly by Eq．（4．5．11）only，according to the consideration
in Section4．5．4．1．Then，As are derived using Eqs．（4．5．9）and（4．5．14）（Fig．4．5．6）．
Uncertainty still remains，however，in determining the parameters，e〉en after Eqs．（4．5．9），（4．5．11），and（4．5．
14）are defined．One uncertainty is in determining what weights are givento Eqs．（4．5．9）and（4．5．14）inderiving
As．The other is that points of sampling are not specifie（1．If we take all the overlapping points of the samples
for Eq．（4．5．11），when most Bs are low，data with a highbeam－altitude may contribute little to determining X．
ThisXmaycauselargeerrorsinareaswithahighsamplingaltitude．Detailedstrategiesforweightingandfor
selecting samples are described in the following sections and appendices。
To better understand the procedure，Fig．4．5．7presents the idea and a schematic explanation of the calibration
forXand∠4．
一85一
Technical Reports of the MRI，No．392000
4．5．4．3、Determination of X by successive approximation
4．5．4．3．1．Relationship between two Xs
The detemination of Xs starts by deriving the relationship between only two radars．
Let the specified radars beαand猷and Xもbe fixed to Xも＊as known．The first estimate of the successive
approximation，Xも＊，is obtained by averaging the data for the last240hours，with higher weighting for the last
6hours．The most appropriate瓦for Xも＊is derived from the actual comparison ofゐ’s for various values of
瓦havingdifferencesofatleastO．005fromeachother．Wenowdescribethemostappropriate＆andゐas瓦＊
（瓦＊）andゐa（瓦＊）二ゐ、（＆＊（Xも＊），Xも＊）ンrespectively．
Taking into account that＆＊derived in this process depends on the value of凡＊，we calculate three different
X益＊’s andゐ、’s for瓦＊一△，X』＊，and凡＊十△，respectively，where△is a small positive number．
4、5．4。3．2．Estimation of a”Xs using the relationship between two radars
Let Xも＊be the minimum ofゐays among the three appropriate candidates in Section4．5．4．3．1．Then，瓦＊is an
appropriate estimate for Xもamong them．However，we may have a different candidate，if the corresponding
radar is changed from4to another．Therefore，relationships with other radars should also be taken into
account．In order to include these relations for many radars in the expression in which calculation of these
parameters is easy，we change the relationship between＆and Xもinto a linear one．In this expression，it is
assumedthat X』c＆nbe represented as a linear function of瓦when Xもis near瓦＊．Under this assumption，the
following equation for X』and瓦is derived with two groups of凡＊and瓦＊producing smallerゐa：
ゆ ゆ ゆ ゆ ・・X、（Xドム）一X、（Xみ） ・ X、冨X、（Xみ）＋ ． ． ｛XドXわ｝ ｛Xみ一△｝一Xみ （4．5．15）
＝CαわXわ＋Dαわ，
Derivingκ
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Radar estimates
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Fig．4．5．6SchematicflowforthealgorithmforderivingXandイ4．ValuesofXarederivedatfirstonlywithradar estimates and archives of X．Detemination of the values of∠4then starts with obtaining a second guess of／1，that
is，the ratio of raingauge measurements to radar estimates calibrated with Xs．Finally，with this ratio and the
relationship among radar estimates over their overlapping area，！1s，are obtained．
一86一
Technical Reports of the MRI，No．392000
where bothゐa（瓦＊一△）andゐ、（Xら＊）are assumed to be smaller thanゐ、（Xも＊十△）．
With Eq．（4．5ユ5），the samevalue as detemined in Section4．5．4．3．1is derivedfor＆when Xもis（瓦＊一△）and
Xも＊．
Equation（4．5．15）canbe obtained for many radars，but瓦＊（Xも＊）and estimates of凡derivedfrom comparison
L （a）
HRADAR A RADAR B
L
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RADAR A RADAR B
Fig，4。5．7 Calibration methodology．Figure4．5．7a shows an areal distribution of precipitation．A large amount is
observed around Radar“A．” F至gures4．5．7b and4．5．7c are cross－sections along the thick line in Fig．4．5．7a．Figure
4．5．7billustrateshowXmodifiestheoriginalradarestimates．Inthefigure，thethinlines，thethicklines，andthe lines with moderate thickness denote the actual intensities，original radar estimates，and estimates modified by．Xs，
respectively．Vertical single lines with arrows show differences between original estimates by Radar A，and those
byRadarB．TheparameterXmodifiestheestimatessuchthatthosedifferencescanbeequalwithoutdependence of location，as vertical double lines with arrows show．On the other hand，the parameter∠4calibrates the estimates
modifiedbyXsuchthatthemodifiedlinesmaybealongtheactualintensities，asinFig．4．5．7c．
一87一
Technical Reports of the MRI，No．392000
with many other radars are not equa1．Therefore，some difference arises between both sides of Eq。（4．5ユ5）if
an estimate of X』，except for凡＊（瓦＊），is substituted for瓦．In view of this situation，we detemine Xsロsuch
that the sum of the differences between the values of both sides of Eq．（4．5．15）for various estimates of Xs is
minimum．
Actually，we obtain the following equation to describe the relationship between Radarσand Radar6in
Section4．5．4．3．1：
L2、6冒ε、み（X、一C、わXみ一P、ゐ）2＋∫、わ（X、一M、み）2＋η、み（Xわ一N、み）2， （4・5・16）
whereε．b，ζゑb，η、b are weighting coefficients，and〃』b and醜b are X』＊and X』＊，respectively．
Equation（4．5．16）indicates：
（1）An apPropriate pair of X』and X』is〃』b and．〈盈b，
（2）Even if they differ from〃直b and Nab，they should satisfy the relationship of the first term described by Cab
and Pab to keepゐsma1L
The last estimates of Xs are derived as those making minimum the total sum of L2ab’s：
五2一》Σ｛εめ（Xα一CめXb－Dαb）2＋‘・6（X召一M“5）2＋．ηαb（Xわ一Nめ）2｝ （4・5ユ7）
Because Eq．（4．5．17）is a quadratic format regarding Xs，the following linear equation is obtained by
differentiating L2with reSpect to Xm：
轟｛（ε励＋∫功み）x遡一ε論xわ＋（一ε励D所ド∫轟）｝
（4．5．18）
＋識｛一ε一6Cわ爪Xわ＋（ε一C島＋ηαゐ）X烈＋（ε一6Cゐ徊Dわ例一ηαδ川Nb醒）｝一〇
Consequently，Xs are detemined by solving the above simultaneous linear equations．The actual weighting
strategy is described in Appendix（1）．
4．5．4．3．3．lteration
The values of Xs detemined in the scheme from Section4．5．4．3．l to Section4．5．4．3．2depend on the first guess
of Xs．To obtain more reliable values，the procedures are repeated three times，replacing the first guesses Xも＊
by the estimates determined by Eq．（4．5．18）．
4，5．4．4．Derivation of A
In order to obtain∠4s satisfying both relationships of Eqs．（4．5．9〉and（4．5．14），we propose the following form
for the target of the least－squares method：
14胃羽、濤）α・｛4一（4＋β ）｝2＋羽γ・（オニーδ・’）2 （4・5・・9）
Then the following linear equation is derived on∠4．m：
爲、羅）α川｛4葡（4＋β ）｝一屍、器9・｛4一（4＋βみ胡た）｝
、 （4．5．20） ＋Σγ卿（4．一δ加’）一〇
2
1t should be noted that Eq．（4。5。7），which is the basis for Eq．（4．5．9），indicates only the ratio of the values of
一88一
Technical Reports of the MRI，No．392000
respective∠4s．For this reason，if the last term in Eq．（4．5．19）were omitted，the determinant for different∠4’ms
in Eq。（4．520）would become zero，which means that∠4s cannot be determi聡ed only by Eq．（4．5．9）．
It should also be noted that if the logarithm were not adopted in Eq．（4．5．19），unsuitable values might be
derived for∠4s。If Eqs，（4．5．8）and（4。5．14）were described without the logarithm and combined together like Eq．
（4．5．19），the term in Eq．（4．5．8）would cause all／1s to become smaller because residues for∠4s in Eq．（4．5．8）are
small only if∠4s are sma11．The result would be that Zls would，in most cases，be smaller than those estimated
from Eq．（4．5．14），unless the relationships in Eqs．（4．5．8〉and（4．5．14）could be completely satisfied with no error．
Consequently，the parameter∠傷is determined by solving the simultaneous linear equations（4．5．20）on∠4s．
The way to detemine an appropriate weighting for the above coefficients is described in Appendix（2）．
4、5、4．5．Process in case of no data for comparison
Determining the parameters for a radar requires some radar echoes over AMeDAS raingauges，and，at the
same time，observed precipitation overlappe（1in some areas by at least two radars．
Actually，however，in some cases these requirements are not satisfied．For example，when a rainfall area
moves eastward from the far northwest to the Kyushu District，only the Fukuoka Radar can detect the area over
the sea（Fig．4．3．1helps show the location of the specified radar）．In this case，neither Eq．（4．5．7）nor Eq．（4．
5．12）proposed for the algorithm is available．After a while，the radar echoes move into the detection area of
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Fig．4．5．8．（a）An example of the field calibrated with the parameters z4and X．
The composite of hourly precipitation estimates by radar at21UTC on27November1990。
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Technical Reports of the MRI，No．392000
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Fig。4。5．8．（b）An example of the field calibrated with the parameters／1and X．
The same as（a）except for the field calibrated with the parameters∠4and X．The discontinuity denoted by“A”is
nearly eliminated．The intense area“B”is not clear in（a）．The area“C”observed by a radar beam at a higher
altitude than“D”is calibrated larger by X，and the difference between these areas has become less．
another radar，Matsue Radar．Then，with Eq．（4．5．7），Xs and the ratio of two且s are determined．When the
area passes over AMeDAS raingauges，∠4s are detemined with all their relationships．When some of the
equations cannot be utilized，unknown parameters are supplemented with the statistical values obtained by
averaging the data for the last240hours，with higher weighting for the last6hours．
It is noteworthy that if only one radar satisfies Eq．（4．5．12），al1∠4s for radars in operation can be analyzed
without statistical data，although all these ra（1ars are needed to satisfy Eq．（4．5．7）．In this case，however，the
accuracynaturallydecreases．
4．5。5．Accuracy
Figure4．5。8a is a gomposite of1－hour precipitation amounts estimated by radar at21UTC on27November
1990。The algorithm of the Forecast Division（1991）was used for compositing，and no calibration is made．In
the chart，there is a clear discontinuity in intensity，denoted by“A．” The left side of thelarea‘‘A”is covered
by three radars，and the right side，by only two radars。The different sensitivity of radar hardware primarily
causes this gap．On the other hand，Fig．4．5．8b shows one calibrated by the algorithm in this study．There is
very little gap in area“A，”mainly because of calibration by the parameter A．The intensive area“B，”which
is not clear in Fig．4。5。8a，is represented distinctly in Fig．4．5．8b after the calibration．The area“C，”which was
一90一
Technical Reports of the MRI，No．392000
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Fig．4。5。9．（a） Effect of the parameter X．
The composite of hourly precipitation estimates by radar at19UTC on16February1993．
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Fig．4．5．9．（b）Effect of the parameter X．
The same as（1），except for the estimates calibrated with∠4and X．
一91一
Technical Reports of the MRI，No．392000
一
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（c）is not recognized in this figure．
The intense area in
一92一
Technical Reports of the MRI，No．392000
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Fig．4．5．9．（e）Effect of the parameter X．
Observation heights of the radar beam of Mt．Fuji Radar．
observed by radar at an altitude higher than area“D，”has been made stronger by X，and the difference in
intensity between these areas has become less．
TheperformanceofXisexplainedinFig．4．5．9．Figure4．5．9aisacompositeof1－hourprecipitationamounts
by radar with no calibration at8UTC on16February1993，while Fig．4．5．9b is modified using the algorithm．
There is a clear difference between the figures in the area denoted by“A．” For detailed verification，original
estimates by two radars for the area“A”are shown in Figs．4．5．9c and4．5．9d．In the estimates by the Mt．Fuji
Radar in Fig．4．5。9c，there is an・intense precipitation area with an arc shape．This intense4rea lies along the
area with low observation height shown in Fig．4．5．9e．It should be noted that the elevation angle for JMA
operational radars is set higher within the tangent line，which the path of the radar beam and the earth7s surface
make，than beyond the line，where radar echoes are free from contamination by sea clutter．As for Mt．Fuji，
there is a large difference of1．4。between the angles on both sides of the target line．The fields of estimates by
theotherradarshavenosuchdistinctintensearea．ThealgorithmforXregardedthattheintenseareawas
made by the influence of veftical variation of precipitation intensity，and the intense area was modified as Fig．
4．5．9b shows．It is noteworthythat althoughthe area south of area“A”is not coveredby anyradars except Mt．
Fuji Radar，the estimates there can be modified by this algorithm．
Figures4．5．10a and4．5．10b illustratevalues ofAs and Xs calculatedfor different radarsfor consecutive4－hour
periods．From18UTC to21UTC on16February1993，a large－scale precipitation area from a synoptic－scale
depression located in the westem part of Japan was moving eastward，accompanied by a weak uniform
precipitation area ahead of it，and a precipitation area with convective moderate cells at its centeL
一93一．
Technical Reports of the MRI，No．392000
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Fig．4翌5．10．！1s and Xs calculated for different radars andfor consecutive hours．
（a）The parameter Xs calculated for4consecutive hours from18UTC on16Febmary1993．The units of X are
inverse square of height in km．In the figure，radars are aligned nearly from southwest to northeast along the
horizontal axis．Their actual locations are shown in Fig．4。3ユ。，，
（b）The parameter。4s for the same condition as（a）．
（c）The same as（a），except for9hours at3－hour intervals．Values for Akita increase with time，while those of Fuドui
increase for the first three samples，but decreased for the last．Only these two radars face the Japan Sea．The
values for the other radars decrease．
（d）The same as（b），except for9hours at3－hour intervals．
ValuesforXareobviouslylargerinthewestempartofJapanthanintheeast，althoughtheyhavefluctua－
tions．TheaverageofthestandarddeviationofXfortheperiodisO．02．Inmanyothercasestudies，morethan
threeofthemodificationfactorscalculatedaboutXindicatedrelativelylargerfluctuations．Thereasonmay
be that the actual vertical profile camot be well described in a single function for such a large ratio．It seems
difficult to estimate the intensity of a radar echo near the ground when the only information is from the intensity
almost at the echo top．
On the other hand．，A has less areal variation in most radars。The fluctuation for、4（1uring the period is also
not larger than O．090f the average of the standard deviation（note that the average of the values is one order
larger than that for X）．
In Figs．4．5．10c and4．5．10d，As and Xs are drawn for9hours at3－hour intervals．Xs in the westem part of
Japan at18UTC was larger than in the eastem part，and became smaller at O3UTC．On the contrary，Xs in
the eastem part became larger with time．In spite of the fluctuations shown in Fig．4．5．10a，this tendency is
一94一
Technical Reports of the MRI，No．392000
clearly recognized，and it agrees with the expectation for the modification by X，that is，1ittle modification for
stratus rainfall with little vertical difference ahead of a depression，and large modification for wide－spread
rainfall with some vertical difference around its center．The differences of。4s among radars are clearly
recognized．In particular，∠4for Mt．Fuji Radar and for Akita radar are different from those for other radars．
There is also a tendency for As to become larger with time，except for radars in eastem Japan．
Consequently，z4s and Xs work well for correcting errors caused by the vertical variation of precipitation
intensity and the difference between radar systems．
To obtain more stable values，averaging for several hours is needed．In Figs．4．5．8and4．5．9，the average of
six estimates calculated at1－hour intervals is employed．
4．5．6．Discussion
4．5．6．1．Assumption for the vertical profile
Inthisstudy，Xisassumedtobethesamevalueoverthewholedetectingrangeofaradar．Precipitationwith
a high echo－top is，however，observed with a smaller calibration factor than that with a lower echo－top．
Actually，echo to－height contributes only a part of the denominator in Eq．（4．5．4），and this part，described as B
in Eq．（4。5．3），is larger for higher echo top－heightsl this situation leads to smaller f in Eq．（4．5．4）．If it is
effective in the detection range of a radar，the assumption for the vertical profile may become more accurate
by introducing an additional parameter，that is，echo top－height．Echo top－height would be useful especially
when precipitation areas would be observed by only one radar over the sea，where the estimation proposed in
this study camot be applied．More investigation is required to clarify the effectiveness and problems that may
arise in adding another parameter．
4．5．6．2，Bright band effect
The model proposed in this study for modifying the vertical difference of precipitation cannot always describe
the bright band effect with reliable accuracy because the thin，distinct，severe reflectivity of the bright band
camotbe expressedbythe parameterX．TheJMAconventionalradarsusedinthisstudyobserveprecipitation
by CAPPI with three or five elevation angles，and hence a distinct bright band near the radar is rarely observed
in this plan－position field．As Andrieu an（1Creutin（1991）stated，in areas over lOO km from the radar，there
is little influence from the bright band．However，further modification is（1esirable for a more reliable estima－
tion，especially for the area near the radar site．It maybe possible to describe thevertical profile withthebright
bandby adding other parameters andmodifying the function proposed inthis study．For example，abe11－shaped
vertical variation changing according to height at which the highest reflectivity is observed owing to the bright
band can be used as an additional coefficient of A，as Gray（1991）proposed．In this case，the intensity of the
coefficient and the elevation of bright band should be treated as parameters to be detemined．This is a theme
for another development．
4，5，7．Conclusions
An algorithm that calibrates radar estimates over the whole detection area using raingauges and more than
one conventional radar was proposed．In this algorithm，two parameters were used to correct two major causes
of errors in estimating precipitation by radar，namely，1）errors arising from the instability of radar hardware，
and2）the difference between the distribution of raindrops near the ground and that observed by radar over the
ground，i．e．errors based upon the vertical profile of precipitation。Using these parameters and the height of the
一95一
Technical Reports of the MRI，No．392000
radar beam over the target，the ratio of the actual field to the radar estimate on a grid of the fie1（iwas described
with a simple function．The two parameters could be calculated every hour with the least－squares method by
comparing radar estimates not only with AMeDAS raingauge measurement，but also with estimates from
different radars．
The use of this algorithm improved not only the discontinuity around borders of the radar coverage in the
composite map，but also the discontinuity found at the borders of observation fields from different elevation
angles in CAPPI of a single radar．The results from many cases showed that the algorithm was effective only
when the calibration factor based on the vertical profile of radar echoes is less than about3．
This modification has few problems for changes in the type of precipitation distribution or changes in season，
because it is determined every hour from current data．This method has another advantage in that it can be
used for data further from the radar than the correction method by observing the distribution of raindrops．
Furthermore，one of the conditions for the algorithm，that there shouldbe some detection areas overlapping each
other among ra（i＆rs located at least some hundreds of km apart，is usually satisfied for most radar networks
already（ieployed for operational use．Thus，this algorithm would help to improve radar estimates of precipita－
tiOn OVer a Wide area．
4．6Radar－estimate calibration by raingauge in view of Z－R relationship modification and appropriate corre－
spondence between calibration targets
Detailed procedures for deriving Radar－AMeDAS precipitation have been described by Forecast Division
（1991）．This section outlines an algorithm that was developed in June1995to improve fields for calibrating
radar precipitation over land areas．
Although radar estimates are improved with the algorithm in Section4．5，more reliable estimates are obtained
on land，or areas where there is a raingauge within a70－km in radius，by a correction method using AMeDAS
raingauge data．
In this algorithm，a value of Radar－AMeDAS precipitation at a pixel g，ノ～g，is described with a calibration
factor，Fg，as follows：
R8一ろEg （4・6・1）
where Eg denotes the radar－precipitation amouht．
The calibration factor Fg is described with the calibration factors瓦derived from comparison between a
raingauge measurement and the corresponding radar precipitation，as follows：
ろ一exp（Σ（ln石）㎎・／Σ照・） （4・62）
”
where VV∫is the weight on F∫for interpolation．
At the second and the third repetition of the successive modification processes，radar precipitation amount Eg，
in Eq．（4．6．1）is replaced by Rg determined with Eq．（4．6．2）in the former process．
4．6．1Weight for interpolation considering precipitation intensity
Weightforinterpolation，骸，includesafactorforthedifferencebetweentheradarprecipitationatthetarget
pixel g and that at the pixel where AMeDAS raingauge J is located，as well as a factor for the distancebetween
一96一
Technical Reports of the MRI，No．392000
0．2
0．1
0
（嫡h）hoo一
＿0．1
＿0．2
230R■’4
400R■。3
●
10 20 30 40（㎜ゾh）Radar precipitation by Z＝200R■。6
Fig．4．6ユApproximations with different algorithms to the observed calibration factors which change depending on the
Iocal variability of precipitation intensity。 The horizontal axis indicates the precipitation rate by a Z－R relationship，
Z＝200ノ～L6，and the vertical axis denotes the calibration factor，that is the rate of the actual precipitation to the radar
estimate by the standard Z－R relationship in a logarithm scale．Dots in the figure indicate calibration factors
calculated by actually comparing radar data withraingauge data．Thebroken line isproducedby interpolatingthose
calibration factors conceming only distance．The solid thin line is produced by interpolation considering distance
and precipitation intensity。 Solid thick lines are calibration factors derived for different Z」ノ～relationship values，
that is the ratio of precipitation estimated with different B andβto that with the standard Z－R relationship for the
same reflectivity factor Z．Variations for so1三d thick lines are from Austin（1987）．
them，as the following equation indicates：
exp（一42／D2）
照認 2君 1＋α（E8／Ef－1）
where
（4。6．3）
4：（iistance between Pixe18’and the pixel where Raingauge歪is located，
Eg：radar precipitation at Pixe19，
瓦：radar precipitation at the pixel where Raingaugeづis located，
．Pゴ：weightbased on the reliability of AMeDAS data and radar data at the pixel where Raingaugeづis located，
Z），α：parameters．
By interpolating F，with PVゴ，Fg’s are different according to the precipitation rate of the pixeL Parameters
P andαchange according to the cycles of the successive modification．In the first cycle，∠）andαare large，
in order thatレV，may be determined mainly by the difference of precipitation intensity．In the later cycles，D
andαare set smaller for calibrating the precipitationtherate ofwhich changes dependingmainly onthe location．
Figure4．6．1compares this effect with other algorithms．The dots indicated in the figure denote the calibra－
tion factors calculated over the respective raingauges．Methods to change B andβcorrespond to solid thick
lines．These solid curves are determined by fixing the two parameters so that the total difference between the
values of the curve and the values indicated by dots maybe the minimum．Interpolation only about the distance
一97一
Technical Reports of the MRI，No．392000
（Makihara et a1．，1995）changes its calibration factor with no relation to the precipitation rate，as describedby
the broken line．The curve which the algorithm in this section describes changes according to the weight on the
intensity of radar precipitation over the dots．The difference between this algorithm and changing of B andβ
is clear at high intensity which is not observed with raingauges。The curve detemined by fixingβandβ
sometimes indicates larger factors for extremely strong rainfa11，while the current algorithm’s values seldom
exceed observed ones．The current algorithm can avoid yielding excessive factors for strong reflections from
very large raindrops or even from large particles of hails。
The weight P，based on the reliability of observation data becomes smaller when：
1）An AMeDAS raingauge or a radar observes very light precipitation，where an error caused by digitization
may occur
2）The altitude at which a radar observes precipitation echo is high
3）The number of AMeDAS raingauges for determining F8is sma11．
4、6．2Correspondence between raingauge measurement and radar precipitation
In detemining F，，it shouldbe taken into account that anAMeDAS measurement does not always correspond
to the ra（1ar precipitation of the same pixel of5km square，because1）there is variability in the distribution of
precipitation even in a5－km pixe1，and2）raindrops aloft are often carried away to another pixel before they
reach the ground．Unsuitable correspondence between an AMeDAS measurement and a radar precipitation
leads to a false calibration factor．To avoid this undesirable situation，we use the following process，taking into
account the eight pixels surrounding the target pixe1：
君・譜∫（C，，C，，q，C．，N） （4．6．4）
where
C，：（AMeDAS measurement，R）／（radar precipitation of the target pixe1，瓦），
C、：R／（maximum radar precipitation among the pixels，E」），
G：R／（minimum radar precipitation among the pixels，E，），
C．：R／（average of ra（1ar precipitation estimates of the pixels），
ノV：mmber of AMeDAS raingauges conceming the calculation of－Fg over the target pixe1．
In the process，C、is used as E when I C，I does not exceed the value of Parameterγ，while other values are
taken into accomt when I C、I exceedsγ，considering that a large I C，I may be due to unsuitable correspon－
dence。For example，when C、exceedsγ，F～is determined by the following equationl
君．＝min（max（1，（フ，δ），max（δ，min（3，max（1，0．5／V））C，），C，） （4．6．5）
where min and max indicate the minimum and the maximum among the values within parentheses，respectively，
andδis a constant。Almost the same restriction is placed on C，below一γ．
Figure4。6．2illustrates this function。In the figure，Ff is equal to C、if C。is in the hatched area，while Fゴ
changes to the value at the top of the hatched area if C，exceeds the area．The adoption of the new value is
equivalent to making the raingauge measurement correspond with the radar precipitation of the intensity
between E∫and E」for obtaining a smaller calibration factor．It is also found by the figure that the fmction
imposes severe restriction when the target has a small number of AMeDAS sites concemed．
一98一
Technical Reports of the MRI，No．392000
C1
つCg巴
顯 γ
δ‘
Number of raingauges conGemed
Fig．4．6．2Schematic diagram showing how to correspond an AMeDAS measurement with the radar observations at the
target and surrounding pixels．The figure illustrates the case when the calibration factor between the AMeDAS
measurement and the radar observation over the target pixel exceeds a criterion valueγ．Hatched area shows the
range where F∫can be allowed．
Consequently，this function makes an AMeDAS raingauge measurement correspond to a radar precipitation
so that1）Radar－AMeDAS may be equal to the AMeDAS measurement if I C，l is underγ，and2）Radar－
AMeDAS may be close to the radar precipitation with small modification by AMeDAS under the restriction of
Eq．（4．6．5）in other cases．
4．7Accuracy of Radar－AMeDAS precipitation
4．7．1 1ntroduction
Radar－AMepAs precipitation，Radar－AMeDAs hereafter provides important information for wamings and
watches for heavy rainfall issued by JMA，because of its detailed continuous distribution of precipitation．
Several reports were presented conceming the accuracy of Radar－AMeDAS on land（e．g．Nyoumura，1985，
Kitabatake et aL，1991）、However，since the algorithm for Radar「AMeDAS has been improved over the past
several years，we will clarify the effect of the new algorithms in this study．
The accuracy over the sea cannot be verifie（i with raingauge，and should be estimated indirectly by other
means．As one of those means，we prepare a distribution of appearance frequencies of Radar－AMeDAS pixels
for different intensities，and compare it with that of AMeDAS．
4．7．2Data
The target ofverification in this study is values of1－hour precipitation amount of Radar－AMeDAS for pixels
of5－km square，which are allocated onto a composite（10main shown in Fig．4．3．1．Operational products of
Radar－AMeDAS after March1993are used for most of the verifications．Products re－analyzed by the
algorithm that has been operational since June1995are also used for the cases before March1993．
A very dense raingauge network of the Tokyo Metropolitan Govemment with average spacing of4。5km is
used for detailed comparison，and raingauge data from other local govemments and from the Ministry of
Constmction are used for verifying severe rainfall over100mm／h．None of those data was used in Radar一
一99一
Technical Reports of the MRI，No．392000
AMeDAS analysis．
For obtaining Radar－AMeDAS，data from JMA conventional radars and AMeDAS raingauges are used。
An estimate of1－hour precipitation by radar，radar precipitation hereafter，is derived from the reflectivity
factor Z observed on a2．5km gridby using a Z－R relationship of200forβand1．6forβ（Marshall and Palmer，
1948）．
The gri（1size for radar precipitation is changed from2．5km to5km before the data are transmitted to the
JMA forecast center by choosing the maximum value among four pixels of2。5km square．
4，7．3Representative precipitation for a5－km square pixel
Radar－AMeDAS should have the following features as a representative value of precipitation for a pixel of
5km square：
1） It should not fail to detect local severe precipitation．
2）It should be“more accurate”with the help of AMeDAS raingauge data than radar precipitation．
Beforeverification，we considertherepresentativevalue of Radar－AMeDASbecausesevereprecipitationcells
such as from thmderstorms sometimes have a large areal variation even in an area of5－km square．
Owing to the variability in the Z－R relationship or the vertical difference of reflectivity from radar precipita－
tion，precipitation needs calibration to achieve sufficient accuracy（Collier et a1。，1975）．Since measurements of
precipitation at AMeDAS raingauges are considered to be random sampling from the actual precipitation
distribution，AMeDAS can be assumed to represent the average of precipitation in a5－km square when
statistically processed onboth space and time．Therefore，calibration with AMeDAS is expected to produce，in
general，Radar－AMeDAS representing the average value in a5－km square．Here，it should be noted that
calibration with AMeDAS includes not only improvement in accuracy of radar precipitation but also modifica－
tion of the representative radar precipitation for a5－km square into the average．
The representative value of radar precipitation for a5－km square，that is the maximurb of four radar
precipitation measurements in a2．5－km square，represents more localized severe precipitation suitable for rea1
－time watching tasked for Radar－AMeDAS．
Under these circumstances，the following two extreme cases of calibration are possible：
1）Whenscarcelycalibrated，Radar－AMeDASrepresentsalmostthemaximumamongfour2．5－kmpixelvalues
in a5－km pixe1．
2）When calibration with AMeDAS is heavily imposed，Radar－AMeDAS becomes statistically the average
precipitation in a5－km pixe1．
For example，1et a5－km pixel consist of three2．5－km pixels of4mm／h and one pixel of20mm／h，and these
radar precipitation estimates be accurate．If there are many AMeDAS raingauges and5－km pixels with the
same condition，the most probable situation is that these AMeDAS raingauges measure values of4mm and20
mm at a frequency rate of three to one．When we adopt calibration in case2），Radar－AMeDAS becomes the
average of8mm，while the case1）provides20mm for Radar－AMeDAS．It should be noted that8mm from
case2）is the correct value for the areal average precipitation in a5－kmpixel andthat20mmbycase1）isbetter
for real time watching for severe precipitation．
The current algorithm arranges both cases depending on the conditions．For localized severe precipitation，
case1）is dominant．In this situation，the algorithm makes an AMeDAS measurement correspond to the radar
一100一
Technical Reports of the MRI，No．392000
precipitation having as small mo（1ification factor as possible among the surromding pixels，taking into account
the location errors and large variability in a5－km pixel（Section4．6．2）．For the other types of precipitation，
the representative value of Radar－AMeDAS is close to one in case2）．There，radar precipitation representing
the maximum of four2．5－km pixels is modified with some nearby AMeDAS measurements into almost the
average of the5－km pixeL Variation in precipitation estimates of neafby pixels and the number of AMeDAS
raingauges used for the calibration of the target pixel are，fundamentally，the parameters for determining how
the weights should be distributed to those cases．
As the result，Radar－AMeDAS represents the following values：
1）For a precipitation with large variation often observed in a severe convective cells，Radar－AMeDAS
represents the maximum of four2．5－km pixels in each5－km pixe1
2）For a large scale precipitation with small variation in distribution，Radar－AMeDAS represents the value
close to the average of precipitation in a5－km pixel
Although less calibration is made in case1）than in case2），calibration on a larger scale is conducted for the
entire detection area，beforehand，forbothcasesusingdata from all oftheAMeDAS stations concemed（Section
4．5）。
4．7．4Detailed accuracy on Iand evaIuated with raingauge data
To assess the accuracy of Radar－AMeDAS in detai1，we compare Radar－AMeDAS withraingaugedataexcept
AMeDAS．In the comparison，a location error of one pixe1（i．e．5km）must be taken into accomt because
raindrops are sometimes advected by wind before reaching the ground，and there is also a possibility of an error
reaching one pixel when Radar－AMeDAS values analyzedfor respective radars are allocated onto the composite
domain．
Raingauge data from the Tokyo Metropolitan Govemment is suitable for this comparison．In Fig．4．7．1，these
raingauge measurements are indicated on Radar AMeDAS chart at6JST on90ctober1992，when an
extratropical depression passed over．It is seen that Radar－AMeDAS values show good agreement with these
raingaUge meaSUrementS．
Figure4．7．2shows a case of a．severe local thunderstorm at22JST on20May1992．A large variation in
precipitation distribution is seen even in a5－km pixe1．For example，in the pixel where Radar－AMeDAS
indicated39mm，a raingauge measured32mm in its top left si（1e and another gauge indicated only6mm in its
bottom right side．Although in the pixel where AMeDAS measured5mm Radar－AMeDAS showed a rather
larger value of30mm，and a raingauge in the next pixel on the right side measured29mm．Thus，with a
location error of one pixel considered，Radar－AMeDAミ』provides good agreement．It is noteworthy that Radar
－AMeDAS agrees with the largest raingauge measurement in a pixel in this case。
Figure4．7．3is a scatter diagram for comparisons between hourly Radar－AMeDAS values and corresponding
raingauge measurements ofthe Tokyo Metropolitan Govemmentfor4months。Araingauge measurementwas
compared with the value that showed the best agreement among the grid containing the gauge site and the eight
surrounding grids．Good agreement is seen also in this figure．
In Table4．7．1，severe rainfalls that measured over100mm／hwithraingauges are compared，Most ofthe radar
data for these cases are obtained at a further distance from radars than the data for the verifications of the
Tokyo area．Half of these cases are within an error of15％。The cases with relatively large－scale distur一
一101一
Technical Reports of the MRI，No。392000
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■
112噂
即
画幻
や
3－1 9
■－
恥8 9
01 玉
9
●
お際
播
P
▽ ・
5曜－
9孔
E5
6．
2孔 5
0雪乙
4■－
8
1
4L
偽
■－
V
囮
16●
P
獅 ぜ
V
Fig．4．7．1Comparison ofRadar－AMeDAS values withraingauge measurements ofthe network oftheTokyo Metropolitan
Govemment at6JST on90ctober1992．Radar－AMeDAS values and raingauge measurements are indicated by a
numeral over the chart for a case when an extratropical depression passed over．The size of each pixel is the same
as in Fig．4．2．2，i．e．almost5km by5km．Points with numbers indicate locations of raingauges，and the numbers for
each point denote the precipitation measured there．Numbers surroundedby a rectangle indicate raingauge readings
by AMeDAS．The exact location ofthe AMeDAS site isonthevertex ofthebottomleftside．The Radar－AMeDAS value for each pixel is indicated in its top left side．It is seen that Radar－AMeDAS values agrees well with the
raingauge meaSurementS．
1 5’E
孔1
．4
聖o鉾 毛 2 6
5 6
7■ ．
8団 も4 。、 1
30
35αN
圃
1●
Fig．4．72Same as Fig。4．7ユexcept for the case of a thunderstorm at22JST on May1992．There is a large variation in
intensity even in a pixeL In the pixel where AMeDAS indicates5mm，Radar－AMeDAS value shows a larger value
of30mm．However，a raingauge in the next pixel on the right side measures29mm．It should be noted that Radar
－AMeDAS agrees with the largest raingauge measurement in the pixel in this case．
一102一
Technical Reports of the MRI，No．392000
（mm／h）
140
120
0 0 0 0
0 8 6 q
芒①目Φ旨器Φε亀壽bo唱蜀国
00
内ごー
● ● ● 0
0
（U∩UO
－ーる5ー一
m（4102－00180
●
●。
．
●．●
●・
・．●鉾髄
OO■
■
0
●
O
9。●●。●●
●●30●●
・説．縞』
●●●銘●．●”
”“驚響箕3
● 器●駅
ぞ脳●。2．．”器”
●鷺o器” 。o● 9
● ●OOO
O
・
09●斜3。
．0
OO●●
3 93り
。 ㎝
． ●03 00
0 ●
●”●●
綴3
鵠．
●O
● OO●
『認
●．
“
．2
蓼。． 0
●o”1
㌦が川…
．野。。
（mm／h）
Radar－AMeDAS precipitation
Fig，4．7．3Scatter diagram for comparisons between hourly Radar－AMeDAS values and raingauge measurements of the
Tokyo Metropolitan Govemment for4months．Data from May to September in1994，that is5，808cases，are used．
A raingauge measurement is compared to the value which showed the best agreement among the pixel including the
gauge site andthe surrounding eightpixels．The figure looks as ifRadar－AMeDAS values areslightlyunderestimat－
ed comparedto raingaugemeasurements．This apparentunderestimation ismainly due to the way ofcorrespondence
between a raingauge measurement and a Radar－AMeDAS value．The figure indicates only whether raingauge
measurements can be detected by Radar－AMeDAS with sufficient accuracy．
bances are estimated within an error of30％．The rest includes three cases with an error of about40％to50
％an（10ne case of70％，all of which arebrought about by localized convection．The last worst case was greatly
affected by attenuation of the radar beam due to a film of water over a radome produced by severe rainfa11．
This evaluation indicates only whether Radar－AMeDAS mfailingly detected such severe precipitation
amounts because those raingauge data are only part of the entire severe precipitation amounts．To determine
the accuracy of Radar－AMeDAS，we also have to know atthe same time，howfewthe overestimates are among
all of the Radar－AMeDAS values．However，it is almost impossible to verify all of severe Radar－AMeDAS
values having spatial continuity by using scattered raingauges．Another approach is necessary for this kind of
verification．
4．7．5Comparison by appearance rates
The density of a raingauge network is，in general，not so high as to correspond with every pixel of Radar－
AMeDAS．Here，by comparing frequencies of the appearance for different intensities，all Radar－AMeDAS
values will be evaluated statistically with AMeDAS．
The distribution of the frequencies of raingauge measurements for（1ifferent intensities is known，in genera1，
to exhibit a gamma distribution（for example Suda，1993）。If the number of AMeDAS raingauges increased to
the spacing of5km，the Qbservation frequencies would increase。In this case，the increase of observation
一103一
Technical Reports of the MRI，No．392000
Table4．7．1Raingauge data for extremely severe rainfall and corresponding Radar－AMeDAS values
Data and Location Raingauge Radar。AMeDAS Situation
time JST ref6cture Value mm1h mm1h1993．6．21．15 Tokyo 112 59 Thunaerstorm H1993．8．1．20 Kagoshima 119 105 Rain band1993．9．3．16 ：Kagoshima 116 100 Typhoon O且1993．11．13．10 Tokushima 115 120 Depression OH：1993．11．13．10 Tokushima 110 140 Depression OH：1993．11．13．10 Tokushima 107 105 Depression OH1993．11．13．10 Tokushima 105 130 Depression OH1994．9．7．0 Osaka 106 64 LOCal COnVeCtiOn1994．9．7．1 Osaka 108 110 ：Local convection
1994．9．22。13 Miyagi 132 66 Thunaer storm1994．9．22．15 Miyagi 134 43 Thunder storm H1994．9．22．16 Mi a1 121 130 Thunder storm H
Note：Marks“0”and“H”at the last of each case indicate existence of orographic effect and
high calibration factor required，respectively．
frequency for each intensity is expected to be proportional to the increase of raingauges because the condition
forsamplingthe actualprecipitationwithAMeDAS raingaugesdoesnotchangeexceptforthesamplingnumber．
As a result，“the ratio of the number of measurements for specific intensity to the total number of measurements
including O”is not influence（1by their spatia1（1ensity。 Hence，by using this ratio，the frequency（1istribution of
Radar－AMeDAS composed of5－kmpixels canbeevaluatedonlandwiththesameconditionasthatofAMeDAS．
On the other hand，there may be some difference between the actual distribution over the sea and that over
land．Almost the same distribution is expected，however，since the period during which those distributions were
collected has a duration of over3months，and the target domain is so large that the sea area can be treated as
being near the land，although some difference may be found in areas affected by orography．In fact，the
appearance frequencies of radar precipitation for the land area and for the sea area which are normalized with
respect to the total number of pixels show only a small difference for each intensity as shown in Fig．4．7．6．
Withthoseconsiderationsinmind，theappearancerateofRadar－AMeDASiscomparedwiththatofAMeDAS
for the land area and for the sea area，as well as for a warm season and for a cold season，respectively．
In addition，verifications are made also for the different heights at which radars observe precipitation echoes．
Some studies pointed out that the intensity of radar precipitation decreases as the observation height increases
owing to the variable vertical profile of a rainfall echo（Joss et a1．，19891Joss，1990）．Although the algorithm
of Radar－AMeDAS compensates for this effect as described in Section4．5to a certain extent，the compensation
is insufficient in extreme cases such as for an area where a radar observes almost the top of rainfa11．The
relationship between these heights and the appearance rate is also investig4ted in the same mamer。
4．7．5．1Method
The number of pixels in which Radar－AMeDAS showed specific intensity are accumulated for different
intensities and heights at which a radar observes precipitation echoes．Data from June to September1993are
accumulated as those of a warm season，and data from December1993to February1994as of a cold season．
Since the total numbers of pixels including no precipitation are different if the observationheights or the regions
are different，we calculate“an appearance rate，”∫，by normalizing with respect to the total number of
一104一
Technical Reports of the MRI，No．392000
［％。］
3鐙8毎詰Φ盆＜
106
105
104
103
102
10
A』
warm season
1km＼§・＝、 ＋2km
＼、墜・ ◇3km 拳、．∋1≧：こ1♪．
隈 ・り◇下・、．、． △ 4km △、 ・・㍉、
＼准、 ゆ一 ・・、 × AMeDAS 、、 じ◇・ ．鴨 曾拳．、． ’．． 欄 ’の◇： ら0、零一．
△、 ・◇． 竜、㌔、 ト ’凸、 ○◇・ 『零、。、 ＼ ・◇． 隔tN ・ム…ぐ ・1魚∴・・、・．、・、・麹，
凸、、 、△ ￥
＼
￥
』◇
～5～10～15～20～25～30～35～40～45～50～55～60～65～70
PreGipitationintensity ［mm／hou r］
Fig．4．7．4Appearance rates of Radar－AMeDAS values for different observation heights．The horizontal axis denotes
precipitation intensity in mm／h，and the vertical axis indicates the appearance rate．The rate at a specific intensity
indicates the ratio of the number of pixels where Radar－AMeDAS values ofthe specific intensity are observed to the
total number of all the pixels．The rates are expressed as common logarithms of PPM，10g（1，000，000！〉，、h。／！〉伽、♂）．
Data from June to September in1993were used．The rate of Radar－AMeDAS values at observation height of2km
and under is almost identical to that of AMeDAS measurements．
frequencies as follows：
め
ノ（h・7）謂s（h・7）／Σs（h・7） （4・乳・）
where
h：height at which a radar observes a precipitation echo
7：precipitation intensity
s：frequency at Intensity7and at Observation height h．
4．7．5．2ResuIts
For detemining the relationship with observation heights，the appearance rates of Radar－AMeDAS for the
warm season are shown in Fig．4．7．4together with that of AMeDAS．The appearance rate of Radar－AMeDAS
at observationheights of2km or less showsfairlygood agreementwiththat ofAMeDAS．The appearancerate
at3km is smaller than that of AMeDAS by about60％for all intensities．
The appearance rates in the cold season show similar differences as in the warm season（Fig．4．7。5）。The
higherrate ofAMeDAS inweaゆrecipitationmightbecausedbyprecipitationwitha lowcloudtopheightwhich
cannot be（1etected by radar in spite that raingauges can detect one。
We，then，evaluate the difference in the appearance rate in terms of intensity，to obtain a standard for
estimating areal precipitation amomts．It is assumed that the appearance rate of AMeDAS is equivalent to the
actual rate。It is also assumedthat the intensity of Radar－AMeDAS values canbe described as afunctionwhich
一105一
Technical Reports of the MRI，No．392000
［％。］ 106
Φ捻』Φ。語』器盆＜
105
104
103
102
10
cold season 、 、．
、”、◎』■
ち ヤし
金、＼ ロ1km△、◆’・、
・＼・＝t・・、図 十 2km 、 ￥・
＼継 ◇3km 〉胤∴・＞ミ、・ △ 4km
へ
＼◆・’桑 ×AMeDAS ペ ヤ
・進＼◇・造爆、ぎ．
’△、’◇㌧・』’＋モト．、．
＼盛、◇・111漢＞．、．
・く’ ・’0φ・∵．｝ヤ ￥△ ・駅．、
￥ 、Y十、 ￥ 縦．．・ ￥ ・ヤ、
～5～10～15～20～25～30～35～40～45～50～55～60～65～70
Precipitationintensity ［mm／hou r］
Fig．4．7．5Same as Fig．4．7．4except for the period from December1993to February1994。
increases monotonically with the increase of the intensity of AMeDAS，although the relationship may not be
proportiona1．
We can then detemine the value of Radar－AMeDAS，ノ～’，so that the sum of the appearance rates of Radar
－AMeDAS from infinity to R’may be the same as that of AMeDAS from infinity to R，as follows：
ざ め
濠∫・（ア）一Σ，∫（h・7）
Where
∫、：appearance rate of AMeDAS
The concept and the assumption of this method are similar to what Rosenfeld et al．（1993）proposed for
deriving the relation between radar－precipitation amounts and raingauge measurements for different types of
precipitation events，although the purpose and the apPlied data are different．
Under the above assumption，the actual intensity of the precipitation which Radar－AMeDAS analyzed asノ～’
is considered to be R．Consequently，1～ソR is the ratio of the intensity ofRadar－AMeDAS to that ofthe actual
precipitation．The result obtained by this method indicates that Radar－AMeDAS at3km underestimates the
intensity by20％at20mm／h，and28％at50mm／h over the sea（Fig．4．7．7）．The small values at low intensity
around5mm／h over the sea are mainly due to failing in detection of precipitation，which disagrees with the
assumption on Eq．（4．7．2）．
For detailed evaluation of the appearance rates at lower observation heights，the ratios of the appearance rate
of Radar－AMeDAS to that of AMeDAS inthewarm season are showninFig．4．7．6bothfor the land andforthe
sea respectively．The ratio for radar precipitation，which is not calibrated with AMeDAS，is also included in
the figure．Figure4．7．6shows that the ratio of the radar precipitation is smaller than that of AMeDAS，
especially at high intensity．Since Radar－AMeDAS shows the ratio much closer to l than radar precipitation，
一106一
Technical Reports of the MRI，No．392000
［％］160
＞↑一ω⊆Oρ⊆一．αO』」⊆一〇一一邸匡
140
120
100
80
60
40
20
0
ギ
warm season
ダ十〇
．；．．fナ＝＝烈曹ニン2∫3臨．、
’鴨十、
隔十・．．嚇
．2rr廠、
’十辱
口十◇△
l and（R－AMeDAS）
sea （R－AMeDAS）
I and（Radar）
sea （Radar）
一峠一 一◇
一… … で
壕… … 盛
く…㎜、、…
鷺｝袴
》… 熔
U㍉ 一 一鴨り
｝覧 一 一隔』
庶…囎
凝8
。＼
■V り
㎜翼
、 ’…
、 1■
魂幅
い＼
…ム．◇…
…
’
訟
…遷◆…
＼・
…＼．…
…ゆ…
ツ…路
◎ム
～5～10～15～20～25～30～35～40～45～50～55～60～65～70
Precipitationintensity（AMeDAS） ［mm／hou r］
Fig．4．7．6Comparison of the appearance rates of Radar－AMeDAS values over the land and over the sea with that of
AMeDAS．Ratios of the appearance rates of Radar－AMeDAS values to those of AMeDAS in the warm season are
shown for the land and for the sea，respectively．The ratios for radar precipitation，which is not calibrated with
AMeDAS，are also included。
［の＜OΦミ虚］
［の＜OΦミ］
1．2
。慧』8慧』8盆＜εo；醒
1．1
1
0．9
0．8
0．7
0．6
0．5
warm
十ρ一
”Φ
season
。』』畢二ニニニニニ墨顎・＋．．陶．．
．φo
’1φ∵ ●・③・9
Q’←噂鴨層『鴨・＋隔鴨舳
．：rもの喝、、”0ギ’湖お峯こ；ニニニニ等二二瓢キ』』』
・●⑥3 9び●。 ●。’0⇔・ ・φ・…〈》・ ・φ．
漁．・．
。’●’。Oo・φ’
口 land
十 sea
◇sea
1km＋2km1．km＋2km
3km
、十
’＜y
～5 ～10 ～15 ～20 ～25～30～35 ～40 ～45～50～55 ～60
precipitationintensity（AMeDAS） ［mm／hour1
Fig．4．7．7Evaluation of Radar－AMeDAS about intensity for estimating areal precipitation amounts．The ratio of
intensity that Radar－AMeDAS indicates to the actual intensity is shown under the assumption that the appearance
rate by AMeDAS is equal to the actual one．It should be noted that Radar－AMeDAS not only seeks to determine
accurate areal precipitation but also must always detect localized severe precipitation。The positive bias over the
land area is caused mainly by this task．Since the figure is indicated by the ratio，the values at higher intensity are
relatively close to l in spite of the larger differences in their apPearance rates．
一107一
Technical Reports of the MRI，No。392000
it is clear that the performance of Radar－AMeDAS surpasses that of radar precipitation．
The ratios of Radar－AMeDAS for the land area exceed l for all intensities，and they become larger as the
intensity increases．This feature is consistent with the result from the consideration onthe representative value
for a5－km pixel in Section．4．7．3．In estimating areal rainfall amounts，Radar－AMeDAS overestimates
AMeDAS by11％at10mm／h and by12％at40mm／h although overestimation decreases to8％at5mm／h
（Fig．4．7．7）．
Onthe otherhand，theratio over thesea isverysimilarto that ofAMeDAS atlowintensity，whileitdecreases
at high intensity over15mm／h．When Eq．（4．7．2）is apPlied，Radar－AMeDAS over the sea gives underestima－
tion of the intensity by2％at10mm／h，and12％at40mm／h under the assumption that the apPearance rate
of AMeDAS is equal to that over the sea．Since the appearance rates around40mm／h are1／100to l／1，0000f
that at5mm／h，the effect of the errors in this intensity on the total areal precipitation over an area is expected
to be under1％．
The following are major reasons accounting for the decrease at high intensity．
1）Local calibration by AMeDAS is not available for Radar－AMeDAS over the sea．The Z－R relationship
used for the JMA radars is thought to cause mderestimation at high intensity for local convections
（Fujiwara1965）．The mderestimation caused by vertical difference of precipitation intensity is also，in
genera1，1arger for the area with higher intensity than the areas with lower intensity surrounding it，These
are considered to lead to the mderestimation of the actual intensity at higher intensities．
2）Orographic effect is less prominent over the sea．This leads to fewer occurrences of severe precipitation
over the sea than over the lan（i．
As for the severe rainfall events in Table4．7．2，five of them are considered to be affected by orography，and
the three remaining had large calibration factors．If those events had occurred over the sea，the five events
would have lower rainfall intensities，which would lea（1to smaller appearance rates at high intensity．The
three remaining would be estimated to have smaller intensities with smaller calibration factors because of
no local calibration．It is difficult to estimate only with those figures，however，the orographic effect might
not be small enough to be neglected in evaluating Radar－AMeDAS over the sea with AMeDAS．
4．7．6Conclusions
The very dense raingauge network of the Tokyo Metropolitan Govemment was used for the detailed
evaluation of Radar－AMeDAS．By evaluating Radar－AMeDAS for a pixel of5km square with several
raingauges located in the pixe1，it was found that Radar－AMeDAS represents almost the average of those
raingauge measurements when precipitation is caused by a large scale disturbance，and that it is close to the
maximum raingauge measurement in the pixel when the precipitation is as localized as scattered thunderstorms．
In a comparison over a period of approximately4months，Radar－AMeDAS exhibited an excellent agreement
if a positioning error of one pixel was allowed。
Radar－AMeDAS values were also compared with AMeDAS raingauge measurements and with radar－precipi－
tation amounts statistically，with respect to the appearance rates，for different precipitation intensities．The
comparisonrevealed that Radar－AMeDAS is much more accuratethanradar precipitation，whichis determined
only with constant Z－R relationship and shows small appearance rates．The rate of Radar－AMeDAS agrees
well with that of AMeDAS if radar observation is made at an altitude below3km．For radar observation at
一108一
Technical Reports of the MRI，No．392000
3km height，the appearance rate decreases to about60％that at2km height．Under the assumption that
AMeDAS represents the actual appearance rate，this rate is equivalent to underestimation of areal precipitation
amounts by20％at20mm／h and28％at50mm／h．
Since Radar－AMeDAS precipitationonlandsometimesrepresentsthemaximum Qfprecipitationinapixelfor
the purpose of unfailingly（1etecting extremely localized severe precipitation，it exhibits higher appearance rates
at higher precipitationratesthanAMeDAS．As a result，inestimating areal rainfall amomts，Radar－AMeDAS
overestimates AMeDAS by11％at10mm／h and12％at40mm／h，although overestimation decreases to8％
at5mm／h．
Radar－AMeDAS over the sea，where there is no local calibration by AMeDAS and little influence of
orography，exhibits smaller appearance rates over15mm／h than AMeDAS at10mm／h in intensity and12％
at40mm／h．This smaller appearance at high intensity is mainly caused by the followings．
1）A Z－1～relationship sometimes unsuitable for convective severe rainfalls，and a vertical precipitationprofile
with characteristic leading to large mderestimation at high intensity are major targets which should be
calibrated on a local scale．These cannot be calibrated on a local scale without AMeDAS．
2）The occurrences of the events of high intensity are fewer due to smaller orographical effect over the sea than
over the land
Because of the effects in2）the actual underestimation of Radar－AMeDAS over the sea is thought to be
smaller than the v甲ues in Fig・4・7・7・In addition・when only a total areal amount for a rather large area is
considered，thQ influence of underestimation at high intensity might be quite small because appearance rates
at high intensity（around40mm／h）are much s；naller than those at lower intensity（around5mm／h）。
Appendix（1）
In this appendix，all the variables are the same as those in Section4．5．4．3．1，and it is，assumed that瓦＊makes
ゐa the minimum among three candidates，and瓦＊一△the second smallest．
The following are some situations when weighting coefficientsε，ζ，ηare determined．
1）When the difference betweenゐa（Xも＊一△）andゐa（瓦＊）is large，．Xもshould be close to瓦＊in order to keep
ゐa small，which leads to large weighting ofζandη．On the other h3nd，when there is little difference
betweenゐ、（瓦＊一△）andゐ、（Xも＊），the increase inゐ、is small as long as Xもis almost between瓦＊一△and
Xも＊十△，and．X』satisfies Eq．（4．5．15）．Therefore，it is more effective to put larger weighting toεthan to
others，in order thatゐfor Radarわand radars other than a may be sma11．
2）Ifゐa（瓦＊）is large，there are two possibilities：（1）．Xも＊is not proper，or（2）the assumption in Eq。（4．5。15）
has some errors．In this case，in order to avoid false modification of these parameters，smaller weighting
is given to the three coefficients．This treatment makes the influence of．L2for these radars on the total
L2in Eq．（4．5．17）small in the early stage of the iteration．
3）When all the data for Radar6are of the same beam height，ゐ，can be resolved only when corresponding
heights of Radar a show different values．Actually，Eq．（4。5．11）can be transformed as follows because B蝕
is independent of pointた：
一109一
Technical Reports of the MRI，No、392000
孚（α・た／Eわた）2 写（Eみ是／α・疋）2
12（X、，Xみ）＝K －1＋κ 一1 ｛孚（ααた／Eゐを）｝2 ｛孚（Eみた／ασた）｝2
This equation does not include瓦．Hence，C、6becomes zero，and Xも＊camot be detemined in this case．
Here，when all the data for Radarわare of nearly the same beam height，the reliability of Xも＊must be low，even
if they may be derived．In this case，the coefficient C、δmay be smal1，and differences amongゐ’s for three
occasions may also be sma11．
In the same way，when all of the beam height values of Radar a are almost the same，the reliability of X』＊
is sma11，and the coefficient C、δis large，although the coefficients may be derived．
Withtheseconsiderations，theactualweightingisdetemined．Specificvaluesforthecoefficientsusedinthis
study are as follows：
（a）
ε＝隅，
∫＝夙聡，
η＝05隅耽，
の 照311」2、（Xb），
ゆ ゆ ロ ％胃｛0．25＋（／2、（Xドム）一12、（Xわ））／12、（Xわ）｝，
were1／WI is limited between O．1and1．0．
（b）When C is nearly O（smaller than1／16），ζis1，whileεandηare O．
（c）When C is over16，ηis1／2，whileεandζare O．
Appendix（2）：Weighting strategy for determining As
Coefficientsαandγused in Eq．（4．5．19）are detemine（1as follows．
（a〉γis set larger when△is derived from the readings of raingauges at the target time，thanwhen derivedfrom
statistical values．
（b）αis fixed to zero when there is no area where the radar observations overlap．
Whenradar echoes are overthesea only，△isnot obtainedatthetargettimelthevaluebasedonthestatistics
for the last240hours is then adopted as its substitute．
WhenΣβ、δ々are actually calculated，weighting is applied to、every point concemed such that the total weight
for every altitude level sliced at500－m intervals is the same，and the number of samples for a radar estimate
less than l mm／h is restricted to half ofthat for the other intensities，so as not to place excessive weight on small
intensities in the analysis for the logarithm．
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111一
気象研究所技術報告一覧表
第1号 バックグラウンド大気汚染の測定法の開発（地球規模大気汚染特別研究班，1978）
Development of Monitoring Techniques for Global Background Air Pollution．（MRI Special Research
Group on Global Atmospheric Pollution，1978）
第2号 主要活火山の地殻変動並びに地熱状態の調査研究（地震火山研究部，1979）
Investigation of Ground Movement andGeothermal State of MainActiveVolcanoes inJapan、（Seismo1－
ogy and Volcanology Research Division，1979）
第3号 筑波研究学園都市に新設された気象観測用鉄塔施設（花房龍男・藤谷徳之助・伴野 登・魚津 博，1979）
On the Meteorological Tower and Its Observational System at Tsukuba Science City．（T．Hanafusa，T．
Fujitani，N．Banno，and H．Uozu，1979）
第4号 海底地震常時観測システムの開発（地震火山研究部，1980）
Permanent Ocean－Bottom Seismograph Observation System．（Seismology and Volcanology Research
Division，1980）
第5号 本州南方海域水温図一400m（又は500m）深と1，000m深一（1934－1943年及び1954－1980年）（海洋研究部，
1981）
Horizontal Distribution of Temperature in400m（or500m）and1，000m Depth in Sea South of Honshu，
Japan and Westem－North Pacific Ocean from1934to1943and from1954to1980．（Oceanographical
Research Division，1981）
第6号 成層圏オゾンの破壊につながる大気成分及び紫外日射の観測（高層物理研究部，1982）
Observations of the Atmospheric Constituents Related to the Stratospheric ozon Depletion and the
Ultraviolet Radiation．（Upper Atmosphere Physics Research Division，1982）
第7号 83型強震計の開発（地震火山研究部，1983）
Strong－Motion Seismograph Mode183for the Japan Meteorological Agency Network．（Seismology
an（1Volcanology Research Division，1983）
第8号 大気中における雪片の融解現象に関する研究（物理気象研究部，1984）
The Study ofMelting of Snowflαkes inthe Atmosphere．（Physicα1Meteorology ResearchDivision，1984）
第9号 御前崎南方沖における海底水圧観測（地震火山研究部・海洋研究部，1984）
Bottom Pressure Observation South off Omaezaki，Central Honsyu．（Seismology and Volcanology
Research Division and Oceanographical Research Division，1984）
第10号 日本付近の低気圧の統計（予報研究部，1984）
Statistics on Cyclones around Japan．（Forecast Research Division，1984）
第11号 局地風と大気汚染質の輸送に関する研究（応用気象研究部，1984）
Observations and Numerical Experiments on Local Circulation and Medium－Range Transport of Air
Pollutions．（Applied Meteorology Research Division，1984）
第12号 火山活動監視手法に関する研究（地震火山研究部，1984）
Investigation on the Techniques for Volcanic Activity Surveillance．（Seismology and Volcanology
Research Division，1984）
第13号 気象研究所大気大循環モデルー1（MRI・GCM－1）（予報研究部，1984）
A Description of the MRI Atmospheric General Circulation Mode1（The MRI・GCM－1）．（Forecast
Research Division，1984）
第14号 台風の構造の変化と移動に関する研究一台風7916の一生一（台風研究部，1985）
A Study onthe Changes ofthe Three－Dimensional Structure andthe Movement Speed ofthe Typhoon
through its Life Time．（Typhoon Research Division，1985）
第15号 波浪推算モデルMRIとMRI－IIの相互比較研究一計算結果図集一（海洋気象研究部，1985）
AnIntercamparisonStudybetweentheWaveModelsMRI andMRI－II－ACompilation of Results一．
（Oceanographical Research Division，1985）
第16号 地震予知に関する実験的及び理論的研究（地震火山研究部，1985）
Study on Earthquake Prediction by Geophysical Method。（Seismology and Volcanology Research
Division，1985）
第17号 北半球地上月平均気温偏差図（予報研究部，1986）
Maps of MonthlyMeanSurface TemperatureAnomalies overthe Northem Hemispherefor1891－1981．
（Forecast Research Division，1986）
第18号 中層大気の研究（高層物理研究部・気象衛星研究部・予報研究部・地磁気観測所，1986）
Studies of the Middle Atmosphere．（Upper Atmosphere Physics Research Division，Meteorological
Sate胱e Research Division，Forecast Research Division，MRI and the Magnetic Observatory，1986）
第19号 ドップラーレーダによる気象・海象の研究（気象衛星研究部・台風研究部・予報研究部・応用気象研究部・
海洋研究部，1986）
Studies on Meteorological and Sea Surface Phenomena by Doppler Radar．（Meteorological Satellite
Research Division，Typhoon Research Division，Forecast Research Division，Applied Meteorology
Research Division，and Oceanographical Research Division，1986）
第20号 気象研究所対流圏大気大循環モデル（MRI・GCM－1）による12年間分の積分（予報研究部，1986）
Mean Statistics ofthe Tropospheric MRI・GCM－I based on12－year Integration．（Forecast Research
Division，1986）
第21号 宇宙線中間子強度1983－1986（高層物理研究部，1987）
Multi－Directional Cosmic Ray Meson Intensity1983－1986．（Upper Atmosphere Physics Research
Division，1987）
第22号 静止気象衛星「ひまわり」画像の噴火噴煙データに基づく噴火活動の解析に関する研究（地震火山研究部，
1987）
Study on Analysis of Volcanic Emptions based on Eruption Cloud Image Data obtained by the
Geostationary Meteorological satellite（GMS）．（Seismology and Volcanology Research Division，1987）
第23号 オホーツク海海洋気候図（篠原吉雄・四竈信行，1988）
Marine Climatological Atlas of the sea of Okhotsk。（Y．Shinohara and N．Shikama，1988）
第24号 海洋大循環モデルを用いた風の応力異常に対する太平洋の応答実験（海洋研究部，1989）
Response Experiment of Pacific Ocean to Anomalous Wind Stress with Ocean General Circulation
Mode1．（Oceanographical Research Division，1989）
第25号 太平洋における海洋諸要素の季節平均分布（海洋研究部，1989）
Seasonal Mean Distribution of Sea Properties in the Pacific．（Oceanographical Research Division，1989）
第26号 地震前兆現象のデータベース（地震火山研究部，1990）
Database of Earthquake Precursors．（Seismology and Volcanology Research Division，1990）
第27号 沖縄地方における梅雨期の降水システムの特性（台風研究部，1991）
Characteristics of Precipitation Systems During the Baiu Season in the Okinawa Area，（Typhoon
Research Division，1991）
第28号 気象研究所・予報研究部で開発された非静水圧モデル（猪川元興・斉藤和雄，1991）
Descriptionof a Nonhydrost段ticModelDeveloped attheForecastResearchDepartmentoftheMRL（M。
Ikawa an（1K．Saito，1991）
第29号 雲の放射過程に関する総合的研究（気候研究部・物理気象研究部・応用気象研究部・気象衛星・観測システ
ム研究部・台風研究部，1992）
A Synthetic Study on Cloud－Radiation Processes．（Climate Research Department，Physical Meteoro1－
ogy Research Department，Applied Meteorology Research Department，Meteorological Satellite and
Observation System Research Department，，and Typhoon Research Department，1992）
第30号 大気と海洋・地表とのエネルギー交換過程に関する研究（三上正男・遠藤昌宏・新野 宏・山崎孝治，1992）
Studies of Energy Exchange Processes between the Ocean－Ground Surface and Atmosphere．（M．
Mikami，M．Endoh，H．Niino，and K．Yamazaki，1992）
第31号 降水日の出現頻度からみた日本の季節推移一30年間の日降水量資料に基づく統計一（秋山孝子，1993）
Seasonal Transition in Japan，as Revealed by Appearance Frequency of Precipitating－Days．一Statistics
of Daily Precipitation Data During30Years一（T．Akiyama，1993）
第32号 直下型地震予知に関する観測的研究（地震火山研究部，1994）
Observational Study on the Prediction of Disastrous Intraplate Earthquakes．（Seismology and Volcano1－
ogy Research Department，1994）
第33号 各種気象観測機器による比較観測（気象衛星・観測システム研究部，1994）
Intercomparisons of Meteorologica10bservation Instruments．（Meteorological Satellite and Observa－
tion System Research Department，1994）
第34号 硫黄酸化物の長距離輸送モデルと東アジア地域への適用（応用気象研究部，1995）
The Long－Range Transport Model of Sulfur Oxides and Its Application to the East Asian Region．
（Applied Meteorology Research Department，1995）
第35号 ウインドプロファイラーによる気象の観測法の研究（気象衛星・観測システム研究部，1995）
Studies on Wind Profiler Techniques for the Measurements of Winds．（Meteorological Satellite and
Observation System Research Department，1995）
第36号 降水・落下塵中の人工放射i生核種の分析法及びその地球化学的研究（地球化学研究部，1996）
Geochemical Studies and Analytical Methods of Anthropogenic Radiomclides in Fallout Samples．
（Geochemical Research Department，1996）
第37号 大気と海洋の地球化学的研究（1995年及び1996年）（地球化学研究部，1998）
Geochemical Study of the Atmosphere and Ocean in1995and1996．（Geochemical Research Department，
1998）
第38号 鉛直2次元非線形間題（金久博忠，1999）
Vertically2－dmensional Nonlinear Problem，（H．Kanehisa，1999）
気象研究所1946（昭和21）年設立
嵩中山所長
夫輝允顯男
禎洋正豊龍
住藤木中房
吉近八田花
博 博
理 理
理博 高島 勉理博’吉田明夫理博 宇治 豪理博 伏見克彦
予報研究部 部長気候研究部 部長台風研究部 部長物理気象研究部 部長
環境・応用気象研究部 部長
気象衛星・観測
システム研究部 部長
地震火山研究部 部長
海洋研究部 部長地球化学研究部 部長
気象研究所技術報告
近・藤洋輝編集委員長
子正和
尚美秀
畠谷枝
北高松
郎子幾文
志
豊直珠孝
上野田田
井清安岡
勝志修博
政正
藤堀内藤
垣
加深上佐
編集委員
事務局
気象研究所技術報告は，1978年（昭和53）年の初刊以来，気象研究所が必要の都度発行する刊行物であ
り，原則として気象研究所職員及びその共同研究者による気象学，海洋学，地震学その他関連の地球科学に
関する技術報告，資料報告および総合報告（以下報告という）を掲載する。
気象研究所技術報告の編集は，編集委員会が行う。編集委員会は原稿の掲載の可否を判定する。
本紙に掲載された報告の著作権は気象研究所に帰属する。本紙に掲載された報告を引用する場合は，出
所を明示すれば気象研究所の許諾を必要としない。本紙に掲載された報告の全部又は一部を複製，転載
翻訳，あるいはその他に利用する場合は気象研究所の許諾を得なければならない。個人が研究，学習，
教育に使用する場合は，出所を明示すれば気象研究所の許諾を必要としない。
気象研究所技術報告 ISSNO386－4049
第39号 平成12年2月29日 発 行
饗簿馨気象研究所 〒305－0052茨城県つくば市長峰1－1
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