QUICK REFERENCE TO DISTRIBUTIONS USED IN FLOOD FREQUENCY ANALYSIS
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1 Quic reference: distriutions used in flood frequency analysis QUICK REFERENCE TO DISTRIBUTIONS USED IN FLOOD FREQUENCY ANALYSIS Normal distriution Cumulative distriution function: 1 ( a) ep π d, < < Cumulative distriution function of the standard normal variale Z: where: Φ( z z a 1 z ep dz π Parameter estimation (method of moments and method of maimum lielihood): a, s Calculation of c.d.f. using tales: a z Φ( Calculation of c.d.f. in MS Ecel: F NORMDIST(,a,,TRUE) or F NORMSDIST( Calculation of the inverse c.d.f. using tales: F z ( a + z Calculation of the inverse c.d.f. in MS Ecel: NORMINV(F,a,) or a + *NORMSINV( 1
2 Quic reference: distriutions used in flood frequency analysis Lognormal distriution Cumulative distriution function: 1 (ln a) ep d, 0 < < π 0 If random variale X is lognormally distriuted, then the random variale: Y ln X is normally distriuted. With transformation Y a Z lognormal distriution ecomes the standard normal distriution. a ln, ln(1 + C v ) Parameter estimation (method of maimum lielihood): a y, S y Calculation of c.d.f. using tales: y a y ln z Φ( Calculation of c.d.f. in MS Ecel: F LOGNORMDIST(,a,) or F NORMSDIST( Calculation of the inverse c.d.f. using tales: F z y a + z ( Calculation of the inverse c.d.f. in MS Ecel: LOGINV(F,a,) or EXP(a + *NORMSINV() NOTE: It is possile to use ase 10 logarithms instead of natural logarithms, i.e. to use the transformation Y log X. Parameter estimates y the method of moments are then given with: a log ln10, v ln(1 + C ln 10 ) However, MS Ecel functions LOGNORMDIST and LOGINV wor with natural logarithms only. If ase 10 logarithms are used, then calculations should e made with transformation Z (log X a)/ and with functions for standard normal distriutions NORMSDIST and NORMSINV. y e
3 Quic reference: distriutions used in flood frequency analysis Pearson type 3 distriution Cumulative distriution function: c 1 c Γ( a) a 1 c ep d 4 a, C s S C s, c a Calculation using tales: Tales for the Pearson type 3 distriution provide relationship etween frequency factor K, values of c.d.f. and coefficient of sewness C s. The frequency factor is defined with K( ( S Note that there is no need to estimate parameters in order to mae distriution calculations using tales. Calculation of c.d.f. using tales: K, S C s Calculation of inverse c.d.f. using tales: F, C s K( ( + S K( Calculation in MS Ecel: MS Ecel features uilt-in functions for the two-parameter gamma distriution, GAMMADIST(,a,,cum) and GAMMAINV(F,a,), whose application is limited to positive values of the sew coefficient C s and parameter. These functions should e used for calculation of the Pearson type 3 distriution in the following way. Calculation of c.d.f. in MS Ecel: for C s > 0 and > 0: F GAMMADIST((-c)/,a,1,TRUE) for C s < 0 and < 0: F 1 GAMMADIST((-c)/,a,1,TRUE) Calculation of inverse c.d.f. in MS Ecel: for C s > 0 and > 0: c + * GAMMAINV(F,a,1) for C s < 0 and < 0: c + * GAMMAINV(1-F,a,1) 3
4 Quic reference: distriutions used in flood frequency analysis Log-Pearson type 3 distriution The easiest way to apply the log-pearson type 3 distriution is to mae logarithmic transformation: Y ln X or Y log X and to apply the Pearson type 3 distriution to the random variale Y. Gumel (EV1) distriution Cumulative distriution function: u F ( ep ep, < < Inverse cumulative distriution function: ( u + [ ln( ln ] Standard Gumel random variale is otained with transformation: X u Y and the standard Gumel distriution is: { [ y] } G( y) ep ep having inverse form: y( ln( ln u S, 0.78S General etreme value (GEV) distriution Cumulative distriution function: c ep 1 1/, 0, for < 0 : for > 0 : > c + / < c + / Inverse cumulative distriution function: ( c + [1 ( ln ] shape parameter is determined numerically from: Γ(1 + 3) + 3Γ(1 + ) Γ(1 + ) Γ (1 + ) f ( ) sgn( ) 3/ [ Γ(1 + ) Γ (1 + )] Parameters and c are then calculated from: 3 C s 4
5 Quic reference: distriutions used in flood frequency analysis S sgn( ) [ Γ(1 + ) Γ c [ 1 Γ(1 + )] Gamma function in MS Ecel: Γ( EXP(GAMMALN() (1 + )] 1/ Eponential distriution Cumulative distriution function: 1 ep a Inverse cumulative distriution function: ( a ln(1 Parameter estimation: a Calculation of c.d.f. in MS Ecel: F EXPONDIST(,a,TRUE) Two-parameter Weiull distriution Cumulative distriution function: 1 ep a Inverse cumulative distriution function: a[ ln(1 ] 1/ shape parameter is determined numerically from: Γ(1 + / ) f + Γ (1 + 1/ ) ( ) 1 C v and parameter a is then calculated from: a Γ( 1+ 1/ ) Gamma function in MS Ecel: Γ( EXP(GAMMALN() Calculation of c.d.f. in MS Ecel: F WEIBULL(,,a,TRUE) 5
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