Variational quality control of non-Gaussian innovations and its parametric optimizations for the GRAPES m3DVAR system

doi:10.1007/s11707-021-0962-1

Front. Earth Sci.

2023, Vol. 17

Issue (2) : 620-631 https://doi.org/10.1007/s11707-021-0962-1

RESEARCH ARTICLE

Variational quality control of non-Gaussian innovations and its parametric optimizations for the GRAPES m3DVAR system

Jie HE¹, Yang SHI², Boyang ZHOU³, Qiuping WANG⁴, Xulin MA⁴(

)

¹. Guangzhou Institute of Tropical and Marine Meteorology, China Meteorological Administration, Guangzhou 510640, China
². Guangdong Meteorological Observatory, Guangzhou 510640, China
³. Qingdao Air Traffic Management Station of Civil Aviation of China, Qingdao 266108, China
⁴. Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Key Laboratory of Meteorological Disaster, Nanjing University of Information Science and Technology, Nanjing 210044, China

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Abstract

The magnitude and distribution of observation innovations, which have an important impact on the analyzed accuracy, are critical variables in data assimilation. Variational quality control (VarQC) based on the contaminated Gaussian distribution (CGD) of observation innovations is now widely used in data assimilation, owing to the more reasonable representation of the probability density function of innovations that can sufficiently absorb observations by assigning different weights iteratively. However, the inaccurate parameters prevent VarQC from showing the advantages it should have in the GRAPES (Global/Regional Assimilation and PrEdiction System) m3DVAR system. Consequently, the parameter optimization methods are considerable critical studies to improve VarQC. In this paper, we describe two probable CGDs to include the non-Gaussian distribution of actual observation errors, Gaussian plus flat distribution and Huber norm distribution. The potential optimization methods of the parameters are introduced in detail for different VarQCs. With different parameter configurations, the optimization analysis shows that the Gaussian plus flat distribution and the Huber norm distribution are more consistent with the long-tail distribution of actual innovations compared to the Gaussian distribution. The VarQC’s cost and gradient functions with Huber norm distribution are more reasonable, while the VarQC’s cost function with Gaussian plus flat distribution may converge on different minimums due to its non-concave properties. The weight functions of two VarQCs gradually decrease with the increase of innovation but show different shapes, and the VarQC with Huber norm distribution shows more elasticity to assimilate the observations with a high contamination rate. Moreover, we reveal a general derivation relationship between the CGDs and VarQCs. A novel schematic interpretation that classifies the assimilated data into three categories in VarQC is presented. They are conducive to the development of a new VarQC method in the future.

Keywords data assimilation variational quality control contaminated Gaussian distribution non-Gaussian distribution innovation

Corresponding Author(s): Xulin MA

Online First Date: 30 June 2022 Issue Date: 04 August 2023

Cite this article:

Jie HE,Yang SHI,Boyang ZHOU, et al. Variational quality control of non-Gaussian innovations and its parametric optimizations for the GRAPES m3DVAR system[J]. Front. Earth Sci., 2023, 17(2): 620-631.

URL:

https://academic.hep.com.cn/fesci/EN/10.1007/s11707-021-0962-1
https://academic.hep.com.cn/fesci/EN/Y2023/V17/I2/620

Fig.1 The fitted Gaussian distributions (histogram) of innovation of brightness temperature from geostationary meteorological satellite (Meteosat-7). The green boxes indicate the non-Gaussian long-tail histograms (distribution). The dashed line represents the approximation between observation and background (ERA5 reanalysis). The x-axis represents the size of the innovation; the y-axis represents frequency. S shows the total sample number of observations at 1200 UTC 10 June 2015.

Fig.2 The distribution (histogram) statistics of normalized innovation by using the horizontal wind (a) and the temperature (b) from sounding and aircraft-reported observations are fitted with the pure Gaussian distribution (red line, Gaussian fit) and the CGD fit (black line). The x-axis represents the size of the innovation; the y-axis represents frequency. S shows the total sample number of observations from August 2013.

Fig.3 Histogram (a) and transformed histogram (b) of temperature departure from analysis from the AIREP observations collected from 1 to 14 August 2013. The slope of the straight lines fitted in (b) defines the standard deviation of the Gaussian curve (dashed line) drawn in (a). The red dashed label represents a Gaussian curve with a mean of 0.06 and a variance of 0.64. The solid red lines are displayed by an expression in

y = a b s (x / 0.77)

. The top S shows the total sample number of observations.

Obs.	Variables	Sample	max(f)	$λ$	$α$	Approximate rejection threshold
Obs.	Variables	Sample	max(f)	$λ$	$α$	Flat-VarQC	BgQC
TEMP	wind u	177089	16162	1.81	4.0	7.24 m/s	7.2?12 m/s
	wind v	177089	15064	1.81	4.0	7.24 m/s	7.2?12 m/s
	rh	101827	7712	9.13	3.5	32.0%	59.5%
	pressure	148670	18451	0.30	2.5	0.75 hPa	0.4?2.3 hPa
AIREP	wind u	170138	18296	1.75	4.0	7.00 m/s	10?16 m/s
	wind v	170138	17416	1.75	4.0	7.00 m/s	10?16 m/s
	temp	169550	17565	0.77	4.0	3.08 K	4.8?5.6 K
SYNOP	rh	64767	5066	9.21	4.0	36.8%	52?92%
SYNOP	pressure	26345	1930	0.60	4.0	2.40 hPa	2.9?3.2 hPa
SATOB	wind u	8984	795	2.27	4.0	9.08 m/s	8?20 m/s
SATOB	wind v	8984	949	2.27	4.0	9.08 m/s	8?20 m/s

Tab.1 Statistics of rejection threshold for Flat-VarQC and BgQC in the GRAPES m3DVAR system. The horizontal wind (u and v), relative humidity (rh), pressure, and temperature (temp) of different observations are obtained from the standard GTS observations over the Chinese mainland (10°N ? 60° N, 70°E ? 140° E). These observations are collected four times every day spanning the period from 1 to 14 August 2013

Parameter names	Parameter values
$ε$	0.002	0.005	0.010	0.020	0.050	0.100	0.200	0.300	0.500
$c$	2.435	2.160	1.945	1.717	1.399	1.140	0.862	0.685	0.436

Tab.2 The corresponding relationship between the contamination rate

ε

and the transition point

c

Fig.4 A schematic diagram of observation classification for variational quality control. The green dots represent the valid observations (VOs); the black dots represent the available observations (AOs) shown as outliers but are not gross errors; the red dots represent the deleterious observations (DOs) shown as outliers but are gross errors. The green circle denotes the perfect threshold for a pure Gaussian error domain, the red circle represents the empirical (extended) rejection threshold in conventional quality control for gross errors. The VarQC pushes the observations (the black dots) in the gray zone iteratively into the green circle to be ingested in the VarQC.

Fig.5 The profiles of the probability density distribution of observation error for the pure Gaussian and two CGDs (Gaussian + flat and Huber norm). The parameters used are

ε = 0.1, d = 5, σ o = 1, c = 1 .14

Fig.6 The cost function, gradient function, and analysis weight function of (a) Flat-VarQC (blue line) and (b) Huber-VarQC (black line). The red lines represent the observational cost function of the pure 3DVAR under the assumption of Gaussian distribution. The parameters are used as shown in Fig. 5.

Fig.7 Same as the analysis weight functions in Fig.6(a) but for the sensitivity to

d

(the width of flat distribution) and

γ

(

γ = ε 2 π / ε 2 π 2 d (1 ? ε) 2 d (1 ? ε)

) in Flat-VarQC. The other parameters are (a)

ε = 0.1

and (b)

d = 5

. The same parameter is

σ o = 1

Fig.8 Same as the analysis weight functions in Fig. 6 but for the sensitivity to the contamination rate (

ε

) in different VarQCs. The parameter

c

corresponding to

ε

is based on Table 2. The other parameters are the same as those of Fig. 5.

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