# A Comparison between 3DVAR and EnKF for Data Assimilation Effects on the Yellow Sea Fog Forecast

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data Assimilation Algorithms

#### 2.1. 3DVAR

**x**, ${\mathbf{x}}^{\mathrm{b}}$, ${\mathbf{x}}^{\mathrm{a}},$ and

**y**be the model state at the beginning of the assimilation window, a background or prior estimate of

**x**, analysis of

**x**and observation, respectively. The cost function for 3DVAR can be defined by

**H**is the observation operator matrix which transforms data from model space to observation space,

**R**is the observation error covariance matrix, and

**B**is the background error covariance matrix. By using the increment formulation [36], the analysis increment and observation innovation are respectively defined as $\mathsf{\delta}x={x}^{\mathrm{a}}-{x}^{\mathrm{b}}$ and $d=y-H({x}^{\mathrm{b}})$. Thus, Equation 1 can be rewritten

**R**is to estimate it statistically for each specific model domain and observation type [37]. However, for many observation types, R can be much more complicated [38] and is difficult to accurately quantify. Thus, in this study,

**R**is assumed to be static and diagonal, and provided prior to the assimilation.

**B**is calculated by statistics, and the NMC method [20] is used here to generate it as follows

_{n}is the time period for statistics, which is usually 15–30 days. For the WRF model, the array size of ${x}^{\mathrm{b}}$ is usually ~1.0 × 10

^{7}for a typical sea fog modeling (e.g., the sizes are respectively ~1.01 × 10

^{7}and ~0.58 × 10

^{7}for the two cases in this study), thus directly solving the inversion of

**B**($\mathrm{i}.\mathrm{e}.,\text{}{B}^{-1}$) requires ${~\text{}\mathrm{O}\text{}(10}^{14})$ times of calculations, which is technically impossible. However,

**B**is usually diagonalized by control variable transform [20], which uses length scale coefficients to store the correlations between different grids and regression coefficients to store the correlations between different variables. Note that these coefficients are regionally averaged to decrease the computational cost and

**B**is therefore static and nearly homogeneous and isotropic.

#### 2.2. EnKF

**R**and

**H**are the same as in Equation (1), ${x}_{\mathrm{i}}^{\mathrm{b}}$ is the i-th ensemble member (let m be their total number, so that i = 1, 2, …, m), ${x}_{\mathrm{i}}^{\mathrm{a}}$ is the i-th updated member, and ${y}_{\mathrm{i}}$ is the i-th perturbed observation for ${x}_{\mathrm{i}}^{\mathrm{b}}$ in a way consistent with

**R**. To avoid calculating the large matrix

**P**

^{b}, EnKF usually approximately calculates

**P**

^{b}

**H**

^{T}and

**HP**

^{b}

**H**

^{T}as

**y**,

**R,**and

**HP**

^{b}

**H**

^{T}become scalers, and

**P**

^{b}

**H**

^{T}is reduced to a vector. This brings the benefit that not only is matrix inversion avoided but also there is no need to simplify background error covariances (e.g., diagonalizing), which results in intact correlations between physical variables. Since

**P**

^{b}

**H**

^{T}is updated along with assimilating observations, it means the background error covariances may vary substantially depending on the flow of the day.

## 3. Numerical Experiments

#### 3.1. Data

#### 3.2. Sea Fog Cases

#### 3.3. Model Configuration

#### 3.4. Design of DA Schemes and Experiments

**B**and

**P**represent the background error covariances for 3DVAR and EnKF, respectively, and obs stands for observations; Each assimilation cycle is connected by the WRF integration (i.e., wrf.exe in Figure 3), and the final assimilation analysis ${x}^{\mathrm{a}}$ or $\overline{{x}^{\mathrm{a}}}$ are the initial conditions for the next WRF forecast.

^{b}_{0}in Figure 3) ensembles for DA-2 and DA-3 were created by a random perturbation method [23,24,58]. In this study, the built-in method known as RANDOMCV in the WRF-DA system, was employed to generate ensemble initial conditions (ICs) by adding random noise to the analysis in the control variable space. The ensemble ICs were generated 6 hours prior to the starting time of the assimilation window, and then the WRF runs with these ensemble ICs were integrated for 6 hours to produce the initial ensemble. The ensemble size was set at 40 (i.e., m = 40 in Figure 3); the GEN_BE tool (Ver 2.0) [59] was employed to generate the static

**B**with the control variables listed in Table 2.

^{−5}). For EnKF, the default value in the GSI/EnKF system for the inflation factor (β = 0.9) was taken, and the localization scale was set to 200 km instead of the default value 500 km, according to the results of the hybrid-3DVAR experiments for the Yellow Sea fog by Wang [60]. The detailed configurations and differences between 3DVAR and EnKF are listed in Table 2, and the observation error variances (

**R**) for each observation type are given in Table 3. Note that the

**R**provided by the GSI/EnKF system in Table 2 may be sub-optimal, because

**R**is often given by the instrument error variance (and with an ad hoc inflation factor for EnKF) to account for representivity errors for a specific model domain and observation type [38].

**B**in Figure 3) by the NMC method [20] with the GEN-BE tool [53]. The background error uses regression coefficients to correlate wind speed, temperature, and pressure, while relative humidity is independent. Note that the flow-dependent background error covariance matrix for EnKF (i.e.,

**P**in Figure 3), in which any two analysis variables are codependent, was statistically calculated based on the ensemble of ${x}^{\mathrm{b}}$ and updated in every cycle.

^{b}## 4. Results

#### 4.1. Methods for Sea Fog Diagnostics and Evaluation

#### 4.2. Evaluation of Assimilation Effect

#### 4.2.1. Verification of Sea Fog Area

#### 4.2.2. Verification by Measurements

#### 4.3. Investigation of Assimilation Effects

#### 4.3.1. Feature of Initial Condition Differences

#### 4.3.2. Reason for the Forecast Improvements

_{2}in Figure 3). Prior to the assimilation of this cycle, the observation innovations and analysis increments of temperature and mixing ratio differ very little between Exp-B1 and Exp-B3 (cf. Figure 11 and Figure 11d; cf. Figure 11b,e). It means that Exp-B1 and Exp-B3 have almost the same backgrounds (i.e., ${x}^{\mathrm{b}}$) of temperature and mixing ratio. However, after assimilating new observations, Exp-B1 and Exp-B3 obtain distinctly different analysis increments (cf. Figure 11c,f). In the significant zone, both Exp-B1 and Exp-B3 get a negative temperature increment, but the amplitude of the former is much smaller than that of the latter; Exp-B1 has a gain of mixing ratio about 0.1 g/kg over the land part of the significant zone, whereas the gain of Exp-B3 increases up to at least 0.2 g/kg almost over the whole significant zone.

#### 4.3.3. Impact of the Background Error Covariances

#### 4.3.4. Sensitivity of Localization Scale and Inflation Factor

## 5. Conclusions

- (a)
- The assimilation effect of EnKF obviously excels that of 3DVAR, performing not only forecasted sea fog but also the distribution of temperature, moisture, and wind in the initial conditions. For the widespread-fog case, the assimilation by EnKF significantly improves the forecasted sea fog area, raising POD and ETS by up to about 57.9% and 55.5%, respectively. Especially for the case that spreads along the coast, the assimilation by EnKF successfully produces the sea fog formation which is completely mis-forecasted by 3DVAR.
- (b)
- The analysis increments strongly depend on the background error covariances. The flow-dependent background error covariances of EnKF out-compete that of 3DVAR, as evidenced by more realistic depiction of sea surface wind for the widespread-fog case and better existence of moisture for the other case in the initial conditions.
- (c)
- Compared with 3DVAR, the multivariate correlations (e.g., correlation between temperature and humidity) in the background error covariances of EnKF play a key role in adjusting/generating moisture through assimilation of temperature. This helps greatly to improve the moisture conditions for sea fog forecast.
- (d)
- A series of sensitivity experiments of EnKF shows that the forecast result is sensitive to ensemble inflation and localization factors, in particular, highly sensitive to the latter. These factors need to be tuned case by case or an adaptive localization method should be considered.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Korea Meteorological Administration (KMA) surface synoptic charts (

**a**,

**c**) and Multifunctional Transport Satellite (MTSAT) visible satellite images (

**b**,

**d**). Upper panels (

**a,b**) are the case C-09 at t at 0000 UTC 10 April 2009, and lower panels (

**c,d**) are for the case C-14 at 0300 UTC 2 April 2014.

**Figure 2.**Weather Research and Forecasting (WRF) domains for the cases: (

**a**) C-09 and (

**b**) C-14. Colors show sea surface temperature (SST) within the D2 domains, and the red and black dots respectively locate radiosonde stations and ship measurement sites. The sea area surrounded by thick dashed lines and coastlines in D2 of (

**a**) is used for evaluating forecasted sea surface winds later.

**Figure 3.**Flowcharts for data assimilation schemes: (

**a**) DA-1, (

**b**) DA-2, and (

**c**) DA-3. See details in the text.

**Figure 4.**Comparison between the forecasted and observed sea fog areas for the case C-09. Panels in the top row (

**a**–

**e**) are the observed fog patches, and panels in the other rows from up to down (

**f**–

**t**) are the forecasted fog patches for Exp-A1, Exp-A2, and Exp-A3, respectively.

**Figure 6.**Vertical profiles of the root-mean-square errors (RMSEs) (solid lines) and biases (dashed lines) between the initial analysis and the radiosonde observations of geopotential height (

**a**,

**d**), temperature (

**b**,

**e**), and mixing ratio (

**c**,

**f**). The upper and lower panels are for the cases C-09 and C-14, respectively.

**Figure 7.**Evaluation result of sea level pressure (SLP) in the initial analysis for the Case C-09. Top panels respectively show (

**a**) KMA surface synoptic chart, (

**b**) RMSE, and (

**c**) bias of SLP by using surface measurements. Bottom panels illustrate SLP of (

**d**) Exp-A1, (

**e**) Exp-A2, and (

**f**) Exp-A3, respectively.

**Figure 8.**Biases between the forecasted and observed mixing ratio for the cases (

**a**) C-09 and (

**b**) C-14. Numbers in parentheses are the mixing ratios of ship measurements. The improvements (%) of Exp-A3 relative to Exp-A1 are in parentheses below the S# (identified mark of ship) along horizontal coordinates, as well as Exp-B3 relative to Exp-B1.

**Figure 9.**Initial differences of temperature (

**a**,

**d**) and mixing ratio (

**b**,

**e**) at 1000 hPa, SLP and 10-m wind (

**c**,

**f**) between Exp-A2 and Exp-A1 (

**a**–

**c**), Exp-A3 and Exp-A1 (

**d**–

**f**), respectively. The dashed frame indicates the significant zone, and the closed thick solid line denotes the forecasted sea fog at 1200 UTC 10 April 2009.

**Figure 10.**As in Figure 9, but for differences between Exp-B2 and Exp-B1 (

**a**–

**c**), Exp-B3 and Exp-B1 (

**d**–

**f**), and the forecasted sea fog at 0100 UTC 02 April 2014.

**Figure 11.**Comparison of observation innovation ($y-H{x}^{b}$) and analysis increment (${x}^{a}-{x}^{b}$) at the lowest model level between Exp-B1 (

**a**–

**c**) and Exp-B3 (

**d**–

**f**) for the last DA cycle. Observation innovations of temperature and mixing ratio before and after assimilation are illustrated in the left and middle panels, respectively; and analysis increments of temperature (colors) and mixing ratio (contours, unit: g/kg) are shown in the right panels. The dots indicate the observation locations, and the dashed frame indicates the significant zone.

**Figure 12.**Time series of the average (

**a**) temperature, (

**b**) mixing ratio, (

**c**) u, and (

**d**) v wind components over the area marked by the dashed frame in Figure 8 for Exp-A1 (red) and Exp-A3 (blue). The bars show the analysis increments within data assimilation window.

**Figure 13.**Comparison between the variance distributions at 1000 hPa for temperature (

**a**,

**d**), relative humidity (

**b**,

**e**) and wind speed (

**c**,

**f**) for 3DVAR (upper; Exp-A1) and EnKF (lower; Exp-A3) of the case C-09. The dashed frame indicates the significant zone.

**Figure 15.**Comparison between (

**a**) 3DVAR and (

**b**) EnKF for u wind component (colors) increments at 1000 hPa in single synthetic observation tests from Exp-AS1 to Exp-AS4. Vectors show background flow.

**Figure 16.**Results of (

**a**) Exp-BS1 and (

**b**–

**d**) Exp-BS2. Colors in (

**a,b**) show temperature increments (K) and contours in (

**b**) represent mixing ratio increments (g/kg); Covariances between temperature at the dot and (

**c**) temperature or (

**d**) RH at each model grid are illustrated.

**Figure 17.**(

**a**–

**y**) Initial differences of mixing ratio at 1000 hPa between the sensitivity experiments and Exp-A1 for the case C-09. The dashed frame indicates the significant zone.

Model Option | Specification | ||
---|---|---|---|

C-09 | C-14 | ||

Domains and Grids | Central point | (34.2°N, 124.1°E) | (35.0°N, 122.0°E) |

Grid number | D1: 166 × 190; D2: 120 × 120 | D1: 60 × 60; D2: 151 × 151 | |

Horizontal resolution | D1: 30 km; D2, 10 km | D1: 30 km; D2: 6 km | |

Vertical grid | 44 $\mathsf{\eta}$ ^{1} with a pressure top at 50 hPa | ||

Time step | Adaptive time step (60–120 s for D1) | ||

PBL scheme | YSU scheme [53] | ||

Cumulus scheme | Kain-Fritsch scheme [54] | ||

Microphysics scheme | Lin (Perdue) scheme [55] | ||

Long-shortwave radiation | RRTMG scheme [56] | ||

Land surface model | Noah [57] |

^{1}$\mathsf{\eta}$ = 1.0000, 0.9975, 0.9925, 0.9850, 0.9775, 0.9700, 0.9540, 0.9340, 0.9090, 0.8800, 0.8506, 0.8212, 0.7918, 0.7625, 0.7084, 0.6573, 0.6090, 0.5634, 0.5204, 0.4798, 0.4415, 0.4055, 0.3716, 0.3397, 0.3097, 0.2815, 0.2551, 0.2303, 0.2071, 0.1854, 0.1651, 0.1461, 0.1284, 0.1118, 0.0965, 0.0822, 0.0689, 0.0566, 0.0452, 0.0346, 0.0249, 0.0159, 0.0076, and 0.0000.

**Table 2.**Specifications of three-dimensional variational assimilation (3DVAR) and ensemble Kalman filter (EnKF).

DA Method | Control Variables | Minimization Iterative No. | Localization and Inflation | Error Covariance Matrix | |||
---|---|---|---|---|---|---|---|

OL | IL | L | β | R | B/Pb | ||

3DVAR | Stream function ($\mathsf{\psi})$, velocity potential $(\mathsf{\chi}$), temperature (T), surface pressure ${(\mathrm{P}}_{\mathrm{s})}$, pseudo relative humidity ${(\mathrm{RH}}_{\mathrm{s}}$) | 3 | 150 | / | static, provided by the GSI/EnKF system | static, calculated statistically by the NMC method using GEN_BE tool | |

EnKF | x-wind component (U), y-wind component (V), temperature (T), perturbation geopotential (PH), pseudo relative humidity ${(\mathrm{RH}}_{\mathrm{s}})$ | / | 200 km | 0.9 | flow-dependent, estimated dynamically based on ensemble |

Observation Type | Variables | |||||
---|---|---|---|---|---|---|

Temperature (K^{2}) | Relative Humidity (%^{2}) | Pressure (hPa^{2}) | Wind Speed (m^{2}/s^{2}) | Brightness Temperature (K^{2}) | ||

Surface measurements | 2.56 | 225.00 | 6.25 | 6.25 | / | |

Radiosonde measurements | 1000 hPa | 2.56 | 225.00 | 1.44 | 1.96 | / |

950 hPa | 2.56 | 225.00 | 1.21 | 2.25 | / | |

900 hPa | 4.00 | 225.00 | 0.81 | 2.25 | / | |

850 hPa | 4.00 | 225.00 | 0.64 | 2.25 | / | |

AMSU-A | / | / | / | / | 0.06–9.00 | |

AMSU-B | / | / | / | / | 7.84–20.25 | |

HIRS-3 | / | / | / | / | 0.13–4.00 | |

HIRS-4 | / | / | / | / | 0.13–6.25 | |

MHS | / | / | / | / | 4.00–6.25 |

Experiment | Case | Assimilation Scheme | Assimilated Observation Type |
---|---|---|---|

Exp-A1 | C-09 | DA-1 | Radiosonde and surface measurements, AMSU-A/B, HIRS-3/4, MHS |

Exp-A2 | DA-2 | ||

Exp-A3 | DA-3 | ||

Exp-B1 | C-14 | DA-1 | Radiosonde and surface measurements |

Exp-B2 | DA-2 | ||

Exp-B3 | DA-3 |

**Table 5.**Statistical results of the experiments. The improvements (%) in Exp-A2 and Exp-A3 relative to Exp-A1 are in parentheses and set in boldface, as well as Exp-B2 and Exp-B3 relative to Exp-B1.

Experiment | Scores | |||
---|---|---|---|---|

POD | FAR | Bias | ETS | |

Exp-A1 | 0.178 | 0.276 | 0.246 | 0.128 |

Exp-A2 | 0.139 (−21.9) | 0.251 (3.5) | 0.186 (−8.0) | 0.102 (−20.3) |

Exp-A3 | 0.281 (57.9) | 0.274 (0.3) | 0.387 (18.7) | 0.199 (55.5) |

Exp-B1 | 0.006 | 0.173 | 0.007 | 0.005 |

Exp-B2 | 0.175 (2816.7) | 0.027 (17.7) | 0.179 (17.3) | 0.152 (2940.0) |

Exp-B3 | 0.605 (9983.3) | 0.304 (−15.8) | 0.869 (86.8) | 0.421 (8320.0) |

Experiment | Case | Location | Assimilation Method | Observation |
---|---|---|---|---|

Exp-AS1 | C-09 | A (125.00° E, 33.00° N) | 3DVAR | 2 m/s u-component wind above the background |

Exp-AS2 | EnKF | |||

Exp-AS3 | B (120.00° E, 25.00° N) | 3DVAR | ||

Exp-AS4 | EnKF | |||

Exp-BS1 | C-14 | C (120.50° E, 36.07° N) | 3DVAR | 2 K temperature below the background |

Exp-BS2 | EnKF |

**Table 7.**Statistical results of the equitable threat score (ETS) for the sensitivity experiments for the case C-09. Their improvements (%) relative to Exp-A3 are in parentheses and set in boldface.

L | 50 km | 100 km | 200 km | 500 km | 800 km | |
---|---|---|---|---|---|---|

β | ||||||

0.6 | 0.252 (26.6) | 0.233 (17.1) | 0.197 (−1.0) | 0.156 (−21.6) | 0.213 (7.0) | |

0.7 | 0.255 (28.1) | 0.239 (20.1) | 0.195 (−2.0) | 0.151 (−24.1) | 0.206 (3.5) | |

0.8 | 0.252 (26.6) | 0.233 (17.1) | 0.193 (−3.0) | 0.144 (−27.6) | 0.197 (−1.0) | |

0.9 | 0.255 (28.1) | 0.229 (15.1) | 0.199 | 0.140 (−29.6) | 0.199 (0.0) | |

1.0 | 0.252 (26.6) | 0.234 (17.6) | 0.197 (−1.0) | 0.136 (−31.7) | 0.179 (−10.1) |

**Table 8.**As in Table 7, but for the sensitivity experiments for the case C-14 relative to Exp-B3.

L | 50 km | 100 km | 200 km | 500 km | 800 km | |
---|---|---|---|---|---|---|

β | ||||||

0.6 | 0.319 (−24.2) | 0.341 (−19.0) | 0.415 (−1.4) | 0.445 (5.7) | 0.420 (−0.2) | |

0.7 | 0.330 (−21.6) | 0.376 (−10.7) | 0.417 (−1.0) | 0.447 (6.2) | 0.403 (−4.3) | |

0.8 | 0.314 (−25.4) | 0.374 (−11.2) | 0.421 (0.00) | 0.454 (7.8) | 0.415 (−1.4) | |

0.9 | 0.322 (−23.5) | 0.389 (−7.6) | 0.421 | 0.452 (7.4) | 0.411 (−2.4) | |

1.0 | 0.322 (−23.5) | 0.391 (−7.1) | 0.425 (1.00) | 0.455 (8.1) | 0.410 (−2.6) |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gao, X.; Gao, S.; Yang, Y. A Comparison between 3DVAR and EnKF for Data Assimilation Effects on the Yellow Sea Fog Forecast. *Atmosphere* **2018**, *9*, 346.
https://doi.org/10.3390/atmos9090346

**AMA Style**

Gao X, Gao S, Yang Y. A Comparison between 3DVAR and EnKF for Data Assimilation Effects on the Yellow Sea Fog Forecast. *Atmosphere*. 2018; 9(9):346.
https://doi.org/10.3390/atmos9090346

**Chicago/Turabian Style**

Gao, Xiaoyu, Shanhong Gao, and Yue Yang. 2018. "A Comparison between 3DVAR and EnKF for Data Assimilation Effects on the Yellow Sea Fog Forecast" *Atmosphere* 9, no. 9: 346.
https://doi.org/10.3390/atmos9090346