Statistical%20Tropical%20Cyclone%20Forecast%20Models

1
Statistical Tropical Cyclone Forecast Models

• Mark DeMaria
• NOAA/NESDIS Center for Satellite Applications and
  Research

AMS Short Course Notes January 21, 2008
2
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

3
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

4
Model Output Statistics (MOS)

• y a1x1 a2x2 aNxN 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

5
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 n1,2 N, Nsample size
• Oiobservations, ynlinear model prediction
• Set ?E/?b 0 and ?E/?ai 0 to get equations for
  regression coefficients

6
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

7
Perfect Prog Approach

• Use observations or analyses for regression model
  development
• Use forecast fields for real-time prediction
• Advantages
• Dont need an archive of forecasts
• Prediction improves as model forecast improves
• Disadvantages
• Model forecast biases not corrected
• Predictor forecast degradation with time not
  included

8
Tropical Cyclone Statistical Model Types

• Statistical
• Use only basic storm information at or before t0
• lat, lon, max winds, Julian Day
• Climatology and Persistence (CLIPER) models
• Statistical-Synoptic
• Add predictors from t0 model fields (analyses)
• Statistical-Dynamical
• Add predictors from model forecasts
• Near all statistical-dynamical TC models use
  perfect-prog approach

9
Long History of NHC Statistical Track Forecast
Models

• Riehl, Haggard, Sanborn (SS)
  1959-1964
• Miller-Moore (SS) 1959-1964
• Travelers-59, -60 (SS) 1959-1964
• NHC-64, 67, 72 (SS) 1964-1988
• NHC-73, 83, 90, 98 (SD) 1973-2006
• HURRAN (S) 1970-1986
• CLIPER (S) 1971-present
• 1970s to early 1990s was Heyday of SS and SD
  track models
• Replaced by 3-D primitive equation models in
  1990s and 2000s
• S statistical, SSstatistical synoptic,
  SDstatistical-dynamical
• Underline still run operationally at NHC

10
Shorter History of NHC Statistical Intensity
Models

• SHIFOR (S) 1988-present
• SHIPS (SS) 1991-1995
• SHIPS (SD) 1996-present
• SHIFOR CLIPER-type intensity model
• SHIPS Statistical Hurricane Intensity
  Prediction Scheme
• S statistical, SSstatistical synoptic,
  SDstatistical-dynamical
• Underline still run operationally at NHC

11
Best Atlantic Intensity Models (48 hr error,
1988-2007)
GFDL NCEP version of GFDL coupled
ocean-atmosphere hurricane model (Experimental
in 1992, Operational in 1995) HWRF follow-on to
GFDL (Operational in 2007)
12
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

13
Case Study The Statistical Hurricane Intensity
Prediction Scheme (SHIPS)

• Original Motivation
• Statistical Philosophy
• Mathematical Formulation
• Predictors
• Model Performance

14
Hurricane Joan 1988
15
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

16
SHIPS Dependent Variable

• Intensity is measured by maximum sustained
  1-minute surface winds (V)
• Predicted quantity is intensity change over give
  forecast interval
• Separate regression equations for 0-6, 0-12, ,
  0-120 hr forecasts
• Sample restricted to storms over water
• 1982-2006 sample
• Kaplan and DeMaria (1995, 2001) inland decay
  model used over land

17
Physical Reasoning for Predictor Selection
Hurricane Katrina August 2005
18
Physical Reasoning for Predictor Selection
Hurricane Debby August 2000
19
2007 Atlantic SHIPS Dependent Variables
(Predictors)

• Climatology and Persistence type (1-4)
• V at t0
• ?V t-12 to t0 hours
• Julian Day variable
• Zonal component of storm motion
• From GFS model analyses or forecasts (5-13)
• 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

20
2007 Atlantic SHIPS Dependent Variables
(Predictors)

• From Reynolds 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)

21
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
• Shearsine(latitude)

22
Statistical Calculations

• Input for each forecast interval
• Dependent variable yn ?Vn n1,2 , N
• N sample size
• Independent variables xjn j1,2 , 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

23
Assume Linear Model

• Yn a1X1n a2X1n aJXJn
• Dont 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

24
Coefficient Calculation

• Set ?E/?aj 0 for j1,2 J
• a C-1b
• a a1, a2, , aJT
• b b1, b2, , bJT
• nN
• bj (1/N)? (XjnOj)
• 
  n1
• 
  nN
• Cij (1/N)? (XinXjn) covariance matrix
  elements
• 
  n1
• 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

25
48 hr SHIPS Normalized Predictor Coefficients
26
Predictor Magnitudes versus Forecast Interval for
Shear and Persistence
27
SHIPS Output to Forecasters 1
ATLANTIC SHIPS INTENSITY
FORECAST
GOES/OHC INPUT INCLUDED
DEAN AL042007
08/17/07 00 UTC TIME (HR)
0 6 12 18 24 36 48
60 72 84 96 108 120 V (KT) NO
LAND 85 87 91 95 99 104 109
110 117 119 124 120 117 V (KT) LAND
85 87 91 95 99 104 109
110 117 119 124 79 84 V (KT) LGE mod
85 86 87 89 91 98 106 113
120 125 127 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
(MEAN625) GOES IR BRIGHTNESS TEMP. STD DEV.
100-300 KM RAD 14.1 (MEAN20.0) GOES IR
PIXELS WITH T lt -20 C 50-200 KM RAD 92.0
(MEAN69.0)
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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.
29
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

30
Mean Absolute Error(2003-2007)
31
Forecast Skill (2003-2007)
32
SHIPS vs Observed 48 hr Intensity Change
(2003-2007)
33
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
• NHCs operational Rapid Intensity Index
• Based on linear discriminant analysis
• Component of the SHIPS model

34
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

35
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)

36
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 lt m, assign to group 2

37
Input for Operational RII

1. Previous 12 hr intensity change
2. 850-200 hPa vertical shear
3. 200 hPa divergence
4. SST potential Initial intensity
5. 850-700 hPa relative humidity
6. GOES TB std deviation (100-300 km)
7. Percent GOES pixels colder than -30oC (50-200 km)

38
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) 0.0 Range-
45.0 to 30.0 Scaled/Wgted Val 0.6/ 0.9
850-200 MB SHEAR (KT) 9.1 Range 35.1 to
3.2 Scaled/Wgted Val 0.8/ 0.6 D200
(107s-1) 46.4 Range -20.0
to 149.0 Scaled/Wgted Val 0.4/ 0.4 POT
MPI-VMAX (KT) 70.6 Range 8.1 to
130.7 Scaled/Wgted Val 0.5/ 1.0 850-700 MB
REL HUM () 72.0 Range 57.0 to 88.0
Scaled/Wgted Val 0.5/ 0.1 area w/pixels
lt-30 C 87.0 Range 17.0 to 100.0
Scaled/Wgted Val 0.8/ 0.5 STD DEV OF IR BR
TEMP 14.1 Range 37.5 to 5.3
Scaled/Wgted Val 0.7/ 0.6 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)
39
Neural Networks
40
Neural Network Transfer Function
T(x)
x
T(x) 1/(1 e-x)
41
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
42
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
43
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

44
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

45
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

46
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.

47
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.