A A.2 ENSO A A.4 85hPa A A
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1 A 1 A A A A A A A.7 EOF A.7.1 REOF A.8 SVD
2 A A.2 ENSO A A.4 85hPa A A A A.8 5hPa A.9 Nino A.1 NinoC SST A A A A A.15 n A.16 1 MSLP EOF A.17 EOF A.18 NDVI SVD
3 A.1 Z A A A A.5 F A A A A
4 A A.1 Pearson x y ( ) n σ x σ y Pearson r = 1 n n i=1 ( x i x σ x )( y i y σ y ) (A.1) r = n i=1 (x i x)(y i y) n i=1 (x i x) 2 n i=1 (y (A.2) i y) 2 r r 1 A
5 S xy = 1 n n i=1 (x i x)(y i y) S xx = 1 n n i=1 (x i x) 2 = σx 2 y = a + bx (A.3) a b y x : b = = n i=1 y ix i ( n i=1 y i)( n i=1 x i)/n n i=1 x2 i ( n i=1 x i) 2 /n n i=1 (y i y)(x i x) n i=1 (x i x)(x i x) (A.4) (A.5) = S xy S xx (A.6) a : n n a = ( y i )/n b( x i )/n i=1 = y bx i=1 S xy xy S xx x 2
6 (A.2 ) : r = = n i=1 (x i x)(y i y) n i=1 (x i x) 2 n i=1 (y i y) 2 (A.7) S xy Sxx S yy (A.8) = S xy σ x σ y (A.9) A.6 A.9 b = r = r σ y σ x Syy S xx (A.1) (A.11) b σ x = σ y = 1 x = ȳ = y = rx (b): b = r σ y σ x = =.198 C/mm 1mm.2 C (A.3).1414( C) 2 ( A.1(a)).7533( C) / % = 18.8% σ 2 ŷ σ 2 y = r 2 r 2 1% = (.433) 2 1% = 18.8% 3
7 Z n 3 Z = ln 1 + r 2 1 r (A.12).433 n = 52 (A.12) 52 3 Z = ln = Z A.1.1 Z 1.1 =.99 A.1.99 Z Z.1 = 2.33 Z t t = r n 2 1 r 2 (A.13) t = = t.5 52 t.5 = t ( n 3) t t (A.12) (A.13) A.2 t 2 ν = n 2 ( ) 4
8 A.1: ( Z ) ( F (z) = 1 2π z e z2 2 dz ) z
9 A.2: ( t ). 9% 95% 99% 99.9% 9% 95% 99% 99.9%
10 A.3: (ν) (n) (r( t)) r( t) < ν/n A.15 ν/n A.16 n 2 ( ) Dawdy Matalas(1964) (ν) ( n) ν n = [1 r x( t)r y ( t)] [1 + r x ( t)r y ( t)] (A.14) Leith(1973) (ν) ( n) ν n = 1 2 ln[r( t)] (A.15) Bretherton (1999) : ν n = [1 r( t)2 ] [1 + r( t) 2 ] (A.16) A.3 x i y i 1 r( t) =.2973 A.15 ν T = n 1 ln[r( t)] = 52ln(.2973) = 32 2 r( t) =.43 ν P = n = 52 ν = ( )/2 = 42 A
11 T ( C) 27 (a) T, P (mm) 12 1 (b) P, (c) T, (d) P, 6 8 T ( C) P (mm) A.1: (a) (b) (c) (d) (outlier ) Pearson r(a, b) =.433; r(c, d) = % ν = 32 99% A.16 ν T = n [1 r( t)2 ] [1 + r( t) 2 = 52 [ ] [ ] =
12 A.2: ENSO , 1999 ENSO ENSO ( 8 N 6 S 93% ) ENSO.38 ENSO.41 ENSO 2 7 ENSO ENSO 7.58 ENSO 2.77( A.2) ENSO 2 N 2 S ENSO ENSO 9
13 A.3 (b) (c) (d) Pearson A.1 r(c, d) =.255 r(a, b) =.433 A.2 99% 95% ( ) A hPa 95% Φ 21/2 Φ 7 1
14 A.3: (a) ;(b) 2 (c) (d) D. L. Hartmann, 22 11
15 6 N.2 4 N N E 6 E 8 E E.2 12 E.2 14 E 16 E A.4: 26 ( ) 85hPa A.2 Spearman (robust) x i y i ( ) n x i y i Dx i Dy i Spearman ( r s ) r s = n n i=1 (Di x D x )(D i y D y ) i=1 (Di x D x ) 2 n i=1 (Di y D y ) 2 (A.17) r s = 1 6 n i=1 (Di x D i y) 2 n(n 2 1) (A.18) (A.18) 12
16 Z Z = r s 1/(n 1) (A.19) ( A.1) ( A.1a,b) r s (a, b) =.47;.433 Z Z =.47 1/(52 1) = A =.95 Z Z Z ( A.1c,d) 95% Spearman r s =.34 Z Z =.34 1/(52 1) = Z.5 = % 13
17 A.3 (harmonic analysis) y t = cos(α) t 9 cos(α) = sin(α + π) 2 α = 1 α A π 2πt n = 2πt t = C C C C= = φ = 2π 7 n = 2π 7 12 = 7π 6 14
18 T ( C) Month A.5: y t = cos( 2πt 12 7π 6 ) y t = y + C cos( 2πt n φ) (A.2) y t = cos( 2πt 12 7π 6 ) (A.21) A.2 y t = y + C cos( 2πt n φ) = y + A cos( 2πt n ) + B sin(2πt n ) (A.22) (A.23) A= C cos(φ) B= C sin(φ) C= A 2 + B 2 15
19 A.4: t y t cos( 2πt ) 12 2πt sin( 12 y tcos( 2πt) 12 y tsin( 2πt) n i= A = 2 n B = 2 n n i=1 n i=1 y t cos( 2πt n ) y t sin( 2πt n ) (A.24) (A.25) A B : arctan( B ), A > A φ = arctan( B ) ± π, A < A π, A = 2 16
20 A.4 A = 2 n B = 2 n n i=1 n i=1 y = = 12.4 C y t cos( 2πt n ) = 2 ( ) = y t sin( 2πt n ) = 2 ( ) = C= B 2 + A 2 = = φ = arctan( B ) ± π =.5436 ± π φ = φ = A y t = cos( 2πt ) = cos( 2πt ) A.21 Matlab 21 17
21 Matlab 21 > load tmpbj21.mat; > n=1:365; > figure;plot(n,tmpbj21); grid on;hold on; > xlabel( Day );ylabel( T ( ^\circc) ); > axis([ length(n) min(tmpbj21)-2 max(tmpbj21)+2]); > ybar=mean(tmpbj21); > A=2/length(n)*tmpbj21*cos(2*pi*n(:)/length(n)); > B=2/length(n)*tmpbj21*sin(2*pi*n(:)/length(n)); > C=sqrt(A^2+B^2); > if A > phase=atan(b/a); end > if A < phase=pi+atan(b/a); end > if A == phase=pi/2; end > yhat=ybar+c*cos(2*pi*n/length(n)-phase); > plot(n,yhat, r--, LineWidth,1.5); > figure;plot(n,tmpbj21-yhat);grid on; > xlabel( Day );ylabel( \DeltaT ( ^\circc) ); 18
22 y t ( ) n/2 y t = y + [C k cos( 2πkt n φ k)] (A.26) k=1 n/2 = y + k=1 [A k cos( 2πkt n A k B k ) + B ksin( 2πkt )] (A.27) n A k B k = 2 n = 2 n n t=1 n t=1 y t cos( 2πkt n ) (A.28) y t sin( 2πkt n ) (A.29) arctan( B k A k ), A k > φ k = arctan( B k A k ) ± π, A k < π, A 2 k = k 21 k = 1 C k = A 2 k + B2 k k = 2 A.6 k = S k = 1 2 C2 k 5 5 S k = 1 2 i=1 5 Ck 2 =.3813 i=1 (.7533 C 2 ).3813/.7533 = 5% 2 94% 19
23 A.5: 2 F k = F k k = F k F k = 1 Ck 2/2 2 (s 2 Ck 2/2)/(n 2 1) (A.3) s 2 ν 1 = 2 ν 1 = n 2 1 F F F k F k A.7 2 k = 1, 3, 16 F A.5 F F 1 95% F.5 = k = k = 1 n = 52 k = 3 16 n = = 17.3 = 3.3 k N 4 N 18 E 18 W 4 5km A hPa (k = 1,..., 4) 1 Matlab Fa=finv(.95,2,49) ν 1 = 2, ν 2 = 49, 95% Fa=3.19 2
24 1.5 k=1 1.5 k=2 T ( C).5 T ( C) k=3 1.5 k=4 T ( C).5 T ( C) T ( C) 1 k= Year T ( C) 2 k= Year A.6: k = 1,..., % k A.7: 2 21
25 A.6 k = A.4 (1) C 2 k ( ) ( A.5) a 1 f = ω k 2π = k n p = 1 f = n k k = 1 p = n = 52 k k 2 k = n = 26 2 k = 27 2 k > n 2 k = n 2 Nyquist n k = n n/2 = 2 1 n n (f) (p) k (k) C 2 k C2 k A.5 C 2 k 22
26 56 (a) Φ 5 * (b) Φ (c) k=1 (d) k= (e) k=3 (f) k= A.8: hPa (a) (Φ 5 ) (b) ( Φ 5) (c) (d)
27 (2) (τ) n n 1 3 τ = n 3 (τ = M) R(τ) = 1 n τ n τ t=1 ( y t y σ y )( y t+τ y σ y ) τ =... M Ŝ k = B M 1 k M [R() + 2 R(τ)cos( πkτ M τ=1 ) + R(M) cos(πk)] (A.31) B k B k = { 1, k = 1,..., M 1 1, k =, M 2 Ŝk Barlett ( ) Hanning Hamming 24
28 S k α =.5 R α = 1+tα n 2 (y n 1 t ) R(1) R α t α α ν = n 2 t α =.5 ν = n 2 = 5 t α = R α = =.2128 n 1 R(1) =.2935 > R α S r = S k 1 R(1) R(1) 2 2R(1)cos( πk M ) 95% S.5 r = S r χ 2.5 (2n M 2 )/M = S k 1 R(1) R(1) 2 2R(1)cos( πk M ) 95% S.5 r = S k χ 2.5 (2n M 2 )/M χ2.5 (2n M 2 )/M χ 2.5 α =.5 ν = (2n M 2 )/M χ2 ( 2 ) S k S r 95% (Sr.5 ) k P k = 2M k k 2 k = 1 2 A.7 τ max = M 2 Matlab α =.5, ν χ 2 Chia=chi2inv(.95,ν) 25
29 A.7 τ function [num,period, Frequency, Density, CL95]=spectrum(x,mLAG) % function for power spectral analysis % usage: [num,period, Frequency, density, cl95]=spectrum(x,mlag) xlen=length(x); SER=x;N=xLEN; mlagwk=mlag;mlen=n;j=mlag;j1=j+1; %calculating auto-covariance coefficient A=.; C=.; for I=1:N A=A+SER(I);end % I A=A/N; for I=1:N SER(I)=SER(I)-A; C=C+SER(I).^2; end % I C=C/N; for L=1:J CC(L)=.; for I=1:N-L CC(L)=CC(L)+SER(I)*SER(I+L); end %I CC(L)=CC(L)./(N-L); CC(L)=CC(L)/C; end %L C=1.; % estimating raw power spectra SPE(1)=.; for L=1:J-1 SPE(1)=SPE(1)+CC(L); end %L SPE(1)=SPE(1)./J+(C+CC(J))./(2*J); for L=1:J-1 % SPE(L+1)=.; for I=1:J-1 SPE(L+1)=SPE(L+1)+CC(I)*cos(pi*L*I/J); end % I SPE(L+1)=2*SPE(L+1)./J+C./J+(-1).^L*CC(J)./J; end % SPE(J1)=.; for I=1:J-1 SPE(J1)=SPE(J1)+(-1).^I*CC(J); end %I SPE(J1)=SPE(J1)/J+(C+(-1).^J*CC(J))/(2*J); %smoothing power spectra PS(1)=.54*SPE(1)+.46*SPE(2); for L=2:J PS(L)=.23*SPE(L-1)+.54*SPE(L)+.23*SPE(L+1); end %L PS(J1)=.46*SPE(J)+.54*SPE(J1); %statistical significence of PS W=.; for L=1:J-1 W=W+SPE(L+1); end %L W=W/J+(SPE(1)+SPE(J1))/(2*J); if (J > fix(n/2)) W=2.57*W; end if(j == fix(n/2)) W=2.49*W; end if(j < fix(n/2) & J > fix(n/3)) W=2.323*W; end if (J == fix(n/3)) W=2.157*W; end if (J < fix(n/3)) W=1.979*W; end %the red noice examination for L=1:J1 SK(L)=W*(1-CC(1).^2)/(1+CC(1).^2-2*CC(1)*cos( *(L-1)/J)); end % L if (CC(1) > & CC(1) >= CC(2) ) %the white noice examination else for L=1:J1 SK(L)=W; end %L end % if %calculating the length of cycle T(1)=NaN; for L=2:J1 T(L)=(2.*J)/(L*1.-1.); end % L num=1:j+1;num=num(:)-1; Period=T(:); Frequency=1./T(:); Density=PS(:); CL95=SK(:); 26
30 ( ) A.9 Nino3 SST ENSO 2 8 ENSO ( ) A.5 k = k = (k) ( A.6) A.5 (Cross spectral analysis ) 27
31 A.6: M = 15 M = 16 k 95% k 95%
32 4 3 SST anomaly( C) Nino3 SST, Nino3 SST, Spectral density.1.5 Spectral density Frequency (month 1 ) Frequency (month 1 ).15 Nino3 SST, Nino3 SST, Spectral density Frequency (month 1 ) Spectral density Frequency (month 1 ) A.9: Nino3 95% 29
33 y t x t R xx (τ) R yy (τ) R xy (τ) 3 R yx (τ) τ = 1... M M x y A.31 S x (k) S y (k) x y { ˆP xy (k) = B k M R xy () + M 1 τ=1 } [R xy (τ) + R yx (τ)] cos( πkτ M ) + R xy(m) cos(πk) k =,..., M B k = { 1, k = 1,..., M 1 1, k =, M 2 P xy () = 1 2 ˆP xy () ˆP xy (1); P xy (k) = 1 4 ˆP xy (k 1) ˆP xy (k) ˆP xy (k + 1); k = 1,..., M 1 P xy (M) = 1 2 ˆP xy (M 1) ˆP xy (M) k P k = 2M k x y ˆQ xy (k) = 1 M M 1 τ=1 [R xy (τ) R yx (τ)] sin( πkτ M ) k = 1,..., M ˆQ xy () = ˆQ xy (M) = Q xy (τ) 3 R xy (τ) = 1 n τ n τ t=1 ( xt x σ x yt+τ y )( σ y ) 3
34 Q xy () = 1 2 ˆQ xy () ˆQ xy (1) = 1 2 ˆQ xy (1); Q xy (k) = 1 4 ˆQ xy (k 1) ˆQ xy (k) ˆQ xy (k + 1); k = 1,..., M 1 Q xy (M) = 1 2 ˆQ xy (M 1) ˆQ xy (M) = 1 2 ˆQ xy (M 1) x y CO 2 xy(k) k = 1,..., M CO 2 xy(k) = P 2 xy(k) + Q 2 xy(k) S x (k)s y (k) (CO 2 xy(k)) x y ( k 2M 2M k ) F = (ν 1)CO2 xy(k) 1 CO 2 xy(k) ν = (2n M )/M α F ν1 = 2 2 ν2 = 2(ν 1) F 4 α F α 2M k x y x y φ(k) φ xy (k) = arctan( Q xy(k) P xy (k) ) φ xy (k) φxy(k)m kπk CROSPEC.FOR (, 1984 ) 5 S 5 N 5hPa 584gpm hPa 585gpm gpm 4 Matlab F a. α =.5, ν1 = 2, ν2 = 3, F α = finv(.95, 2, 3) =
35 % ( ) A.1: NinoC SST SST (τ) ENSO NinoC A.1 95% 46 SST SST 6 A.6 32
36 A.7: T( C) C Year A.11: / ( A.7) A (w) 33
37 (w) x y t = w k = w k 2L + 1 L k= L w k x t+k f y x f C y (f) C x (f) H(f) = C y(f) C x (f) H (FIR finite impulse response) H(f) = w + 2 L w k cos(2kπf t) k=1 (A.32) f w k k w, t 1 1 t 1 A.32 H(f) = w + 2 L w k cos(2kπf) k= ( A.12) f = f =
38 1.8 H(f) f A.12: ( ) n n! B k = =, k =, 1,..., n k k!(n k)! 3 n = 3 1 = 2 B = 2!! 2! = 1 B 1 = 2! 1! 1! = 2 B 2 = 2! 2!! = ( 1/4 2/4 1/4) n = /16 4/16 6/16 4/16 1/16 A.8 n = A.13 9 (f >.3) f <.1 6% ( ) Matlab nchoosek(n,k) nchoosek(2,)=1; nchoosek(4,2)=6 35
39 (a).8 (b) w k.2.15 H(f) k f A.13: 9 (a) (b) (2L + 1 1)/2 = L 2L L L L L (Gaussian filter) A.9 A % 5% A f s f > f s f f s { 1, f f s H(f) =, f s < f
40 A.8: n k = Bk wk Bk wk Bk wk Bk wk Bk wk Bk wk
41 A.9: n
42 1 n=5 1 n=7 H(f).5 H(f) n=9 1 n=13 H(f).5 H(f) n=15 1 n=17 H(f).5 H(f) f f A.14:, 39
43 1.2 1 n=2 n= n=2 n=4.8.8 H(f).6.4 H(f) f f A.15: n, Lanczos w k = + H(f) cos(2πfk) df f n f t = t 2n [ ] w k = 1 n H() + 2 H( t 2n 2n ) cos(2πk t 2n ) t=1 f s =.25 n = 5 w 4 = w 3 =.12 w 2 = w 1 =.32 w =.5 w 1 = w 1 =.32 w 2 = w 2 = w 3 = w 3 =.12 w 4 = w 4 = n = 5 n = 2 ( A.15) Lanczos sin(πk/n) πk/n w k = sin( πk n ) πk n w k, 1 k n (n-1) 4
44 n y n = K J a k x n k + b j y n j k= j=1 Matlab Butterworth [b,a]=butter(ord,wa) ord wa 1 1 π Nyquist (1) 1.1 Nyquist.5 wa =.1/.5 =.2 (2) ord = 9 [b,a]=butter(ord,wa) b a (3) b a [hw,w]=freqz(b,a) hw w w (f = w ) plot(w/2/pi,hw) 2π (4) b a yt=filtfilt(b,a,x)yt yt Butterworth (1) wa =.1/.5 =.2, wa =.2/.5 =.4 (2) ord= 9 [b,a]=butter(9,[.2.4]) b a (3) b a [hw,w]=freqz(b,a) plot(w/2/pi,hw) (4) b a yt=filtfilt(b,a,x) 41
45 A.7 EOF (empirical orthogonal function, EOF) (eigenvector analysis) (principal component analysis, PCA) Lorenz 195 EOF X m n X X T C m m = 1 n X XT X C X ( C 1) C C (λ 1,...,m ) V m m m m C m m V m m = V m m m m = λ 1... λ λ m λ λ 1 > λ 2 >... > λ m X λ 42
46 EOF λ 1 EOF V EOF 1 = V (:, 1) λ k V k EOF k = V (:, k) EOF X ( ) P C m n = Vm m T X m n P C PC(1,:) EOF X EOF (PC) EOF PC X X = EOF P C EOF X EOF PC 1 P C P n CT = PC E E T = I I 1 EOF C Matlab EOF 1 [EOF,E]=eig(C) EOF E P C = EOF T X ( 1km NDVI ) m C 2 X X = U V T ( ) 43
47 C = 1 n XXT C λ = nλ C = XX T C λ = λ X 1 m Xi 2 = i=1 m λ k = k=1 m P Ck 2 X λ k k=1 λ k m i=1 λ i 1% EOF North (1982) 95% λ = λ 2 λ N ( 4 ) λ λ A EOF 44 N
48 [E,V]=eig(C) X % X=[ ; ]; X(1,:)=X(1,:)-mean(X(1,:)); X(2,:)=X(2,:)-mean(X(2,:)); X X= %%% co-variance matrix C=X*X /5; C= [EOF,E]=eig(C); % V: eigenvectors; E: eigenvalues PC=EOF *X; %% reverse the order E=fliplr(flipud(E)) lambda=diag(e); % retain eigenvalues only EOF=fliplr(EOF) PC=flipud(PC) EOF= E= PC= %%check EOF*EOF % = I EOF PC*PC /5 % = lambda PC EOF*PC % =X X
49 [U,S,V]=svd(X) X X=[ ; ]; X(1,:)=X(1,:)-mean(X(1,:)); X(2,:)=X(2,:)-mean(X(2,:)); X [U,S,V]=svd(X); U= S= V= EOF=U; PC=S*V ; PC= E=S.^2/5; %=lambda E E= EOF*PC % =X X
50 EOF1 26.1% Eigenvalue Number PC# Year hpa A.16: 1 EOF. (a) 95% (b) (c) (d) +σ (hpa) NCEP/NCAR 47
51 X EOF P C EOF P C (k) = P C(k) λk EOF (k) = EOF (k) λ k PC 1 X P C X A.16(d) North (1) (2) 16 EOF ( A.17) (3) 1 SLP m n m m A = 1 n XX B = XX 48
52 (a) EOF1 9.7% (b) EOF1 15.8% (c) EOF1 13.4% (d) EOF1 11.3% A.17: EOF (a) (b) (c) (d)
53 C = X X C = X X, X X XX C V = V V C V C V a = X V V a C n V k V k = 1 λk V a (:, k) V P C = V T X EOF m m EOF m n n X 5 2 XX 5 2 X=[ ] [V1,E1]=eig(X*X ); %% V1=fliplr(V1);%% E1=fliplr(flipud(E1));%% V1=
54 E1= [V2,E2]=eig(X *X);%% V2=fliplr(V2);%% E2=fliplr(flipud(E2));%% V2= E2= E1 E2 XX Va=X*V2; %% V_k1=Va(:,1)/sqrt(E2(1,1)); XX V_k2=Va(:,2)/sqrt(E2(2,2));.49 51
55 XX 1 n XX hPa (Φ) EOF m = 1512 n = 54 C = ΦΦ T C = Φ Φ NDVI 2 EOF A.7.1 REOF : varimax.m A.8 SVD EOF X EOF(combined empirical orthogonal function CEOF) (singular value decomposition SVD) (canonical correlation analysis CCA) SVD SVD SVD SVD 52
56 X Y m n p n C = 1 XY T C m p n SVD C = U V T U X V Y γ Matlab [U,S,V]=svd(C) X A = U T X Y B = V T Y γ X Y γ 2 γ 2 1% NDVI NDVI SVD NDVI A.18 NDVI % % NDVI NDVI NDVI NDVI NDVI 53
57 EU NDVI EU WP PNA NDVI NAO % NDVI WA PNA SO EA NP ( :,. NDVI. 22, 57(5),55-514) 54
58 7N 6N 5N 4N (A) SVD 1 3N 15W 1W 5W 5E 1E 15E 7N 6N 5N 4N (B) SVD 2 3N 15W 1W 5W 5E 1E 15E 7N 6N 5N 4N (C) SVD 3 3N 15W 1W 5W 5E 1E 15E 7N 6N 5N 4N 3N 7N 6N 5N (D) SVD 4 15W 1W 5W 5E 1E 15E 4N (E) SVD 5 3N 15W 1W 5W 5E 1E 15E 7N 6N 5N 4N 3N 7N 6N 5N 4N 3N (F) SVD 6 15W 1W 5W 5E 1E 15E (G) SVD 7 15W 1W 5W 5E 1E 15E NDVI A.18: NDVI SVD
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