Development of a new humidity retrieval algorithm

One-dimensional assimilation
method for the humidity
estimation with the wind profiling
radar data using the MSM
forecast as the first guess
Jun-ichi Furumoto, Toshitaka Tsuda, Hiromu Seko,
Kazuo Saito
Introduction

The turbulence echo power intensity with wind profiling
radar is closely related with the refractive index gradient
squared (M2), which is largely depends on the vertical
humidity gradient in the moist atmosphere. Using the
relations, humidity profiles can be estimated from a wind
profiling radar data, if the sign of the radar-derived |M| is
determined.

Furumoto et al. (in print) has employed one-dimensional
assimilation method to estimate humidity profiles with the
MU radar-RASS measurements, complementary GPSderived precipitable water vapor and 12-hourly radiosonde
results.

Aiming at the estimation without simultaneous radiosonde
data, this study estimates humidity profiles with a first
guess from MSM forecast.
Basic Principle of humidity estimation with wind
profiling radar
Specific humidity is derived from the relation between turbulence echo power and
height potential of refractive index.
| M | 1/ 2 N 1/ 3
M: Height potential of refractive index
η: Echo power intensity
N: Brunt-Vaisala frequency squared
ε: Turbulence energy dissipation rate
p  N2
q N2
1 dq 

M   K 0
 K1
 K2
T
g
T g
T dz 
p: Pressure
K0 ,K1 K2 : Constant
q: Specific humidity
z: Height
g: Gravity acceleration rate
2
2



T
1
dT


q( z )   2  1.65 M 
    2 dz  02 q0

z0
p
7800 dz



z
 : Potential temperature, Γ: Dry adiabatic lapse rate
q0, θ0: Boundary value at the height of z=z0
•The time-interpolated radiosonde results are used as the boundary value at the height of
z=z0
•The sign of the radar-derived |M| is determined to agree the integrated water vapour with
GPS result and the constraint of time continuity of q.
Variational method
Variational method is a data assimilation technique to determine the most
reasonable atmospheric state based on maximum likelihood estimation.
The observation operator H, is defined to convert the atmospheric state vector
to the observational one as:
y  Hx
x: state vector consisting of the atmospheric state variables
y: observation vector consisted of observed variables
The analysis vector xa is determined as x when the conditional probability of x given
the first guess (xb) and observation results (yo) has its maximum value.
xa is obtained as x to minimize the cost function J(x) as :
1
1
J (x) = (x  x b ) T B 1 (x  x b ) + (y  y o ) T R 1 (y  y o )
2
2
B: background covariance metrics
R: observation covariance metrics
If J(x) is differentiable, xa can
be derived by minimizing J(x)
using a quasi-Newton
method.
Expansion of 1D-Var for humidity estimation
•
•
When the absolute value of |M| is assimilated directly into the background
atmospheric state, J(x) has many local minima, and it is very difficult to find the
global minimum using finite computer resources.
To reduce the calculation cost of the assimilation, a new cost function was
formulated by considering the statistical probability (Pr(z)) of the sign of |M|.
Determination of sign of |M|
1
J  (i )  0.5  y (i)  y 0 (i)  R 1 (i, i)  y(i)  y 0 (i) 
J1 (x)  (x  x b )T B 1 (x  x b )  J o1 (x)
2
1
J
(
i
)


0.5
y
(
i
)

y
(
i
)
R
(i, i)  y (i)  y 0 (i) 



0
n


J o1 (x)  ln   Pr(i)  exp  J  (i)   (1  Pr(i))  exp  J  (i)    R-1(i,i) : the (i,i)-th component of R-1
 i 1

Genetic algorithm (GA) is used to find the global minimum
Pr(z) is calculated data from almost 1500 radiosondes launched since 1986.
After the sign of M was determined, y0 after the previous step,
is again assimilated using the general cost function.
J 2 (x) = 0.5(x  x0 )T B-1 (x  x0 ) + 0.5(y  y0 )T R-1 (y  y0 )
The quasi-Newton method (BFGS method) is employed for the optimization
Background and observation vector
The variational method was applied to the assimilation of the MU radar-RASS
observation results for the period from July 29 to August 5, 1999
x  [ p0 , T0 , T1,..., TN 1,RH0 ,RH1,...,RHN 1 ]
y  [| M 0 |,| M1 |,...,| M N 1 |,IWVGPS ]
p0: pressure at the lowest height.
Ti: temperature at the j-th height
RHi: relative Humidity at the j-th height
IWVGPS: Integrated Water Vapor with GPS
By assimilating the IWVGPS
together with the radar-derived |M|,
the signs of |M| are constrained.
• The first guess of the atmospheric state vector was
obtained from the MSM forecast obtained every hour.
• The background error variances in the operational
forecast model was used in this study.
• The observational error variance was calculated from the
statistics of the difference between the radar-derived M
and MSM forecast.
Observation error covariance (R)

Histogram of the difference radar-derived M and radiosonde value
•
•
•
•
thin solid: s.d. of the difference.
thick solid: the s.d. of the difference
approximated to the exponential function.
dashed line: observation error of radiosonde
measurement.
dot dash line: observation error variance
Time-height structure of specific humidity

Balloon
observation

First guess
from MSM

Analysis
Estimation with the forecast of prediction model
The forecast of the operational Meso-Scale Model (MSM) of the Japan
Meteorological Agency (JMA) used as the first guess, instead of the timeinterpolation of radiosonde data.
The forecast error used at JMA is employed as the background error.
.
q profile
Difference from radiosonde
Bias error averaged for 6 profiles
Dotted: MSM
Black solid: analysis
Red: radiosonde result
Both bias and
random errors in the
analysis are smaller
than these in the
first guess.
Random error averaged for 6 profiles
15LT
Jul. 29, 2002
The discrepancy in the analysis is smaller
than that in the first guess below 3.0 km.
Conclusion

Aiming at the precise estimation of humidity profiles with
the wind profiling radar, the humidity estimation method with
wind profiling radar data was developed. One-dimensional
assimilation method was employed to determine the sign of
the radar-derived refractive index gradient. The MSM
forecast was used for the first guess of the assimilation
algorithm.

Time-height structure of humidity profile has successfully
obtained with the MU radar-RASS measurement data. The
retrieval results shows the improvement of precision from
the first guess of MSM forecasts.

Download Report