Research on optimization method of evaporation duct prediction model based on particle swarm algorithm

The sea surface roughness parameterization and universal stability functions, as key components of the evaporation duct prediction models rooted in the Monin-Obukhov similarity theory, dictate the model performance which further impacts the efficiency and accuracy of offshore electromagnetic applications. In this paper, layered meteorological and hydrological observations are collected during two cruises and processed to obtain the reference modified refractivity profiles close to the sea surface, and then particle swarm algorithm is utilized to optimize the parameters of the sea surface roughness parameterization and universal stability functions. The results show that compared with the pre-optimization model, the prediction accuracy of the optimized model is improved by 5.09% and 8.12% under stable conditions, and by 9.97% and 31.51% under unstable conditions for observation dataset from each cruise, which proves the feasibility of the proposed method for evaporation duct prediction model optimization.


Introduction
Influenced by the air-sea momentum and heat flux as well as the advection of both dry-hot and moistcold air masses over the sea and land, the tropospheric atmosphere over the sea is prone to atmospheric duct phenomena, which leads to anomalous refraction of electromagnetic waves.In particular conditions, electromagnetic waves from shipborne radar and other radio-frequency devices can become trapped into the duct layer, achieving "over-the-horizon" propagation with minimal transmission losses [1].Besides, trapping electromagnetic waves into the duct layer can also eliminate parts of the echo region, giving rise to the "blind zone."The formation of the evaporation duct, a unique type of atmospheric duct over air-sea interface, is largely attributed to the rapid drop in atmospheric refractivity close to the sea surface.This decline, triggered by water vapor evaporation, makes electromagnetic waves bend more towards the sea surface, exhibiting a curvature greater than the Earth's surface.Remarkably, the occurrence probability of the evaporation duct phenomenon can exceed 85%, and some regions even observe persistent evaporation duct.Recognizing, predicting and utilizing evaporation duct is crucial for accurately estimating radar detection ranges over sea and identifying detection blind spots.
The prevalent method for predicting the evaporation duct primarily relies on models based on the Monin-Obukhov similarity theory.These models aim to forecast and diagnose the refractivity profiles close to sea surface, subsequently output evaporation duct parameters.Inputs for these models are twofold: the average value of meteorological and hydrological measurements as well as the height of the sensor installation above the sea surface.The model's outputs detail near-surface wind speed, temperature, humidity and refractivity profiles, and both the height and strength of the evaporation duct.Notably, these output values play a pivotal role in numerical simulations of electromagnetic wave propagation over sea.Models based on the Monin-Obukhov similarity theory adopt iterative methods to determine the intermediate variables such as Monin-Obukhov length and friction velocity.Difference among these models are often attributed to the selection of sea surface roughness parameterization and universal stability functions [2].
Some studies make use of shipborne observation data to improve the prediction accuracy [10].Numerical simulations have been carried out for optimization of the sea surface roughness and universal stability functions tailored for specific areas.Moreover, a few studies employ experimental data from sea surface electromagnetic wave propagation to tune parameters of the universal stability function [11].
In this study, we exploit shipborne platform to gather meteorological and hydrological observations at various heights above the sea surface to establish reference profiles of atmospheric refractivity.The Particle Swarm Optimization (PSO) algorithm was subsequently deployed to fine-tune the parameters associated with the sea surface roughness and the universal stability functions of the evaporation duct prediction model, in order to improve the prediction accuracy for specific sea areas.The results prove the feasibility of the particle swarm optimization algorithm in enhancing the model performance.
The paper is organized as follows: The second section introduces theoretical frameworks and relevant models, then the model optimization algorithm is proposed in the third section.In the fourth section, details of the experiments and the data analysis are provided.Finally, the fifth section is the conclusions.

Monin Obukhov's similarity theory
The evaporation duct prediction model is anchored in the physical principles governing the marine atmospheric boundary layer, with its theoretical framework rooted in the Monin-Obukhov similarity theory.This theory, introduced in 1954, service as a cornerstone for contemporary micrometeorological studies.In scenarios characterized by static and horizontally uniform conditions, atmospheric stability is typically described using the stability parameter [2], defined as . Specifically, an 0   indicates a stable atmospheric state, while an 0   suggests an unstable state.
( ) where 0.4  = is the von Ká rmá n constant; g is the gravitational acceleration; * u is the friction velocity, * T , * q are the scalar parameters of temperature and specific humidity, T is the atmospheric temperature (K).

Sea surface roughness.
The sea surface roughness quantifies the level of irregularities of the airsea interface.This concept encompasses different aspects: momentum roughness o z , scalar roughness ot z and oq z [12].The roughness is important as it directly dictates the mass and energy exchange processes over boundary.While direct measurements on board are difficult, it's common to derive roughness based on wind speed, wave period and etc., translating to what's known as the sea surface roughness parameterization.For example: where  is the Charnock coefficient, v is the kinematic viscosity and The universal stability function serves as a link, associating observations from a specified height close to the surface with the meteorological parameter.Within the context of the atmospheric stability condition, this function derives corrective values which modify meteorological profiles deduced under neutral condition based on the Monin-Obukhov similarity theory.Consequently, a precise vertical distribution of attributes for wind speed, temperature and humidity under stable or unstable condition is achieved.The universal stability functions differ in their specific formulaic representations and inherent parameter values.The selection of these formulae and parameters typically relays on empirical insights [13].

Evaporation duct prediction model
Commonly used evaporation duct prediction models such as COARE [3], BYC [1], MM5REV [9], NPS [5-7], ECMWF [8] and etc.While all these models draw their foundation from the Monin-Obukhov similarity theory, their predictive accuracy differs in regional climate characteristics.The suitability of these models in different sea areas have been extensively studied to offer tailored models or application recommendations [14].Distinctions among these models lies in the sea surface roughness parameterization and universal stability functions, which is a critical determinant of the prediction accuracy of the atmospheric refractivity profile.
The atmospheric refractivity is typically denoted as: where T is the atmospheric temperature (K), P is the total atmospheric pressure (hPa), e is the water vapor pressure (hPa).Considered the Earth's curvature the modified refractivity M is presented as: where R is the average radius of earth, z (m) is the altitude.
The evaporation duct prediction model outputs the modified refractivity profile to diagnose for duct parameters.The modified refractivity profile from model model M , is evaluated by its root mean square deviation from the measured environmental profile measure M .The smaller the deviation rms M  , the higher the model prediction accuracy: ,, where i denotes the order of the measurement level, I is the total number of measurement levels, j is the sequence of the sample and J signifies the total sample count.For the jth sample at the ith layer, , ij measure M is the observed modified refractivity and , ij model M is the corresponding predicted value of model.

Shipboard data acquisition experiment
High-precision meteorological and hydrological observations closed to the sea surface are critical for evaporation duct prediction model optimizations.To ensure the data quality, high-precision meteorological and hydrological sensors are mounted on different decks of the R/V "Xiangyanghong 18".The sensor installations are illustrated in Figure 1 and the specifications of each type of sensor are outlined in Table 1.In this research, observation data from the "Xiangyanghong 18" during spring and autumn cruises in 2021, coded as NORC2021-02-1 and NORC2021-02-2 respectively and depicted in Figure 2, are exploited for evaporation duct prediction model optimization.

Observation data process
The data sampling frequency of the onboard meteorological and hydrological sensors is 1Hz.Before integrating observation data into the evaporation duct prediction model, it's necessary to process the instantaneous observations to obtain the bulk parameters that encapsulate the regional environmental conditions.In this study, a 10-minute moving average window is applied.For wind observation data, it's firstly decomposed into north and east components and then averaged and combined again to produce the mean wind speed and direction.For sea surface roughness parameterizations which rely on wave parameters such as significant wave height, wave period and wave direction, the required parameters are retrieved from the ERA5 reanalysis dataset [15].For layered observations of temperature and humidity, least squares fitting [1] is used to derive the referenced modified refractivity profile in each time instant to act as baselines in the following optimization procedures.where z is the height from the sea surface, i denotes the order of the measurement level and 0 f , 1 f , 2 f are the fitting coefficients.
For the convenience of subsequent optimization, a constraint was introduced to ensure the consistency that the modified refractivity from model at the height of ith level is equal to the one from corresponding measurements.

Evaporation duct prediction model selection
The sea surface roughness parameterization and universal stability functions are combined and then the model prediction accuracy ( ) is evaluated based on the NORC2021-02-1 observation dataset to select the best combination with highest prediction accuracy as the model to be optimized.

Figure 3. Comparison of Evaporation duct Prediction Models
As shown in Figure 3, the combination of TY01 sea surface roughness parameterization and the G07 universal stability function achieves the highest prediction accuracy are selected to optimize the model.

Evaporation duct prediction model optimization
where a=1200, b=4.5, s h is the significant wave height and w l is the dominant wavelength.

Universal stability function:
For stable conditions: For unstable conditions: M unstable where M  and H  are universal stability function.For the TY01 sea surface roughness parameterization scheme and the G07 universal stability function, the optimized parameters and corresponding numeric range are shown in Table 2.In this study, we exploit the particle swarm optimization (PSO) algorithm [16] to refine parameters associated with the sea surface roughness parameterization and the universal stability function.The PSO algorithm stands out for its minimal parameters, fast convergence, and straightforward implementation, making it especially effective for continuous optimization challenges.
where k is the iteration number.The inertia weight is represented as  , while 1 c and 2 c are the cognitive and social factors respectively.The particle count is given by f.The position of the fth particle in a Ddimensional space is described by fD x , with its velocity in the same space represented as fD v .The optimal position discovered by the fth particle in the D-dimensional space is symbolized as fD p .Meanwhile, the swarm's overall best-found position in this space is indicated by gD p .In this work, the number of particles is set to 100, and the number of iterations is 100.The algorithm employs an inertia weight of 0.8, a cognitive factor of 0.8 and a social factor of 1.5. .Through iterative processes, the optimal parameter values are identified to enhance the prediction accuracy.

Optimization results analysis
Results obtained after optimization are shown in Table 3.The universal stability functions have greater impacts on the model prediction accuracy than the sea surface roughness parameterization in each dataset.Moreover, the model prediction accuracy varies significantly across different sea areas and seasons.
Table 3 The proposed PSO-based optimization method yields better results under unstable conditions than under stable conditions, reflecting the suitability of the sea surface roughness parameterization and universal stability function in certain sea area.Among various parameters to be optimized,

Summary
This study presents a feasible approach to optimize the evaporative duct prediction model based on the particle swarm algorithm.Leveraging layered meteorological and hydrological observations on shipborne platform, reference modified refractivity profiles close to the sea surface are obtained.Using the particle swarm optimization algorithm, the parameters associated with the sea surface roughness parameterization scheme and the universal stability function are fine-tuned.Compared with the model performance before optimization, the prediction accuracy is improved by 5.09% and 8.12% in two cruises datasets under stable conditions.While in unstable conditions, the enhancements are 9.97% and 31.51%.The results present substantial improvements in the model prediction accuracy, which indicates better model suitability in certain sea areas and seasons.

Figure 2 .
Figure 2. Cruise trajectories, "  " marks the start point and " o " marks the end point.

3. 4 . 1 .
Parameters to be optimized.The TY01 Sea Surface roughness parameterization and G07 universal stability function associated formulae and parameters are presented below:Momentum roughness length:

=
as a criterion, each dataset is divided into stable and unstable parts for model optimization under different stability conditions.The PSO algorithm evaluates particle fitness based on models performance, followed by M  , while the sea surface roughness o z and ot z have lesser effects.Due to the substantial benefits of optimizing H

Table 1 .
Sensor specification and installation locations.

Table 2 .
Parameters to optimize and corresponding numeric range
o z , ot z , M  ,