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pdf_utilities.F90
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!-------------------------------------------------------------------------
! $Id$
!===============================================================================
module pdf_utilities
implicit none
private ! Set default scope to private
public :: mean_L2N, &
mean_L2N_dp, &
stdev_L2N, &
stdev_L2N_dp, &
corr_NL2NN, &
corr_NL2NN_dp, &
corr_NN2NL, &
corr_LL2NN, &
corr_LL2NN_dp, &
corr_NN2LL, &
compute_mean_binormal, &
compute_variance_binormal, &
calc_comp_corrs_binormal, &
calc_corr_chi_x, &
calc_corr_eta_x, &
calc_corr_rt_x, &
calc_corr_thl_x, &
calc_xp2
contains
!=============================================================================
pure function mean_L2N( mu_x, sigma2_on_mu2 ) &
result( mu_x_n )
! Description:
! For a lognormally-distributed variable x, this function finds the mean of
! ln x (mu_x_n) for the ith component of the PDF, given the mean of x (mu_x)
! and the variance of x (sigma_sqd_x) for the ith component of the PDF. The
! value ln x is distributed normally when x is distributed lognormally.
! References:
! Garvey, P. R., 2000: Probability methods for cost uncertainty analysis.
! Marcel Dekker, 401 pp.
! -- App. B.
!-----------------------------------------------------------------------
use constants_clubb, only: &
one ! Constant(s)
use clubb_precision, only: &
core_rknd ! Variable(s)
implicit none
! Input Variables
real( kind = core_rknd ), intent(in) :: &
mu_x, & ! Mean of x (ith PDF component) [-]
sigma2_on_mu2 ! Ratio: sigma_x^2 / mu_x^2 (ith PDF component) [-]
! Return Variable
real( kind = core_rknd ) :: &
mu_x_n ! Mean of ln x (ith PDF component) [-]
! Find the mean of ln x for the ith component of the PDF.
mu_x_n = log( mu_x / sqrt( one + sigma2_on_mu2 ) )
return
end function mean_L2N
!=============================================================================
pure function mean_L2N_dp( mu_x, sigma2_on_mu2 ) &
result( mu_x_n )
! Description:
! For a lognormally-distributed variable x, this function finds the mean of
! ln x (mu_x_n) for the ith component of the PDF, given the mean of x (mu_x)
! and the variance of x (sigma_sqd_x) for the ith component of the PDF. The
! value ln x is distributed normally when x is distributed lognormally.
! This function uses double precision variables.
! References:
! Garvey, P. R., 2000: Probability methods for cost uncertainty analysis.
! Marcel Dekker, 401 pp.
! -- App. B.
!-----------------------------------------------------------------------
use constants_clubb, only: &
one_dp ! Constant(s)
use clubb_precision, only: &
dp ! double precision
implicit none
! Input Variables
real( kind = dp ), intent(in) :: &
mu_x, & ! Mean of x (ith PDF component) [-]
sigma2_on_mu2 ! Ratio: sigma_x^2 / mu_x^2 (ith PDF component) [-]
! Return Variable
real( kind = dp ) :: &
mu_x_n ! Mean of ln x (ith PDF component) [-]
! Find the mean of ln x for the ith component of the PDF.
mu_x_n = log( mu_x / sqrt( one_dp + sigma2_on_mu2 ) )
return
end function mean_L2N_dp
!=============================================================================
pure function stdev_L2N( sigma2_on_mu2 ) &
result( sigma_x_n )
! Description:
! For a lognormally-distributed variable x, this function finds the standard
! deviation of ln x (sigma_x_n) for the ith component of the PDF, given the
! mean of x (mu_x) and the variance of x (sigma_sqd_x) for the ith component
! of the PDF. The value ln x is distributed normally when x is distributed
! lognormally.
! References:
! Garvey, P. R., 2000: Probability methods for cost uncertainty analysis.
! Marcel Dekker, 401 pp.
! -- App. B.
!-----------------------------------------------------------------------
use constants_clubb, only: &
one ! Constant(s)
use clubb_precision, only: &
core_rknd ! Variable(s)
implicit none
! Input Variables
real( kind = core_rknd ), intent(in) :: &
sigma2_on_mu2 ! Ratio: sigma_x^2 / mu_x^2 (ith PDF component) [-]
! Return Variable
real( kind = core_rknd ) :: &
sigma_x_n ! Standard deviation of ln x (ith PDF component) [-]
! Find the standard deviation of ln x for the ith component of the PDF.
sigma_x_n = sqrt( log( one + sigma2_on_mu2 ) )
return
end function stdev_L2N
!=============================================================================
pure function stdev_L2N_dp( sigma2_on_mu2 ) &
result( sigma_x_n )
! Description:
! For a lognormally-distributed variable x, this function finds the standard
! deviation of ln x (sigma_x_n) for the ith component of the PDF, given the
! mean of x (mu_x) and the variance of x (sigma_sqd_x) for the ith component
! of the PDF. The value ln x is distributed normally when x is distributed
! lognormally.
! This function uses double precision variables.
! References:
! Garvey, P. R., 2000: Probability methods for cost uncertainty analysis.
! Marcel Dekker, 401 pp.
! -- App. B.
!-----------------------------------------------------------------------
use constants_clubb, only: &
one_dp ! Constant(s)
use clubb_precision, only: &
dp ! double precision
implicit none
! Input Variables
real( kind = dp ), intent(in) :: &
sigma2_on_mu2 ! Ratio: sigma_x^2 / mu_x^2 (ith PDF component) [-]
! Return Variable
real( kind = dp ) :: &
sigma_x_n ! Standard deviation of ln x (ith PDF component) [-]
! Find the standard deviation of ln x for the ith component of the PDF.
sigma_x_n = sqrt( log( one_dp + sigma2_on_mu2 ) )
return
end function stdev_L2N_dp
!=============================================================================
pure function corr_NL2NN( corr_x_y, sigma_y_n, y_sigma2_on_mu2 ) &
result( corr_x_y_n )
! Description:
! For a normally-distributed variable x and a lognormally-distributed
! variable y, this function finds the correlation of x and ln y (corr_x_y_n)
! for the ith component of the PDF, given the correlation of x and y
! (corr_x_y) and the standard deviation of ln y (sigma_y_n) for the ith
! component of the PDF. The value ln y is distributed normally when y is
! distributed lognormally.
! References:
! Garvey, P. R., 2000: Probability methods for cost uncertainty analysis.
! Marcel Dekker, 401 pp.
! -- Eq. B-1.
!-----------------------------------------------------------------------
use constants_clubb, only: &
max_mag_correlation, & ! Constant(s)
zero
use clubb_precision, only: &
core_rknd ! Variable(s)
implicit none
! Input Variables
real( kind = core_rknd ), intent(in) :: &
corr_x_y, & ! Correlation of x and y (ith PDF component) [-]
sigma_y_n, & ! Standard deviation of ln y (ith PDF component) [-]
y_sigma2_on_mu2 ! Ratio: sigma_y^2 / mu_y^2 (ith PDF component) [-]
! Return Variable
real( kind = core_rknd ) :: &
corr_x_y_n ! Correlation of x and ln y (ith PDF component) [-]
! Find the correlation of x and ln y for the ith component of the PDF.
! When sigma_y = 0 and mu_y > 0, y_sigma2_on_mu2 = 0. This results in
! sigma_y_n = 0. The resulting corr_x_y_n is undefined. However, the
! divide-by-zero problem needs to be addressed in the code.
if ( sigma_y_n > zero ) then
corr_x_y_n = corr_x_y * sqrt( y_sigma2_on_mu2 ) / sigma_y_n
else ! sigma_y_n = 0
! The value of sqrt( y_sigma2_on_mu2 ) / sigma_y_n can be rewritten as:
! sqrt( y_sigma2_on_mu2 ) / sqrt( ln( 1 + y_sigma2_on_mu2 ) ).
! This can be further rewritten as:
! sqrt( y_sigma2_on_mu2 / ln( 1 + y_sigma2_on_mu2 ) ),
! which has a limit of 1 as y_sigma2_on_mu2 approaches 0 from the right.
! When sigma_y_n = 0, the value of corr_x_y_n is undefined, so set it
! to corr_x_y.
corr_x_y_n = corr_x_y
endif ! sigma_y_n > 0
! Clip the magnitude of the correlation of x and ln y in the ith PDF
! component, just in case the correlation (ith PDF component) of x and y and
! the standard deviation (ith PDF component) of ln y are inconsistent,
! resulting in an unrealizable value for corr_x_y_n.
if ( corr_x_y_n > max_mag_correlation ) then
corr_x_y_n = max_mag_correlation
elseif ( corr_x_y_n < -max_mag_correlation ) then
corr_x_y_n = -max_mag_correlation
endif
return
end function corr_NL2NN
!=============================================================================
pure function corr_NL2NN_dp( corr_x_y, sigma_y_n, y_sigma2_on_mu2 ) &
result( corr_x_y_n )
! Description:
! For a normally-distributed variable x and a lognormally-distributed
! variable y, this function finds the correlation of x and ln y (corr_x_y_n)
! for the ith component of the PDF, given the correlation of x and y
! (corr_x_y) and the standard deviation of ln y (sigma_y_n) for the ith
! component of the PDF. The value ln y is distributed normally when y is
! distributed lognormally.
! This function uses double precision variables.
! References:
! Garvey, P. R., 2000: Probability methods for cost uncertainty analysis.
! Marcel Dekker, 401 pp.
! -- Eq. B-1.
!-----------------------------------------------------------------------
use constants_clubb, only: &
max_mag_correlation, & ! Constant(s)
zero_dp
use clubb_precision, only: &
dp ! double precision
implicit none
! Input Variables
real( kind = dp ), intent(in) :: &
corr_x_y, & ! Correlation of x and y (ith PDF component) [-]
sigma_y_n, & ! Standard deviation of ln y (ith PDF component) [-]
y_sigma2_on_mu2 ! Ratio: sigma_y^2 / mu_y^2 (ith PDF component) [-]
! Return Variable
real( kind = dp ) :: &
corr_x_y_n ! Correlation of x and ln y (ith PDF component) [-]
! Find the correlation of x and ln y for the ith component of the PDF.
! When sigma_y = 0 and mu_y > 0, y_sigma2_on_mu2 = 0. This results in
! sigma_y_n = 0. The resulting corr_x_y_n is undefined. However, the
! divide-by-zero problem needs to be addressed in the code.
if ( sigma_y_n > zero_dp ) then
corr_x_y_n = corr_x_y * sqrt( y_sigma2_on_mu2 ) / sigma_y_n
else ! sigma_y_n = 0
! The value of sqrt( y_sigma2_on_mu2 ) / sigma_y_n can be rewritten as:
! sqrt( y_sigma2_on_mu2 ) / sqrt( ln( 1 + y_sigma2_on_mu2 ) ).
! This can be further rewritten as:
! sqrt( y_sigma2_on_mu2 / ln( 1 + y_sigma2_on_mu2 ) ),
! which has a limit of 1 as y_sigma2_on_mu2 approaches 0 from the right.
! When sigma_y_n = 0, the value of corr_x_y_n is undefined, so set it
! to corr_x_y.
corr_x_y_n = corr_x_y
endif ! sigma_y_n > 0
! Clip the magnitude of the correlation of x and ln y in the ith PDF
! component, just in case the correlation (ith PDF component) of x and y and
! the standard deviation (ith PDF component) of ln y are inconsistent,
! resulting in an unrealizable value for corr_x_y_n.
if ( corr_x_y_n > real( max_mag_correlation, kind = dp ) ) then
corr_x_y_n = real( max_mag_correlation, kind = dp )
elseif ( corr_x_y_n < -real( max_mag_correlation, kind = dp ) ) then
corr_x_y_n = -real( max_mag_correlation, kind = dp )
endif
return
end function corr_NL2NN_dp
!=============================================================================
pure function corr_NN2NL( corr_x_y_n, sigma_y_n, y_sigma2_on_mu2 ) &
result( corr_x_y )
! Description:
! For a normally-distributed variable x and a lognormally-distributed
! variable y, this function finds the correlation of x and y (corr_x_y) for
! the ith component of the PDF, given the correlation of x and ln y
! (corr_x_y_n) and the standard deviation of ln y (sigma_y_n) for the ith
! component of the PDF. The value ln y is distributed normally when y is
! distributed lognormally.
! References:
! Garvey, P. R., 2000: Probability methods for cost uncertainty analysis.
! Marcel Dekker, 401 pp.
! -- Eq. B-1.
!-----------------------------------------------------------------------
use constants_clubb, only: &
max_mag_correlation, & ! Constant(s)
zero
use clubb_precision, only: &
core_rknd ! Variable(s)
implicit none
! Input Variables
real( kind = core_rknd ), intent(in) :: &
corr_x_y_n, & ! Correlation of x and ln y (ith PDF component) [-]
sigma_y_n, & ! Standard deviation of ln y (ith PDF component) [-]
y_sigma2_on_mu2 ! Ratio: sigma_y^2 / mu_y^2 (ith PDF component) [-]
! Return Variable
real( kind = core_rknd ) :: &
corr_x_y ! Correlation of x and y (ith PDF component) [-]
! Find the correlation of x and y for the ith component of the PDF.
! When sigma_y = 0 and mu_y > 0, y_sigma2_on_mu2 = 0. This results in
! sigma_y_n = 0. The resulting corr_x_y and corr_x_y_n are undefined.
! However, the divide-by-zero problem needs to be addressed in the code.
if ( sigma_y_n > zero ) then
corr_x_y = corr_x_y_n * sigma_y_n / sqrt( y_sigma2_on_mu2 )
else ! sigma_y_n = 0
! The value of sigma_y_n / sqrt( y_sigma2_on_mu2 ) can be rewritten as:
! sqrt( ln( 1 + y_sigma2_on_mu2 ) ) / sqrt( y_sigma2_on_mu2 ).
! This can be further rewritten as:
! sqrt( ln( 1 + y_sigma2_on_mu2 ) / y_sigma2_on_mu2 ),
! which has a limit of 1 as y_sigma2_on_mu2 approaches 0 from the right.
! When sigma_y_n = 0, the value of corr_x_y is undefined, so set it
! to corr_x_y_n.
corr_x_y = corr_x_y_n
endif ! sigma_y_n > 0
! Clip the magnitude of the correlation of x and y in the ith PDF component,
! just in case the correlation (ith PDF component) of x and ln y and the
! standard deviation (ith PDF component) of ln y are inconsistent, resulting
! in an unrealizable value for corr_x_y.
if ( corr_x_y > max_mag_correlation ) then
corr_x_y = max_mag_correlation
elseif ( corr_x_y < -max_mag_correlation ) then
corr_x_y = -max_mag_correlation
endif
return
end function corr_NN2NL
!=============================================================================
pure function corr_LL2NN( corr_x_y, sigma_x_n, sigma_y_n, &
x_sigma2_on_mu2, y_sigma2_on_mu2 ) &
result( corr_x_y_n )
! Description:
! For lognormally-distributed variables x and y, this function finds the
! correlation of ln x and ln y (corr_x_y_n) for the ith component of the
! PDF, given the correlation of x and y (corr_x_y), the standard deviation
! of ln x (sigma_x_n), and the standard deviation of ln y (sigma_y_n) for
! the ith component of the PDF. The value of ln x (or ln y) is distributed
! normally when x (or y) is distributed lognormally.
! References:
! Garvey, P. R., 2000: Probability methods for cost uncertainty analysis.
! Marcel Dekker, 401 pp.
! -- Eq. C-3.
!-----------------------------------------------------------------------
use constants_clubb, only: &
one, & ! Constant(s)
zero, &
max_mag_correlation
use clubb_precision, only: &
core_rknd ! Variable(s)
implicit none
! Input Variables
real( kind = core_rknd ), intent(in) :: &
corr_x_y, & ! Correlation of x and y (ith PDF component) [-]
sigma_x_n, & ! Standard deviation of ln x (ith PDF component) [-]
sigma_y_n, & ! Standard deviation of ln y (ith PDF component) [-]
x_sigma2_on_mu2, & ! Ratio: sigma_x^2 / mu_x^2 (ith PDF component) [-]
y_sigma2_on_mu2 ! Ratio: sigma_y^2 / mu_y^2 (ith PDF component) [-]
! Return Variable
real( kind = core_rknd ) :: &
corr_x_y_n ! Correlation of ln x and ln y (ith PDF component) [-]
! Local Variable
real( kind = core_rknd ) :: &
log_arg ! Input into the ln function [-]
! Find the correlation of ln x and ln y for the ith component of the PDF.
! When sigma_x = 0 and mu_x > 0, x_sigma2_on_mu2 = 0. This results in
! sigma_x_n = 0. The resulting corr_x_y_n is undefined. The same holds
! true when sigma_y = 0 and mu_y > 0. However, the divide-by-zero problem
! needs to be addressed in the code.
if ( sigma_x_n > zero .and. sigma_y_n > zero ) then
! corr_x_y_n = log( one + corr_x_y * sqrt( exp( sigma_x_n**2 ) - one ) &
! * sqrt( exp( sigma_y_n**2 ) - one ) ) &
! / ( sigma_x_n * sigma_y_n )
log_arg = one + corr_x_y * sqrt( x_sigma2_on_mu2 * y_sigma2_on_mu2 )
corr_x_y_n = log( log_arg ) / ( sigma_x_n * sigma_y_n )
else ! sigma_x_n = 0 or sigma_y_n = 0
! The value of corr_x_y_n is undefined, so set it to corr_x_y.
corr_x_y_n = corr_x_y
endif ! sigma_x_n > 0 and sigma_y_n > 0
! Clip the magnitude of the correlation of ln x and ln y in the ith PDF
! component, just in case the correlation (ith PDF component) of x and y,
! the standard deviation (ith PDF component) of ln x, and the standard
! deviation (ith PDF component) of ln y are inconsistent, resulting in an
! unrealizable value for corr_x_y_n.
if ( corr_x_y_n > max_mag_correlation ) then
corr_x_y_n = max_mag_correlation
elseif ( corr_x_y_n < -max_mag_correlation ) then
corr_x_y_n = -max_mag_correlation
endif
return
end function corr_LL2NN
!=============================================================================
pure function corr_LL2NN_dp( corr_x_y, sigma_x_n, sigma_y_n, &
x_sigma2_on_mu2, y_sigma2_on_mu2 ) &
result( corr_x_y_n )
! Description:
! For lognormally-distributed variables x and y, this function finds the
! correlation of ln x and ln y (corr_x_y_n) for the ith component of the
! PDF, given the correlation of x and y (corr_x_y), the standard deviation
! of ln x (sigma_x_n), and the standard deviation of ln y (sigma_y_n) for
! the ith component of the PDF. The value of ln x (or ln y) is distributed
! normally when x (or y) is distributed lognormally.
! This function uses double precision variables.
! References:
! Garvey, P. R., 2000: Probability methods for cost uncertainty analysis.
! Marcel Dekker, 401 pp.
! -- Eq. C-3.
!-----------------------------------------------------------------------
use constants_clubb, only: &
one_dp, & ! Constant(s)
zero_dp, &
max_mag_correlation
use clubb_precision, only: &
dp ! double precision
implicit none
! Input Variables
real( kind = dp ), intent(in) :: &
corr_x_y, & ! Correlation of x and y (ith PDF component) [-]
sigma_x_n, & ! Standard deviation of ln x (ith PDF component) [-]
sigma_y_n, & ! Standard deviation of ln y (ith PDF component) [-]
x_sigma2_on_mu2, & ! Ratio: sigma_x^2 / mu_x^2 (ith PDF component) [-]
y_sigma2_on_mu2 ! Ratio: sigma_y^2 / mu_y^2 (ith PDF component) [-]
! Return Variable
real( kind = dp ) :: &
corr_x_y_n ! Correlation of ln x and ln y (ith PDF component) [-]
! Find the correlation of ln x and ln y for the ith component of the PDF.
! When sigma_x = 0 and mu_x > 0, x_sigma2_on_mu2 = 0. This results in
! sigma_x_n = 0. The resulting corr_x_y_n is undefined. The same holds
! true when sigma_y = 0 and mu_y > 0. However, the divide-by-zero problem
! needs to be addressed in the code.
if ( sigma_x_n > zero_dp .and. sigma_y_n > zero_dp ) then
corr_x_y_n &
= log( one_dp + corr_x_y * sqrt( x_sigma2_on_mu2 * y_sigma2_on_mu2 ) ) &
/ ( sigma_x_n * sigma_y_n )
else ! sigma_x_n = 0 or sigma_y_n = 0
! The value of corr_x_y_n is undefined, so set it to corr_x_y.
corr_x_y_n = corr_x_y
endif ! sigma_x_n > 0 and sigma_y_n > 0
! Clip the magnitude of the correlation of ln x and ln y in the ith PDF
! component, just in case the correlation (ith PDF component) of x and y,
! the standard deviation (ith PDF component) of ln x, and the standard
! deviation (ith PDF component) of ln y are inconsistent, resulting in an
! unrealizable value for corr_x_y_n.
if ( corr_x_y_n > real( max_mag_correlation, kind = dp ) ) then
corr_x_y_n = real( max_mag_correlation, kind = dp )
elseif ( corr_x_y_n < -real( max_mag_correlation, kind = dp ) ) then
corr_x_y_n = -real( max_mag_correlation, kind = dp )
endif
return
end function corr_LL2NN_dp
!=============================================================================
pure function corr_NN2LL( corr_x_y_n, sigma_x_n, sigma_y_n, &
x_sigma2_on_mu2, y_sigma2_on_mu2 ) &
result( corr_x_y )
! Description:
! For lognormally-distributed variables x and y, this function finds the
! correlation of x and y (corr_x_y) for the ith component of the PDF, given
! the correlation of ln x and ln y (corr_x_y_n), the standard deviation of
! ln x (sigma_x_n), and the standard deviation of ln y (sigma_y_n) for
! the ith component of the PDF. The value of ln x (or ln y) is distributed
! normally when x (or y) is distributed lognormally.
! References:
! Garvey, P. R., 2000: Probability methods for cost uncertainty analysis.
! Marcel Dekker, 401 pp.
! -- Eq. C-3.
!-----------------------------------------------------------------------
use constants_clubb, only: &
one, & ! Constant(s)
zero, &
max_mag_correlation
use clubb_precision, only: &
core_rknd ! Variable(s)
implicit none
! Input Variables
real( kind = core_rknd ), intent(in) :: &
corr_x_y_n, & ! Correlation of ln x and ln y (ith PDF component) [-]
sigma_x_n, & ! Standard deviation of ln x (ith PDF component) [-]
sigma_y_n, & ! Standard deviation of ln y (ith PDF component) [-]
x_sigma2_on_mu2, & ! Ratio: sigma_x^2 / mu_x^2 (ith PDF component) [-]
y_sigma2_on_mu2 ! Ratio: sigma_y^2 / mu_y^2 (ith PDF component) [-]
! Return Variable
real( kind = core_rknd ) :: &
corr_x_y ! Correlation of x and y (ith PDF component) [-]
! Find the correlation of x and y for the ith component of the PDF.
! When sigma_x = 0 and mu_x > 0, x_sigma2_on_mu2 = 0. This results in
! sigma_x_n = 0. The resulting corr_x_y and corr_x_y_n are undefined. The
! same holds true when sigma_y = 0 and mu_y > 0. However, the
! divide-by-zero problem needs to be addressed in the code.
if ( sigma_x_n > zero .and. sigma_y_n > zero ) then
! corr_x_y = ( exp( sigma_x_n * sigma_y_n * corr_x_y_n ) - one ) &
! / ( sqrt( exp( sigma_x_n**2 ) - one ) &
! * sqrt( exp( sigma_y_n**2 ) - one ) )
corr_x_y = ( exp( sigma_x_n * sigma_y_n * corr_x_y_n ) - one ) &
/ sqrt( x_sigma2_on_mu2 * y_sigma2_on_mu2 )
else ! sigma_x_n = 0 or sigma_y_n = 0
! The value of corr_x_y is undefined, so set it to corr_x_y_n.
corr_x_y = corr_x_y_n
endif ! sigma_x_n > 0 and sigma_y_n > 0
! Clip the magnitude of the correlation of x and y in the ith PDF component,
! just in case the correlation (ith PDF component) of ln x and ln y, the
! standard deviation (ith PDF component) of ln x, and the standard deviation
! (ith PDF component) of ln y are inconsistent, resulting in an unrealizable
! value for corr_x_y.
if ( corr_x_y > max_mag_correlation ) then
corr_x_y = max_mag_correlation
elseif ( corr_x_y < -max_mag_correlation ) then
corr_x_y = -max_mag_correlation
endif
return
end function corr_NN2LL
!=============================================================================
elemental function compute_mean_binormal( mu_x_1, mu_x_2, mixt_frac ) &
result( xm )
! Description:
! Computes the overall grid-box mean of a binormal distribution from the
! mean of each component
! References:
! None
!-----------------------------------------------------------------------
use clubb_precision, only: &
core_rknd ! Constant
use constants_clubb, only: &
one ! Constant
implicit none
! Input Variables
real( kind = core_rknd ), intent(in) :: &
mu_x_1, & ! First PDF component mean of 'x' [?]
mu_x_2, & ! Second PDF component mean of 'x' [?]
mixt_frac ! Weight of the first PDF component [-]
! Output Variables
real( kind = core_rknd ) :: &
xm ! Mean of 'x' (overall) [?]
!-----------------------------------------------------------------------
!----- Begin Code -----
xm = mixt_frac * mu_x_1 + ( one - mixt_frac ) * mu_x_2
return
end function compute_mean_binormal
!=============================================================================
elemental function compute_variance_binormal( xm, mu_x_1, mu_x_2, &
stdev_x_1, stdev_x_2, &
mixt_frac ) &
result( xp2 )
! Description:
! Computes the overall grid-box variance of a binormal distribution from the
! variance of each component.
! References:
! None
!-----------------------------------------------------------------------
use clubb_precision, only: &
core_rknd ! Constant
use constants_clubb, only: &
one ! Constant
implicit none
! Input Variables
real( kind = core_rknd ), intent(in) :: &
xm, & ! Overall mean of 'x' [?]
mu_x_1, & ! First PDF component mean of 'x' [?]
mu_x_2, & ! Second PDF component mean of 'x' [?]
stdev_x_1, & ! Standard deviation of 'x' in the first PDF component [?]
stdev_x_2, & ! Standard deviation of 'x' in the second PDF component [?]
mixt_frac ! Weight of the first PDF component [-]
! Output Variables
real( kind = core_rknd ) :: &
xp2 ! Variance of 'x' (overall) [?^2]
!-----------------------------------------------------------------------
!----- Begin Code -----
xp2 = mixt_frac * ( ( mu_x_1 - xm )**2 + stdev_x_1**2 ) &
+ ( one - mixt_frac ) * ( ( mu_x_2 - xm )**2 + stdev_x_2**2 )
return
end function compute_variance_binormal
!=============================================================================
subroutine calc_comp_corrs_binormal( xpyp, xm, ym, mu_x_1, mu_x_2, & ! In
mu_y_1, mu_y_2, sigma_x_1_sqd, & ! In
sigma_x_2_sqd, sigma_y_1_sqd, & ! In
sigma_y_2_sqd, mixt_frac, & ! In
corr_x_y_1, corr_x_y_2 ) ! Out
! Description:
! Calculates the PDF component correlations of variables x and y, where
! x and y are both distributed as two-component normals (or binormals).
! The PDF component correlations are set equal to each other.
!
! The overall covariance of x and y, <x'y'>, can be expressed in terms of
! PDF parameters by integrating over the PDF:
!
! <x'y'> = INT(-inf:inf) INT(-inf:inf) ( x - <x> ) ( y - <y> ) P(x,y) dy dx;
!
! where <x> is the overall mean of x, <y> is the overall mean of y, and
! P(x,y) is the equation for the two-component normal PDF of x and y.
!
! The integral is evaluated, and the equation for <x'y'> is:
!
! <x'y'> = mixt_frac * ( ( mu_x_1 - <x> ) * ( mu_y_1 - <y> )
! + corr_x_y_1 * sigma_x_1 * sigma_y_1 )
! + ( 1 - mixt_frac ) * ( ( mu_x_2 - <x> ) * ( mu_y_2 - <y> )
! + corr_x_y_2 * sigma_x_2 * sigma_y_2 );
!
! where mu_x_1 is the mean of x in the 1st PDF component, mu_x_2 is the mean
! of x in the 2nd PDF component, mu_y_1 is the mean of y in the 1st PDF
! component, mu_y_2 is the mean of y in the 2nd PDF component, sigma_x_1 is
! the standard deviation of x in the 1st PDF component, sigma_x_2 is the
! standard deviation of x in the 2nd PDF component, sigma_y_1 is the
! standard deviation of y in the 1st PDF component, sigma_y_2 is the
! standard deviation of y in the 2nd PDF component, corr_x_y_1 is the
! correlation of x and y in the 1st PDF component, corr_x_y_2 is the
! correlation of x and y in the 2nd PDF component, and mixt_frac is the
! mixture fraction (weight of the 1st PDF component).
!
! This equation can be rewritten as:
!
! <x'y'> = mixt_frac * ( mu_x_1 - <x> ) * ( mu_y_1 - <y> )
! + mixt_frac * corr_x_y_1 * sigma_x_1 * sigma_y_1
! + ( 1 - mixt_frac ) * ( mu_x_2 - <x> ) * ( mu_y_2 - <y> )
! + ( 1 - mixt_frac ) * corr_x_y_2 * sigma_x_2 * sigma_y_2.
!
! Setting the two PDF component correlations equal to each other
! (corr_x_y_1 = corr_x_y_2), the equation can be solved for the PDF
! component correlations:
!
! corr_x_y_1 = corr_x_y_2
! = ( <x'y'> - mixt_frac * ( mu_x_1 - <x> ) * ( mu_y_1 - <y> )
! - ( 1 - mixt_frac ) * ( mu_x_2 - <x> ) * ( mu_y_2 - <y> ) )
! / ( mixt_frac * sigma_x_1 * sigma_y_1
! + ( 1 - mixt_frac ) * sigma_x_2 * sigma_y_2 );
!
! where -1 <= corr_x_y_1 = corr_x_y_2 <= 1.
!
! When sigma_x_1 * sigma_y_1 = 0 and sigma_x_2 * sigma_y_2 = 0, at least one
! of x or y are constant within each PDF component, and both PDF component
! correlations are undefined.
! References:
!-----------------------------------------------------------------------
use grid_class, only: &
gr ! Variable type(s)
use constants_clubb, only: &
one, & ! Variable(s)
zero
use clubb_precision, only: &
core_rknd ! Variable(s)
implicit none
! Input Variables
real ( kind = core_rknd ), dimension(gr%nz), intent(in) :: &
xpyp, & ! Covariance of x and y (overall) [(x units)(y units)]
xm, & ! Mean of x (overall) [x units]
ym, & ! Mean of y (overall) [y units]
mu_x_1, & ! Mean of x (1st PDF component) [x units]
mu_x_2, & ! Mean of x (2nd PDF component) [x units]
mu_y_1, & ! Mean of y (1st PDF component) [y units]
mu_y_2, & ! Mean of y (2nd PDF component) [y units]
sigma_x_1_sqd, & ! Variance of x (1st PDF component) [(x units)^2]
sigma_x_2_sqd, & ! Variance of x (2nd PDF component) [(x units)^2]
sigma_y_1_sqd, & ! Variance of y (1st PDF component) [(y units)^2]
sigma_y_2_sqd, & ! Variance of y (2nd PDF component) [(y units)^2]
mixt_frac ! Mixture fraction [-]
! Output Variables
real ( kind = core_rknd ), dimension(gr%nz), intent(out) :: &
corr_x_y_1, & ! Correlation of x and y (1st PDF component) [-]
corr_x_y_2 ! Correlation of x and y (2nd PDF component) [-]
where ( sigma_x_1_sqd * sigma_y_1_sqd > zero &
.or. sigma_x_2_sqd * sigma_y_2_sqd > zero )
! Calculate corr_x_y_1 (which also equals corr_x_y_2).
corr_x_y_1 &
= ( xpyp &
- mixt_frac * ( mu_x_1 - xm ) * ( mu_y_1 - ym ) &
- ( one - mixt_frac ) * ( mu_x_2 - xm ) * ( mu_y_2 - ym ) ) &
/ ( mixt_frac * sqrt( sigma_x_1_sqd * sigma_y_1_sqd ) &
+ ( one - mixt_frac ) * sqrt( sigma_x_2_sqd * sigma_y_2_sqd ) )
! The correlation must fall within the bounds of
! -1 <= corr_x_y_1 (= corr_x_y_2) <= 1.
where ( corr_x_y_1 > one )
corr_x_y_1 = one
elsewhere ( corr_x_y_1 < -one )
corr_x_y_1 = -one
endwhere
elsewhere ! sigma_x_1^2 * sigma_y_1^2 = 0 and sigma_x_2^2 * sigma_y_2^2 = 0.
! The correlation is undefined (output as 0).
corr_x_y_1 = zero
endwhere
! Set corr_x_y_2 equal to corr_x_y_1.
corr_x_y_2 = corr_x_y_1
return
end subroutine calc_comp_corrs_binormal
!=============================================================================
pure function calc_corr_chi_x( crt_i, cthl_i, sigma_rt_i, sigma_thl_i, &
sigma_chi_i, corr_rt_x_i, corr_thl_x_i ) &
result( corr_chi_x_i )
! Description:
! This function calculates the correlation of extended liquid water mixing
! ratio, chi (old s), and a generic variable x, within the ith component of
! the PDF. The variable chi can be split into mean and turbulent
! components, such that:
!
! chi = <chi> + chi';
!
! where < > denotes a mean field an ' denotes a turbulent component.
!
! The linearized equation for chi' is given in Larson et al. (2001), where
! within the ith component of the PDF:
!
! chi_(i)' = Coef_rt(i) * r_t(i)' - Coef_thl(i) * th_l(i)'.
!
! The equation for chi' can be multiplied by x'. The equation becomes:
!
! chi'x'_(i) = Coef_rt(i) * r_t'x'_(i) - Coef_thl(i) * th_l'x'_(i).
!
! Averaging both sides, the covariance <chi'x'> is given by the equation:
!
! <chi'x'_(i)> = Coef_rt(i) * <r_t'x'_(i)> - Coef_thl(i) * <th_l'x'_(i)>.
!
! This equation can be rewritten as:
!
! sigma_chi(i) * sigma_x(i) * corr_chi_x(i)
! = Coef_rt(i) * sigma_rt(i) * sigma_x(i) * corr_rt_x(i)
! - Coef_thl(i) * sigma_thl(i) * sigma_x(i) * corr_thl_x(i).
!
! This equation can be solved for corr_chi_x(i):
!
! corr_chi_x(i)
! = Coef_rt(i) * ( sigma_rt(i) / sigma_chi(i) ) * corr_rt_x(i)
! - Coef_thl(i) * ( sigma_thl(i) / sigma_chi(i) ) * corr_thl_x(i).
!
! The correlation of chi and x within the ith component of the PDF is
! calculated.
! References:
! Eq. (13) and Eq. (14) of Larson, V. E., R. Wood, P. R. Field, J.-C. Golaz,
! T. H. Vonder Haar, W. R. Cotton, 2001: Systematic Biases in the
! Microphysics and Thermodynamics of Numerical Models That Ignore
! Subgrid-Scale Variability. J. Atmos. Sci., 58, 1117--1128,
! doi:https://doi.org/10.1175/1520-0469(2001)058%3C1117:SBITMA%3E2.0.CO;2.
!
! Eq. (A29) of Griffin, B. M., 2016: Improving the Subgrid-Scale
! Representation of Hydrometeors and Microphysical Feedback Effects Using a
! Multivariate PDF. Doctoral dissertation, University of
! Wisconsin -- Milwaukee, Milwaukee, WI, Paper 1144, 165 pp., URL
! http://dc.uwm.edu/cgi/viewcontent.cgi?article=2149&context=etd.
!-----------------------------------------------------------------------
use grid_class, only: &
gr ! Variable type(s)
use constants_clubb, only: &
zero, & ! Constant(s)
chi_tol, &
max_mag_correlation
use clubb_precision, only: &
core_rknd ! Variable(s)
implicit none
! Input Variables
real( kind = core_rknd ), dimension(gr%nz), intent(in) :: &
crt_i, & ! Coefficient of r_t for chi (old s) (ith PDF comp.) [-]
cthl_i, & ! Coefficient of th_l for chi (ith PDF comp.) [(kg/kg)/K]
sigma_rt_i, & ! Standard deviation of r_t (ith PDF component) [kg/kg]
sigma_thl_i, & ! Standard deviation of th_l (ith PDF component) [K]
sigma_chi_i, & ! Standard deviation of chi (ith PDF component) [kg/kg]
corr_rt_x_i, & ! Correlation of r_t and x (ith PDF component) [-]
corr_thl_x_i ! Correlation of th_l and x (ith PDF component) [-]
! Return Variable
real( kind = core_rknd ), dimension(gr%nz) :: &
corr_chi_x_i ! Correlation of chi and x (ith PDF component) [-]
! Calculate the correlation of chi and x in the ith PDF component.
where ( sigma_chi_i > chi_tol )
corr_chi_x_i = crt_i * ( sigma_rt_i / sigma_chi_i ) * corr_rt_x_i &
- cthl_i * ( sigma_thl_i / sigma_chi_i ) * corr_thl_x_i
elsewhere ! sigma_chi_i = 0
! The standard deviation of chi in the ith PDF component is 0. This
! means that chi is constant within the ith PDF component, and the ith
! PDF component covariance of chi and x is also 0. The correlation of
! chi and x is undefined in the ith PDF component, so a value of 0 will
! be used.
corr_chi_x_i = zero
endwhere
! Clip the magnitude of the correlation of chi and x in the ith PDF
! component, just in case the correlations and standard deviations used in
! calculating it are inconsistent, resulting in an unrealizable value for
! corr_chi_x_i.
where ( corr_chi_x_i > max_mag_correlation )
corr_chi_x_i = max_mag_correlation
elsewhere ( corr_chi_x_i < -max_mag_correlation )
corr_chi_x_i = -max_mag_correlation
endwhere
return
end function calc_corr_chi_x
!=============================================================================
pure function calc_corr_eta_x( crt_i, cthl_i, sigma_rt_i, sigma_thl_i, &
sigma_eta_i, corr_rt_x_i, corr_thl_x_i ) &
result( corr_eta_x_i )
! Description:
! This function calculates the correlation of the variable that is
! orthogonal to extended liquid water mixing ratio in a PDF transformation,
! eta (old t), and a generic variable x, within the ith component of
! the PDF.
! References:
! Eq. (A30) of Griffin, B. M., 2016: Improving the Subgrid-Scale
! Representation of Hydrometeors and Microphysical Feedback Effects Using a
! Multivariate PDF. Doctoral dissertation, University of
! Wisconsin -- Milwaukee, Milwaukee, WI, Paper 1144, 165 pp., URL
! http://dc.uwm.edu/cgi/viewcontent.cgi?article=2149&context=etd.
!-----------------------------------------------------------------------
use grid_class, only: &
gr ! Variable type(s)
use constants_clubb, only: &
zero, & ! Constant(s)
eta_tol, &
max_mag_correlation
use clubb_precision, only: &
core_rknd ! Variable(s)