Download Statistical Tropical Cyclone Forecast Models

Statistical Tropical Cyclone
Forecast Models
Mark DeMaria
NOAA/NESDIS Center for Satellite
Applications and Research
AMS Short Course Notes
January 21, 2008
Outline
• Introduction and Terminology
• Short history of NHC statistical TC models
• The SHIPS intensity model
– Application of linear regression
• The TC rapid intensity index
– Application of discriminant analysis
• Advanced fitting techniques
– Neural Networks and Genetic Algorithms
• Class exercise
Why Use Statistical Models?
• Standard NWP model limitations
–
–
–
–
–
Grid resolution
Predictability
Physical parameterizations
Treatment of terrain, local effects
Model biases
• Statistical Models
– Model Output Statistics (MOS)
– Perfect Prog
• Both based on linear regression
– Classification
• Linear discriminant analysis
Model Output Statistics (MOS)
• y = a 1x 1 + a 2x 2 + … a N x N + b
• y = Predicted quantity (dependent variable)
– Surface temp, precipitation amount and type, visibility,
etc
• xi, i = 1, 2 … N
– Quantities from model forecast related to y
• Independent variables
– Can also include past data and climate input, latitude,
longitude, Julian Day, etc
• ai, b = regression coefficients
MOS Regression Coefficients
• Training sample
– Several years of model forecasts
– “Ground truth” observations
– Independent validation data (if possible)
• Can use cross validation if necessary
• Least-squares fit
E = ½(yn-On)2 n=1,2 … N, N=sample size
Oi=observations, yn=linear model prediction
Set E/b =0 and E/ai = 0 to get equations for
regression coefficients
MOS Development
• MOS Advantages
– Direct relationship between predicted variable and
model forecasts
– Model biases corrected
– Takes into account forecast degradation with time
• MOS Disadvantages
– Modelers almost never leave their models alone
– Data and assimilation changes can also impact model
performance and bias
– Model forecast archive files are very large
“Perfect Prog” Approach
• Use observations or analyses for regression
model development
• Use forecast fields for real-time prediction
• Advantages
– Don’t need an archive of forecasts
– Prediction improves as model forecast improves
• Disadvantages
– Model forecast biases not corrected
– Predictor forecast degradation with time not included
Tropical Cyclone Statistical Model Types
• Statistical
– Use only basic storm information at or before t=0
• lat, lon, max winds, Julian Day
– Climatology and Persistence (CLIPER) models
• Statistical-Synoptic
– Add predictors from t=0 model fields (analyses)
• Statistical-Dynamical
– Add predictors from model forecasts
– Near all statistical-dynamical TC models use perfectprog approach
Long History of NHC Statistical
Track Forecast Models
•
•
•
•
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•
•
Riehl, Haggard, Sanborn
Miller-Moore
Travelers-59, -60
NHC-64, 67, 72
NHC-73, 83, 90, 98
HURRAN
CLIPER
(SS)
(SS)
(SS)
(SS)
(SD)
(S)
(S)
1959-1964
1959-1964
1959-1964
1964-1988
1973-2006
1970-1986
1971-present
• 1970’s to early 1990’s was “Heyday” of SS and SD track models
• Replaced by 3-D primitive equation models in 1990’s and 2000’s
•
•
S = statistical, SS=statistical synoptic, SD=statistical-dynamical
Underline= still run operationally at NHC
Shorter History of NHC
Statistical Intensity Models
• SHIFOR
• SHIPS
• SHIPS
•
•
•
•
(S)
(SS)
(SD)
1988-present
1991-1995
1996-present
SHIFOR = CLIPER-type intensity model
SHIPS = Statistical Hurricane Intensity Prediction Scheme
S = statistical, SS=statistical synoptic, SD=statistical-dynamical
Underline= still run operationally at NHC
Best Atlantic Intensity Models
(48 hr error, 1988-2007)
1.2
SHIFOR
SHIPS
GFDL
20
07
20
06
20
05
20
04
20
03
20
02
20
01
20
00
19
99
19
98
19
97
19
96
19
95
19
94
19
93
19
92
19
91
19
90
19
89
19
88
0
Year
GFDL = NCEP version of GFDL coupled ocean-atmosphere hurricane model
(Experimental in 1992, Operational in 1995)
HWRF = follow-on to GFDL (Operational in 2007)
Current Statistical TC Forecast
Techniques used by NHC
•
CLIPER and SHIFOR
– Track and intensity forecast skill baseline models (regression)
•
SHIPS
– SD intensity model (regression)
•
LGEM
–
•
hybrid dynamical, statistical model (regression for model growth rate)
Rapid Intensity Index
– Discriminant analysis technique for classification
•
Annular Hurricane Index
– Discriminant analysis technique for classification
•
Wind radii CLIPER
– NESDIS version with idealized vortex (least squares for vortex fit)
– NHC version (regression)
•
Rainfall CLIPER
– Climatological rainfall rate along forecast track (least squares)
•
Tropical cyclone formation probability product
– NESDIS product with discriminant analysis technique
•
Wind probability products
– Monte Carlo technique to estimate probability of 34, 50 and 64 kt winds
Case Study: The Statistical
Hurricane Intensity Prediction
Scheme (SHIPS)
•
•
•
•
•
Original Motivation
Statistical Philosophy
Mathematical Formulation
Predictors
Model Performance
Hurricane Joan 1988
Statistical “Philosophy”
• Use physical reasoning to select predictors
– Especially for higher-order terms (quadratic, etc)
• Require statistical significance at 1% level
• Normalize variables so prediction coefficients are in units
of standard deviations
• Backwards stepwise procedure
• Include at least one ENSO cycle in developmental
sample
• Perfect prog approach
• Test on independent cases
• Bill Gray, AT796 Tropical Meteorology
– “Look at your data”
SHIPS Dependent Variable
• Intensity is measured by maximum sustained 1minute surface winds (V)
• Predicted quantity is intensity change over give
forecast interval
– Separate regression equations for 0-6, 0-12, …, 0120 hr forecasts
• Sample restricted to storms over water
– 1982-2006 sample
• Kaplan and DeMaria (1995, 2001) inland decay
model used over land
Physical Reasoning for
Predictor Selection
Hurricane Katrina August 2005
Physical Reasoning for
Predictor Selection
Hurricane Debby August 2000
2007 Atlantic SHIPS Dependent
Variables (Predictors)
• Climatology and Persistence type (1-4)
–
–
–
–
V at t=0
V t=-12 to t=0 hours
Julian Day variable
Zonal component of storm motion
• From GFS model analyses or forecasts (5-13)
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–
–
–
–
–
–
–
850-200 hPa vertical shear (0-500 km avg)
200 hPa divergence (0-1000 km avg)
850 hPa vorticity (0-1000 km avg)
200 and 250 hPa temperature (200-800 km avg)
700-500 avg hPa relative humidity (200-800 km avg)
Vertical instability parameter (200-800 km avg)
850 hPa tangential wind change (0-600 km avg, 0 to fcst time)
Pressure where environmental winds best match storm motion
2007 Atlantic SHIPS Dependent
Variables (Predictors)
• From Reynold’s SST fields (14)
– Maximum Potential Intensity at storm center
minus initial intensity
• From satellite data (15-16)
– Std Deviation of IR brightness T (100-300 km)
– Oceanic Heat Content at storm center (from
satellite altimetry)
2007 Atlantic SHIPS Dependent
Variables (Predictors)
• Quadratic terms (17-21)
– Square of SST potential
– V(0)* V (t=-12)
– V(0)*Shear
– V(0)*GOES TB Std Dev
– Shear*sine(latitude)
Statistical Calculations
• Input for each forecast interval
– Dependent variable yn = Vn n=1,2 …, N
N = sample size
– Independent variables xjn j=1,2 …, J
J = no. of predictors (21)
• Find sample mean and std deviation of yn and xjn
• Calculate normalized dependent and
independent variables
_
Yn = (yn-y)/y , Similarly for xjn
Assume Linear Model
• Yn = a1X1n + a2X1n + … aJXJn
– Don’t need constant term with normalized input
• Compare model predictions (Yn ) with
observed intensity changes from NHC best
track (On)
• Find coefficients ai to minimize model error
1
E
2N
n N
2
(
Y

O
)
 n n
n 1
Coefficient Calculation
• Set E/aj = 0 for j=1,2 … J
a = C-1b
a = [a1, a2, …, aJ]T
b = [b1, b2, …, bJ]T
n=N
bj = (1/N) (XjnOj)
n=1
n=N
Cij = (1/N) (XinXjn) = covariance matrix elements
n=1
• Use standard statistical tests to calculate P-values for
coefficients
– Probability that the coefficient is significantly different than zero
• Model R2 = Percent of variance of observations explained by
the model
Normalized Coefficient
-0.2
-0.4
GOES Tb Std Dev * Vo
Steering layer press
GFS Vortex Tend
Vor 850
Div 200
Rel Hum
Ocean Heat Content
GOES Cold Pixel Count
Predictor
Vert Instab
T 250
T 200
Shear * sin(lat)
Shear * Vo
Shear
SST potential **2
SST potential
Vo
Zonal motion
Julian Day
Persistence * Vo
-0.6
Persistence
48 hr SHIPS Normalized
Predictor Coefficients
1.2
1
0.8
0.6
0.4
0.2
0
Predictor Magnitudes versus Forecast
Interval for Shear and Persistence
0.7
0.5
0.4
Persistence
Shear
0.3
0.2
0.1
Forecast Interval (hr)
120
114
108
102
96
90
84
78
72
66
60
54
48
42
36
30
24
18
12
0
6
Normalized Coefficient
0.6
SHIPS Output to Forecasters 1
* ATLANTIC SHIPS INTENSITY FORECAST
*
GOES/OHC INPUT INCLUDED
* DEAN
AL042007 08/17/07 00 UTC
TIME (HR)
0
V (KT) NO LAND 85
V (KT) LAND
85
V (KT) LGE mod 85
6
87
87
86
12 18 24 36 48 60
91 95 99 104 109 110
91 95 99 104 109 110
87 89 91 98 106 113
*
*
*
72
117
117
120
84
119
119
125
96 108 120
124 120 117
124 79 84
127 81 94
SHEAR (KTS)
14 10
9
6
6 5
3
6
6
7
9
9
7
SHEAR DIR
274 266 263 199 198 310 123 338 333 64 84 66 46
SST (C)
28.6 28.6 28.7 28.7 28.8 29.0 28.9 29.3 29.6 30.1 30.0 29.2 29.9
POT. INT. (KT) 149 149 150 150 151 154 153 160 165 173 172 157 170
ADJ. POT. INT. 158 155 156 155 155 158 157 164 167 173 168 152 164
200 MB T (C)
-52.8 -52.9 -52.8 -52.1 -52.0 -52.4 -51.7 -51.9 -51.2 -51.1 -50.3 -50.2 -49.5
TH_E DEV (C)
11 11 11 12 11
8
10 10 11
8
11
9
10
700-500 MB RH 58 59 60 60 61 63 63 62 62
59
64 66 67
GFS VTEX (KT) 17 19 20
22 22 20 21 19 23 23
28
25 26
850 MB ENV VOR 14 14 14 29 28 59 94
73 79 65
84
80 87
200 MB DIV
54 52 48 61 17 31 49
64 76 60
77
81 48
LAND (KM)
504 415 427 436 328 277 182 100 164 352 150 -17 249
LAT (DEG N)
14.0 14.3 14.6 14.9 15.1 15.6 16.2 16.9 17.8 18.8 19.9 21.2 22.5
LONG(DEG W) 57.7 59.7 61.6 63.5 65.3 68.7 72.3 76.1 79.8 83.1 85.9 88.9 92.1
STM SPEED (KT) 21 19 19 18 17 17
18 19 17 15 15 16 16
HEAT CONTENT
70 70 72 66 72 55 104 95 134 130 134 29 62
FORECAST TRACK FROM OFCI
INITIAL HEADING/SPEED (DEG/KT):275/ 22
CX,CY: -21/ 2
T-12 MAX WIND: 85
PRESSURE OF STEERING LEVEL (MB): 622 (MEAN=625)
GOES IR BRIGHTNESS TEMP. STD DEV. 100-300 KM RAD: 14.1 (MEAN=20.0)
% GOES IR PIXELS WITH T < -20 C 50-200 KM RAD: 92.0 (MEAN=69.0)
SHIPS Output to Forecasters 2
INDIVIDUAL CONTRIBUTIONS TO INTENSITY CHANGE
6 12 18 24 36 48 60 72 84 96 108 120
------------------------------------------------------------------SAMPLE MEAN CHANGE
1. 2. 3. 4. 6. 8. 9. 10. 11. 11. 12. 13.
SST POTENTIAL
2. 5. 7. 9. 10. 9. 6. 3. 1. 0. -3. -6.
VERTICAL SHEAR
-1. -2. -2. -2. 0. 2. 5. 7. 9. 9. 10. 11.
PERSISTENCE
0. -1. -1. -1. -1. -1. -1. -1. -1. -1. 0. 0.
200/250 MB TEMP.
0. 0. -1. -1. -2. -2. -3. -4. -5. -6. -7. -8.
THETA_E EXCESS
0. 0. 0. 0. 0. 0. -1. -1. -1. -2. -2. -2.
700-500 MB RH
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -1. -1.
GFS VORTEX TENDENCY 0. 1. 2. 2. 1. 2. 0. 3. 3. 7. 4. 4.
850 MB ENV VORTICITY
0. 0. 0. 0. 0. 1. 2. 2. 3. 3. 4. 4.
200 MB DIVERGENCE
0. 0. 1. 1. 1. 2. 3. 4. 4. 5. 6. 5.
ZONAL STORM MOTION
0. 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
STEERING LEVEL PRES
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
DAYS FROM CLIM. PEAK
0. 0. 0. 0. 0. 0. 0. 0. 0. -1. -1. -1.
-----------------------------------------------------------------SUB-TOTAL CHANGE
2. 5. 10. 14. 19. 23. 24. 29. 31. 35. 32. 29.
SATELLITE ADJUSTMENTS -----------------------------------------------------------------MEAN ADJUSTMENT
0. 0. 0. 0. -1. -1. -1. -1. -1. -1. -1. -2.
GOES IR STD DEV
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
GOES IR PIXEL COUNT
0. 1. 1. 1. 1. 0. 0. 0. 0. 0. 0. 0.
OCEAN HEAT CONTENT
0. 0. 0. 0. 0. 1. 2. 3. 5. 6. 5. 4.
-------------------------------------------------------------------TOTAL ADJUSTMENT
0. 0. 0. 0. 0. 1. 1. 2. 3. 4. 3. 3.
-------------------------------------------------------------------TOTAL CHANGE (KT)
2. 6. 10. 14. 19. 24. 25. 32. 34. 39. 35. 32.
Forecast Evaluation
• Evaluation of SHIPS, GFDL, NHC Official and
SHIFOR forecasts for 5-year Atlantic sample
(2003-2007)
• Compare forecasts with NHC best track
intensities
– Mean Absolute Error
– Bias
• usually small for statistical models
• Skill
– Percentage reduction in forecast error relative to a
baseline forecast
– Climatology and Persistence model (SHIFOR) used
for skill baseline
Mean Absolute Error
(2003-2007)
30
Intensity Error (kt)
25
SHIFOR
GFDL
SHIPS
20
NHC
15
10
5
0
12
24
36
48
60
72
Forecast Interval (hr)
84
96
108
120
Forecast Skill
(2003-2007)
30
Forecast Skill (%)
20
10
0
12
24
36
48
60
72
84
96
108
120
-10
NHC
SHIPS
-20
GFDL
-30
Forecast Interval (hr)
SHIPS vs Observed 48 hr
Intensity Change (2003-2007)
125
100
2
R = 0.5721
Observed Intensity Change (kt)
75
50
25
0
-125
-100
-75
-50
-25
0
25
50
-25
-50
-75
-100
-125
SHIPS Forecast Intensity Change (kt)
75
100
125
Rapid Intensity Index (RII)
• Rapid Intensification (RI) defined by percentiles
of intensity change PDF
– 95th percentile of Atlantic sample = 30 kt
– 90th percentile of Atlantic sample = 25 kt
• Classification problem
– How do you separate the two groups?
• RI from non-RI
• NHC’s operational Rapid Intensity Index
– Based on linear discriminant analysis
– Component of the SHIPS model
Linear Discriminant Analysis
• Developmental data
– Group classification
• Is this an RI case or not?
– Observations that help to distinguish between
the two groups (discriminators xj)
• SHIPS predictors for the 24 hour forecast
• Discriminant function
– Linear combination of discriminators
d = a1x1 + a2x2 + … aJxJ
Discriminant Weights
• Choose weights to maximize separation of mean
inputs between the two groups
• Maximize [aT(x1 - x2)]2/(aTCpoola)
a = (a1, a2 … aJ)T
x1= (x11, x21, … xJ1)T (group 1 means)
x2= (x12, x22, … xJ2)T (group 2 means)
Cpool = common covariance matrix for
the two groups
• Optimal weights:
a = [Cpool]-1(x1 - x2)
Group Estimation
• Average distance between the two groups
m = ½(aTx1 + aTx2)
• For given xo, calculate discriminate value
do = aTxo
If do ≥ m, assign to group 1
If do < m, assign to group 2
Input for Operational RII
1.
2.
3.
4.
5.
6.
7.
Previous 12 hr intensity change
850-200 hPa vertical shear
200 hPa divergence
SST potential – Initial intensity
850-700 hPa relative humidity
GOES TB std deviation (100-300 km)
Percent GOES pixels colder than -30oC
(50-200 km)
RII Output to Forecasters
** 2007 ATLANTIC RAPID INTENSITY INDEX AL042007 DEAN
08/17/07 00 UTC **
( 25 KT OR MORE MAX WIND INCREASE IN NEXT 24 HR)
12 HR PERSISTENCE (KT):
850-200 MB SHEAR (KT) :
D200 (10**7s-1)
:
POT = MPI-VMAX (KT) :
850-700 MB REL HUM (%):
% area w/pixels <-30 C:
STD DEV OF IR BR TEMP :
0.0 Range:- 45.0 to 30.0 Scaled/Wgted Val:
9.1 Range: 35.1 to 3.2 Scaled/Wgted Val:
46.4 Range: -20.0 to 149.0 Scaled/Wgted Val:
70.6 Range: 8.1 to 130.7 Scaled/Wgted Val:
72.0 Range: 57.0 to 88.0 Scaled/Wgted Val:
87.0 Range: 17.0 to 100.0 Scaled/Wgted Val:
14.1 Range: 37.5 to 5.3 Scaled/Wgted Val:
Scaled RI index= 4.4 Prob of RI= 30% is 2.4 times the sample mean(12%)
Discrim RI index= 4.2 Prob of RI= 29% is 2.4 times the sample mean(12%)
0.6/
0.8/
0.4/
0.5/
0.5/
0.8/
0.7/
0.9
0.6
0.4
1.0
0.1
0.5
0.6
Neural Networks
Neural Network Transfer Function
T(x)
x
T(x) = 1/(1 + e-x)
Example
• Start with training data consisting of
observed intensity change (y) predicted by
shear (w) and SST potential (x)
Intensity
Shear
SST Potential
…
…
…
y
w
x
Example
• Have neural network with inputs w,x, two
hidden nodes and an output y
• h₁ = a₁T(x) + a₂T(w)
• h₂ = a₃T(x) + a₃T(w)
• y = b₁T(h₁) + b₂T(h₂)
w
h1
y
x
h2
Genetic Algorithms
• General search algorithms inspired by
biology
• Solutions to problems are encoded
• Encoded solutions can be thought of as the
DNA of the solution
• Initial population of randomly generated
solutions is generated
• Each generation, solutions are evaluated
using a “fitness” function
• Solutions with better fitness functions have a
higher probability to breed
Genetic Algorithms
• Breeding performed by mixing solution
encodings
• Encodings in the population can be randomly
altered to mutate the population
• Optionally, the lowest performing members of
the population can be culled and replaced
• Mutation and culling helps prevent getting
stuck in local minima and maxima
• Process continued until a desired fitness has
been reached or until a set number of
generations have passed
Example
• Define error function: E = ∑(y - O) ²
• Encoding for GA is simply a list of neural
network weights
• Randomly generate a population of neural
network weights and run the GA using the
error function as the fitness function
• Breeding performed by swapping random
elements of two sets of network weights
Summary
• NHC has long history of operational statistical
tropical cyclone models
• Statistical track models replaced by dynamical
models
• Intensity, structure, genesis models still used
• Most developed from “perfect prog” approach
• Most use multiple regression (e.g., SHIPS) or
discriminant analysis (e.g., RII)
• Statistical “by-products” also useful to
forecasters
• More sophisticated methods under development
– Neural networks and genetic algorithms.
References
•
•
•
•
•
DeMaria, M., M. Mainelli, L.K. Shay, J.A. Knaff and J. Kaplan, 2005: Further
Improvements in the Statistical Hurricane Intensity Prediction Scheme (SHIPS). Wea.
Forecasting, 20, 531-543.
DeMaria, M., and J.M. Gross, 2003: Hurricane! Coping with Disaster, edited by
Robert Simpson, Chapter 4: Evolution of Tropical Cyclone Forecast Models.
American Geophysical Union, ISBN 0-87590-297-9, 360 p.
Kalnay, E., 2003: Atmospheric Modeling, Data Assimilation and Predictability.
Cambridge University Press, ISBN 0-521-79629-6, 341 p.
Russell S and P. Norvig 2003:. Artificial Intelligence: A Modern Approach, Second
Edition. Upper Saddle River, New Jersey: Pearson Education Inc, 1047 p.
Wilks, D.S., 2006: Statistical Methods in the Atmospheric Sciences, 2nd Edition.
Academic Press, ISBN 13: 978-0-12-751966-1, 627 p.