A statistical nowcasting method for an urban wind field at rooftop - - PowerPoint PPT Presentation

A statistical nowcasting method for an urban wind field at rooftop level

1

Ziv Klausner, Eyal Fattal Applied mathematics department

Urban wind field: varying homogeneity

15-8-98 2100 15-8-98 0800

2

Ongoing network of weather stations

Therefore, our approach:

– Define a distribution – Choose a representative sample – Nowcasting using statistical inference

Ideal Reality

4

Population’s spatial distribution

Define the wind vector, in time t, at a point (x, y) as a random variable The population of wind vectors in time t: wind in all possible points (x, y) in a given area A Parameters:

• Expectancy (2D vector)
• Covariance (2x2 matrix)

y x

t ,

U

t



t



 

 

A y x F

y x

t P

  , ;

,

U

A sample of the spatial distribution

5

Given the population, a representative sample can be

• chosen. It consists of n weather stations:

From it, the following statistics are calculated:

• spatial average
• covariance matrix

These are estimators for and

t



t



       

var cov , cov , var

t t t t t t t

u u v u v v        S

t t t

u v        U

 

n x n x x x x x

t t t S

F

, 2 , 2 1 , 1

..., , , U U U 

6

Tolerance region

In terms of repeated sampling, for the i-th sample:

• Ri - tolerance region constructed to assure: E[pi] = P
• pi - actual proportion contained in Ri

We choose to use elliptic Rt in u, v space: D2 – Mahalanobis distance for P

   

 

 

P D R

t t t T t t t t

y x y x

2 1

, ,

;    



U U U U U S

D2(P) for the case of bivariate normal distribution

7

For a population ( and are known): For a sample (around for E[P]):

   

2 , 1

2 2

P P D   

t



t



t

U

        

2 2 , 2 ; 1 1 1 2 ,

1 2

     



n n n P F n n n P D

Chew, Journal of American Statistical Association,1966

Application for the metropolitan area of Tel-Aviv

8

Photographs by Yan Nasonov, Felix Rubinstein, Remi Jouan, EdoM, Paul Simpson, Cccc3333, distributed under a CC-BY 2.0 license.

Stages

9

• 1. Choosing a sample
• 2. Sample’s representativeness examination
• 3. Tolerance region estimation: D2=f(E[P])

Choosing a sample

10

Available DB:

• metropolitan Tel-Aviv
• ~20 weather stations
• rooftop level

Based on meteorological considerations - entry of climatic phenomena to the area:

• sea breeze
• slope winds
• weather fronts

Therefore, weather stations on the area’s perimeter (“fence”)

Sample’s representativeness examination

For each t we’ve calculated:

• spatial average
• covariance matrix

For every Ut we’ve calculated: Once for and another for

11

  

  

t t t T t t t

y x y x

D U U U U U   



, ,

1 2

S

t t t

u v        U

       

var cov , cov , var

t t t t t t t

u u v u v v        S

P t

F U

y x  ,

S t

F U

y x  ,

Sample’s representativeness examination – D2 distributions

12

Tolerance region: D2(E[P])

13

─── D2=2.5, E[P] = 52% ─── D2=5.1, E[P] = 69% ─── D2=7.7, E[P] = 79%

Tolerance region: Empirical vs theoretical

14

 Winter  Spring  Summer  Autumn

Learning: 2 months of each season

15

R2 = 0.9039 R2 = 0.96086  0400  0800  1200  2100

Validation:

• n the remaining month

16

 0400  0800  1200  2100

Discussion

The model:

• Inherent wind speed-

direction correlation

• Includes a distinct

directionality

17

─── D2=2.5, E[P] = 52% ─── D2=5.1, E[P] = 69% ─── D2=7.7, E[P] = 79%

Summary

• Wind field is represented by 4 perimeter stations
• Nowcasting based on O/L measurements

Advantages:

• No need for further historical data
• Relevant for all the area (rooftop level)
• Speed-direction correlation, distinct directionality

18

19