Combined wind profile characteristics based on wind parameters joint probability model in a mountainous gorge

Long-span bridges in mountainous areas are greatly disturbed by wind, and the wind field at the mountain gorge bridge site is extremely complex. Therefore, it is of great engineering significance to accurately evaluate the wind field characteristics of this kind of terrain. In this paper, to enhance understanding of this kind of wind field, the wind field in a mountainous gorge is measured for a long time using wind radar, and the mean wind parameters are statistically analyzed. The results show that the mean wind parameters vary greatly under different wind directions, and the wind speed profile does not meet the power-law model. Therefore, a mixed model suitable for the wind speed profile in a mountainous gorge is proposed. Additionally, GEV distribution and Logistic distribution are found to be suitable for describing the distribution characteristics of wind speed and angle of attack, respectively. In addition, considering the correlation between wind parameters, this paper also constructs the joint probability model of wind speed and angle of attack at different heights by the Copula function. Thus, a combined wind parameters profile model is developed under different exceedance probabilities based on the inverse first-order reliability method (IFORM). This study can provide a reference for the construction of the joint probability model of wind parameters.


Introduction
With the development of China's economy, the investment and construction proportion of infrastructure in Western China is further expanded, and more and more bridges are built in western mountainous areas (Lystad et al. 2018). In recent years, the bridges built are also developing towards a longer, larger, and more flexible tendency, making such bridges more sensitive to the wind (Wang et al. 2016;Li et al. 2011). Due to the large undulation of mountain terrain and the aerodynamic interference between different features of the topography, the wind in the atmospheric boundary layer will be affected by terrain, surface roughness, and some thermal factors; thus, the wind field in the bridge site area of complex terrain become extremely complex (Ma et al. 2019). The wind environment parameters in the existing specifications or standards are mostly obtained based on the good climate environment model of the inland plain open area, which makes it of great engineering significance to accurately evaluate the wind field characteristics of such terrain (Hu et al. 2021). Therefore, many scholars have researched the wind field characteristics under complex terrain conditions by using the terrain model in the wind tunnel test, CFD numerical simulation, and field measurement methods and have achieved many remarkable results (Li et al. 2014;Quan et al. 2020).
According to the actual mountain terrain conditions, much literature studies the wind field characteristics of this terrain by placing two-dimensional and three-dimensional terrain scale models with different scales in the wind tunnel (Lubitz and White, 2007;Kozmar et al. 2016;Kim et al. 2017). The results show that the mountain gorge has a locking effect on the wind direction, and the convex mountain body has a certain acceleration effect on the airflow, especially on the top of the mountain, but the increased degree is related to the airflow direction (Chen et al. 2017). At the same time, the slope, surface roughness, and shielding effect of mountains have different effects on the 10-min mean wind speed and turbulence intensity of the wind field. In addition, the strong non-uniform characteristics and the variability of the wind speed profile of wind fields in mountainous and gorge areas are also verified by topographic wind tunnel tests (Tang et al. 2020). Nevertheless, for some more complex mountainous terrain, building the corresponding terrain model becomes very complex, and the landform complexity can not be well-reflected. Therefore, some scholars have also studied many two-dimensional and three-dimensional terrain wind fields combined with CFD numerical simulation and compared them with the wind tunnel test results . The results show that the CFD method can draw the same conclusion as the wind tunnel test when using the appropriate calculation model (Cao et al. 2012;Tamura and Kareem 2013). It is worth noting that the research has shown that the wind parameter profile in the mountainous gorge area does not meet the power-law or logarithmic law distribution, but there is no in-depth analysis of its reasonable profile model (Holtslag 1984; Barnéoud and Ek 2020;Chen et al. 2020). Therefore, it is still necessary to use the measured data to study the wind field of this complex terrain, which has important reference value for further understanding and summarizing the wind field characteristics of the deep gorge area.
At present, many scholars have measured and studied different complex wind fields and achieved many breakthrough results (Li et al. 2017a, b;Fenerci and Øiseth 2018;Wang 2022). Davenport (1960Davenport ( , 1964 previously proposed the power-law model of wind speed profile and surface roughness based on strong wind observation records in wind fields at different locations and heights. Three different types of land surfaces were described by this method, and this lays a good foundation for the study of the wind field in the mountain gorge area. Thus, many researchers have measured different mountainous terrain, including the U-shaped gorge (Wang et al. 2021a, b), V-shaped gorge , and Y-shaped river channel (Huang et al. 2019), and others. The results show that the wind field in mountainous areas is greatly constrained by local terrain, which is different from that in open flat areas (Hui et al. 2009;Li et al. 2017a, b;Wang et al. 2020a, b). The main performance is that the mainstream wind direction is consistent with the gorge direction, and the wind parameters corresponding to different wind directions are different (Zhang et al. , 2022. At the same time, the influence of thermal and local shielding effects on wind characteristics in mountainous areas is also found and proved , which reveals the causes of strong daily wind, and greatly promotes the understanding of wind fields in mountainous areas. However, the above-measured analysis mostly focuses on the research of a single point or several measuring points, and the research on wind field profiles is relatively few. Existing studies have shown that the wind parameter profile of this kind of wind field is quite different from the conventional power-law model (Tamura et al. 2001;Zhang et al. 2020). Consequently, it is necessary to study its wind field profile model further. It should be noted that angle of attack, as one of the most important wind parameters in the wind-resistant design of bridges, plays a decisive role in the wind-resistant performance of bridges (Cheynet et al. 2016). Existing studies have shown a strong correlation between the angle of attack and wind speed in the mountain bridge site, and the dispersion and range of attack angle gradually decrease with the increase of wind speed ). In addition, obviously different from the flat area, the angle of attack at high wind speed is still in a large range, rather than 0 . Conversely, there is a lack of consideration of the correlation between them, limiting the development of design wind speed standards for bridges in mountainous areas to a certain extent. Therefore, it is necessary to study the correlation between wind speed and angle of attack.
Therefore, to build a method that can comprehensively and reasonably measure the influence of wind direction on wind speed and angle of attack profile and the correlation characteristics between wind speed and angle of attack. In this paper, based on the two-year observation of the wind field in a V-shaped gorge mountain area by using wind radar, the correlation between the wind parameters at different heights is studied. Thus, it can provide help for the follow-up research on the correlation between multi-dimensional wind parameters, and provide an important reference for the reliable design of bridges in mountainous areas and the determination of composite wind speed. It is also of great significance for the further improvement of wind resistance design specifications of bridges in mountainous areas. Based on the measured samples, this paper first investigates the actual distribution of wind speed samples in each wind direction and investigates the statistical model of wind field profile in a deep-cut gorge (in Sect. 2). Then, the reasonable probability distribution model of wind parameters at different heights is studied, and the joint probability model of wind speed and angle of attack is constructed by the Copula method (in Sect. 3). Finally, based on the statistical model of wind profile and joint probability model, the wind speed angle of attack profile under different probabilities is resampled (in Sect. 4).

Field measurement site
LongJiang Bridge, as a key project of the S10 expressway between Baoshan city and Tengchong city, is a single-span suspension bridge with a main span of 1196 m and an anchor span of 320 m. As can be seen from Fig. 1, the bridge towers on the Baoshan side and Tengchong side are 178.7 m and 137.7 m, respectively, and the deck of the bridge is 285 m from the bottom of the gorge. The bridge site area is a typical V-shaped deep-cutting gorge of 1100 m, located in the mountainous area of Southwest China. It can be seen from Fig. 1 that the slope gradient of the two sides is generally 30 ~ 50°, and the majority of the terrain is a steep slope with developed vegetation. Generally, the terrain is a roughly step-like distribution with a wide platform on the gorge's Baoshan (East) side. On the Tengchong (West) side, some cliffs with a height of 30 ~ 40 m appear in some local areas, and their gradients reach as high as 60°. The local terrain at the bridge site is shown in Fig. 2, the orientation of the river is 26.5° ~ 206.5°, and the bridge axis is 127° ~ 307°. In addition, there are three East-West gorges near the bridge axis, a ridge on the Tengchong side on the downstream, and a mountain with a height of 1704 m located on the Baoshan side.

Layout of the monitor system
To measure the wind speed profile, wind direction, and angle of attack near the midspan, a wind radar profiler system (see Fig. 1) was installed 190 m below the designed height of the bridge deck in the gorge. Its working frequency is from 1650 to 2750 Hz, and the detection height ranges from 30 to 1000 m. Besides, the range of horizontal wind speed is 0 ~ 50 m/s (the corresponding accuracy is 0.1 ~ 0.3 m/s), and the range of perpendicular to wind speed is 0 m/s ~ 10 m/s (the corresponding accuracy is 0.03 ~ 0.1 m/s). Figure 3 shows the field installation location of the wind radar profiler system, and the timespan of the field measurement is set as 10 min. The data returned by the instrument include the perpendicular to distance from the instrument, the average wind speed of U 1 (East-West), U 2 (North-South), and U 3 (perpendicular to direction) over 10 min, and the corresponding mean square deviations in three directions for 10 min.

Recorded data and analysis process
Due to the corresponding data obtained from the wind radar is not the final required wind parameters, the relevant wind parameters need to be converted. The calculation formulas of 10-min mean wind speed, wind direction, and angle of attack can be expressed as Eqs.
(1) ~ (3), respectively. At the same time, there is invalid data in the collected data samples. When the ground is high, the number of samples is limited in that the reflection path is long, and the observation signal is weak. When the ground is low, the accuracy is poor since the reflected signals from the ground, surrounding buildings, and mountains. Therefore, this paper only analyzes the samples in 50 m ~ 400 m. ( Components of the wind radar system 1 3 here, U is the 10-min average wind speed, unit: m/s; U 1 , U 2 , and U 3 represent the measured wind speed in the east, north, and vertical directions, respectively, unit: m/s; φ is the 10-min mean wind direction, unit (°), and 0° and 180° represent the north and south directions, respectively; α is the angle of attack, unit (°) and the value is positive when the flow direction is upward.
Considering the absence and distortion of the measured data, this paper first identifies the invalid data. When the data missing rate of the measured wind speed profile exceeds 5% or three or more consecutive data are missing, it will be discarded; otherwise, linear interpolation will be performed. Then, the distortion judgment is performed on the data of the whole section. If the data does not meet the principle of three times standard deviation, the distorted data is deleted and the data missing the judgment cycle is re-entered. When the data of the whole section is continuous and effective, the calculation of relevant wind parameters can be carried out. The specific flow is shown in Fig. 4.
Furthermore, to fully clarify the influence of terrain on wind profile, the sample is divided into four sectors according to the actual terrain conditions , as shown in Fig. 5a. Where the sector S1 (wind direction range: 337.5° ~ 67.5°) and the sector S3 (wind direction range: 157.5° ~ 247.5°) belong to the direction along the gorge, sector S2 (wind direction range: 67.5° ~ 157.5°) and sector S4 (wind direction range: 247.5° ~ 337.5°) belong to the direction perpendicular to the gorge. Considering that the profile characteristics of small wind speed are very discrete, it is very important to select a reasonable wind speed threshold to screen the samples. Records of lower wind speeds tend to be severely unstable due to rapid changes in temperature and wind direction, and different scholars choose different wind speed thresholds as filtering conditions according to the actual needs (Fenerci and  Here, the wind speed at the height of 50 m is higher than 1 m/s as the sample filtering condition to ensure the sufficiency and reliability of the samples. When the profile data does not meet this condition, it will not be used for subsequent analysis. Figure 5b shows the distribution of samples at the height of 50 m in each sector, and the number in the box in the figure represents the number of samples of each sector filtered. It can be seen from the figure that the number of samples along the gorge (S1 and S3 sectors) is significantly higher than those perpendicular to the gorge (S2 and S4 sectors), mainly caused by the shelter of the terrain. Therefore, the subsequent analysis will be carried out on this basis unless otherwise specified. Figure 5c is the overall analysis process of this paper. It includes the following steps: (1) Analyze the evolution law of wind speed with wind direction, normalize it, and then explore the analytical model of normalized wind speed profile; (2) Based on step 1, the relationship between the parameters in the analytical model and the reference value of wind speed is further clarified, and the parametric normalized wind speed profile analytical model is obtained; (3) The evolution law of wind speed and angle of attack with height is explored, and the reasonable edge distribution model of wind speed and angle of attack at each height is obtained; (4) Based on the edge distribution model, a two-dimensional joint probability model of wind speed and angle of attack is constructed by copula method; (5) Combined with the joint probability model, the inverse first-order reliability method (IFORM) is used to resample the wind speed and angle of attack profiles at different heights under different probabilities.

Evolution with wind direction
The wind speed profiles of different sectors are statistically analyzed, and the results are shown in Fig. 6. It can be seen from the figure that the wind speed profiles belonging to the same sector have the same shape, while the profile shape between different sectors is quite different, which is mainly caused by the terrain differences corresponding to different wind directions. This difference made the incoming flow along the gorge mainly affected by the topography of the upper and lower reaches of the river gorge, and the wind speed will accelerate obviously. The mountains block the incoming flow perpendicular to the gorge, and the number and size of wind speed samples were greatly limited. In addition, by analyzing the wind speed profile patterns under different wind sectors, it is found that there is an essential difference in the wind speed profile with conventional flat areas (Wang 2021). The mean value of the wind speed profile in each sector has an obvious inflection point on the curve. There is a significant S-type feature at 50-30 m and an obvious logarithmic law feature after 230 m. After analyzing the local topography of the bridge site, it can be explained that the range of 50-30 m is greatly affected by the local terrain around the bridge site (230 m is just the platform height of the local terrain), and thus an obvious reverse shear phenomenon exists in the wind speed profile. When it is higher than the terrain around the bridge site, the space is relatively open, and the influence effect of the terrain on the profile is significantly weakened. Therefore, combined with some current research results (Sathe and Bierbooms 2007;Ricciardelli et al. 2019;Wang et al. 2021a, b), this paper proposes a wind speed profile model by combining S-type and logarithmic law profile functions to describe the mean value of wind speed profile (Mixed wind speed profile model, called MWSPM). As shown in Fig. 7, f(z) and g(z) in the figure are respectively the cubic polynomial function describing the S-shaped curve (see Eq. (4)) and the logarithmic function describing the logarithmic law curve (see Eq. (5)). The two functions are continuous at the dividing point z r (taking the value according to the actual terrain, z r in this paper is 230 m). To further explore  Fig. 7) to eliminate the difference between wind speed profiles under different wind speeds. At the same time, to compare the difference between the mixed wind speed profile model (MWSPM) and the conventional power-law model (expressed by Eq. (6)), the results obtained by the two methods are compared at the height of 400 m. The coefficient of determination R 2 (see Eq. (7)) is used to judge the fitting quality of mixed profile mode (MWSPM), and the fitting results are shown in Fig. 8. It can be seen from the figure that the R 2 of each sector is greater than 0.98, indicating that the mixed wind speed profile model (MWSPM) proposed in this paper can better characterize the wind speed profile characteristics of bridge sites in the mountainous area.
(4)   Fig. 8 Evolution of wind speed ratio with wind direction here, p i (i = 1, 2, 3, 4, 5) is the parameter to be fitted, when z = z 0 , p 4 = f (z 0 ) = 1 ; r 0 is the fair value at the dividing point z r ; α 0 is the roughness coefficient in the power-law model of wind speed profile; ŷ j and y j are the fitted value and observation value of the sample, respectively. m indicates the number of samples. Figure 9 and Table 1 show the distribution of the fitted parameters in each sector. It can be concluded that the parameters fitted by S1 and S3 sectors along the gorge are the same, and the samples of the two sectors can be combined and analyzed when the number of samples is insufficient. The results of the S2 and S4 sectors are quite different due to the large dispersion of samples from the blocking effect of mountains. It is worth noting that the red line in the figure is the result of the wind speed ratio at 400 m height calculated by the conventional power-law model. The results show that the calculated values of the MWSPM are less than those of the power-law method. Thus, the construction height of the bridge should be selected in the section with low wind speed in the S-shaped curve of MWSPM when conditions permit. Considering that most of the construction height of the  bridge is near the platform, it is conservative in calculating the design reference wind speed of the bridge site by using the conventional power law or logarithmic law. It can be further seen from the figure that the roughness coefficient obtained by fitting is significantly higher than that in conventional flat areas. It is due to the complex landform, and the wind speed threshold also controls the wind speed ratio. Therefore, it is necessary to explore further the effect of the threshold of sample screening on the wind speed profile.

Evolution with wind speed threshold
To explore the control effect of the wind speed threshold on the wind speed distribution scale, this paper also takes the wind speed at the height of 50 m as the reference value U 50 and studies different wind speed profile characteristics with an interval of 1 m/s. In addition, the samples in the same straight line direction are combined and analyzed to ensure a sufficient number of samples, and the results are shown in Fig. 10. Figure 10a-f are the wind speed ratio profiles when the wind speed is 2-5 m/s at the height of 50 m along with the gorge sector (S1 + S3) and perpendicular to the gorge sector (S2 + S4), respectively. The results show that the applicability of the MWSPM model is not affected by various filtering conditions. Compared with the sector of the S1 and S3, the scale of the wind speed ratio profile perpendicular to the gorge sector (S2 + S4) gradually decreases with the increase of the wind speed threshold. Thus, selecting a representative wind speed threshold becomes the key factor in characterizing the wind profile scale. At the same time, the surface roughness coefficient is the key parameter in bridge windresistant design; thus, the roughness coefficient corresponding to the wind speed at different wind speed thresholds in different sectors is further analyzed. It can be seen from Fig. 10a-f that under the same wind speed threshold, the roughness coefficient in the direction along the gorge (S1 + S3) is smaller than that in the direction perpendicular to the gorge (S2 + S4). This is mainly because the lower surface along the gorge is mainly water surface, and the surface roughness is small; The vertical direction of the gorge is greatly affected by the mountain terrain and surface vegetation on both sides of the gorge, and the surface roughness is large. However, it should be noted that the wind speed in the direction perpendicular to the gorge is generally smaller than that in the direction along the gorge. In addition, the roughness coefficient decreases with the increase of wind speed in both the direction along the gorge and the direction perpendicular to the gorge and finally tends to a stable value. This is mainly because when the wind speed is large, the air within the scale of the bridge site is affected by the surface vegetation, which reduces part of the friction force acting on the air, and reduces the relative change rate of the pressure gradient between the heights.
Moreover, the fitting parameters of MWSPM in different wind speed ranges under different wind sectors are explored, as shown in Fig. 11. The parameters are simulated by Eq.(8) and evaluated by the coefficient of determination R 2 . The results indicate that the parameters tend to be stable with the increase in wind speed. Therefore, the wind speed profile under the design reference wind speed can be deduced using the results obtained in Fig. 11. It is worth noting that, in the actual design wind speed standard, the selection of wind field profile also determines the final wind speed profile scale by integrating the complexity of landform around the bridge site, surface type, historical extreme value data, and the importance of bridge construction, for determining a more reasonable wind speed threshold.

3
where a i and b i are the parameter to be fitted.

Distribution model of wind speed
Section 3.1 obtained different situations of wind speed ratio profiles, so it is necessary to further determine the design reference wind speed in the bridge site area for the design wind speed standard (Davenport 1960). There may be differences in the distribution models of wind parameters at different heights. Thus, it is of great reference significance to explore the distribution profiles of wind parameters at different heights to determine the design value of wind speed at bridge sites in mountainous areas.
According to some current research results (Carta and Ramírez 2007), the probability density functions of GEV (see Eq.(9)), Gamma (see Eq.(10)), Lognormal (see Eq.(11)), and two-parameter Weibull distribution (see Eq. (12)) are selected to analyze the filtered samples from Sect. 2.3. To obtain the parameters, the maximum likelihood method (MLE) with the advantages of simple principle is selected for parameter estimation. Same as Sect. 3, the coefficient of determination R 2 is used to judge the fitting quality of the data, as shown in Fig. 12. Compared with the other three distribution functions, the GEV distribution function is more consistent with the measured wind speed distribution in different sectors, and R 2 at each height in other sectors exceeds 0.95. Therefore, GEV distribution can describe the wind speed distribution of wind fields in mountainous areas. Figure 13 is the GEV distribution parameter of each height, and the shape parameter k gradually decreases from positive to negative with the increase of height.  Fig. 11 Evolution of wind speed ratio with wind speed range of at reference height z r It indicated that the wind speed distribution characteristics in mountainous areas would change significantly with height, and the GEV distribution function can well capture this characteristic. In addition, position parameter c and scale parameter γ keep a strong positive correlation with the height. It is worth noting that their evolution law is highly similar to the law of wind speed profile. where v represents the sample value of the wind speed variable. k, γ, and c are the shape, position (describing the starting position of the distribution curve), and scale parameters of each distribution function, respectively, which are the parameters to be fitted.

Distribution model of angle of attack
The angle of attack is an important mean wind parameter in the bridge design standard of wind speed. It is also of great significance to study the distribution of the angle of attack at different heights. Some existing studies show that the angle of attack can be described by Gaussian distribution and Gumbel distribution . However, the applicability of the Gaussian distribution may be limited due to the differences in the topography of the bridge site; thus, this paper also selects the Logistic distribution with a longer tail than the Gaussian distribution to explore the angle of attack distribution. The probability density functions (PDF) corresponding to each distribution are shown in Eqs.(13) ~ (15). The same parameter estimation method and fitting evaluation index as Sect. 3.2.1 are selected to study the angle of attack distribution characteristics.
(9) f (v ;k, , c) = 1 3 here, α is the angle of attack; k and c are the shape and scale parameters of each distribution function, which are the parameters to be fitted. Figure 14 shows the fitting results corresponding to the three distribution functions. It can be seen from the figure that in the area below 100 m, the Gaussian distribution and Logistic distribution can achieve the same effect, but with the increase in height, the Logistic distribution represents great advantages. Consequently, Logistic distribution is suitable for characterizing the distribution of the angle of attack at various heights in similar areas. Further, the variation of Logistic distribution parameters along the height is investigated, as shown in Fig. 15. Compared with the perpendicular to the gorge direction, the correlation between the shape and position parameters of the angle of attack along the gorge is significantly weakened, mainly due to the relatively large wind speed along the gorge and the relatively concentrated distribution of the angle of attack. In addition, with the increase in height, the constraint effect of the gorge will gradually decrease, and the shape parameters of each sector will eventually tend to the same stable value.

Construction approach of joint distribution model
In the design wind speed standard of bridges in mountainous areas, the correlation between wind speed and angle of attack has not been fully considered, which makes the wind resistance design of the bridge conservative. Therefore, after obtaining the distribution characteristics of wind speed and angle of attack corresponding to each height in different sectors, the joint distribution model of the two can be obtained according to Eq. here, F(v) and f(v) are the GEV distribution and corresponding probability density function of wind speed, respectively. F(α) and f(α) are the Logistic distribution function and corresponding probability density function of attack angle, respectively. f(α|v) is the conditional probability density function of the angle of attack at different wind speeds. f(α,v) is the joint probability density function of wind speed and angle of attack. C α,v is the Copula probability density function connecting the edge distributions F(α) and F(v), which can be expressed by Eq.(18). Where C φ is the Copula function (Nelsen 2006).
There are many kinds of Copula functions, and the distribution characteristics they can describe are limited (Wei et al. 2021); thus, it is necessary to select the optimal Copula based on the actual distribution characteristics. Figure 16a-d shows the actual distribution of wind speed and angle of attack along with the gorge sectors, including asymmetric accumulation type (Fig. 16a,b), single tail convex type (Fig. 16c) and asymmetric depression type (Fig. 16d). It can be seen from the figure that the joint distribution between wind speed and angle of attack shows strong asymmetric characteristics, and the Archimedean Copula function can describe this complex dependence. Therefore, this paper mainly where, φ is the parameter to be fitted corresponding to each Copula function, and the Gumbel, Clayton, and Frank correspond to [1, ∞), [1, ∞) and (− ∞, ∞)\{0}.

Parameters estimation and model evaluations
The maximum likelihood method (MLE) is used to estimate the parameters of the three Copula functions to be fitted. The corresponding likelihood function is shown in Eq. (22), and the parameters φ by the MLE can be expressed as Eq. (23). Furthermore, the concept of empirical Copula is introduced in this paper, as shown in Eq. (24), to evaluate different Copula function models describing the actual wind speed and angle of attack distribution. where n is the sample length and Ĉ n (u, w) is the empirical Copula function. I [·] is an illustrative function, and if F( i )≤u , then At the same time, Bayesian Information Criterion (BIC) is introduced to measure the Copula model's complexity and goodness of fit to minimize the possibility of overfitting. Thus, the optimal model is R 2 maximum and BIC minimum (Wang et al. 2020a, b). Thus, this paper only analyzes the samples along the gorge direction (S1 + S3 sectors) to control the length of the paper. Figure 17a-i are the joint probability models of wind speed and angle of attack corresponding to 100 m, 190 m (at bridge deck height), and 230 m (at platform height z r ) under different Copula functions. It can be seen from the figure that the distribution of wind speed and angle of attack at different heights are quite different due to the influence of local terrain. The angle of attack at 100 m height gradually changes from a positive angle of attack to a negative angle of attack with the increase of wind speed, while

(c)
Frank H= 100 m θ = 0.6577 BIC= -3.58 10 3 Fig. 17 Combined wind parameters profile result under different exceedance probabilities the opposite characteristic is shown at 190 m height, and it is more evenly distributed at 230 m. It is worth noting that when the wind speed at the bridge deck height is in the range of 8-2 m/s, the angle of attack is positive and kept in a large range, which is extremely unfavorable to controlling vortex-induced vibration of the main beam. According to R 2 and BIC of different Copula functions at various heights in Fig. 18a,b, it can be concluded that Frank Copula is more consistent with the distribution of wind speed and angle of attack. Therefore, in this paper, the Frank Copula function describes the joint distribution of wind speed and angle of attack. The corresponding fitting parameters and the joint probability model corresponding to each height are shown in Figs. 18c and 19, respectively.
where, φ is the fitting parameter of the Copula function; n is the sample length, and n φ is the number of φ (n φ = 1); F c and F e are the corresponding probability values of fitting Copula function and empirical Copula function at variable X i , respectively.

Simulation of combined wind parameters profile model via probability resampling
Section 4.2 has obtained the joint distribution model of wind speed and angle of attack along the gorge. With the MWSPM in Sect. 3.2, the combined wind parameters profile model of wind speed and angle of attack under different exceedance probabilities λ can be calculated. Thus, taking the wind speed at 50 m height as 15 m/s as an example, the wind speed at each height can be obtained, as shown in Fig. 20a. Then, as shown in Fig. 20b,c, the angle of attack distribution under different exceedance probabilities λ can be obtained by combining the  (Haselsteiner et al. 2017). Figure 20d shows the angle of attack distribution corresponding to the current wind speed profile under different λ (0.05 -0.45 from outside to inside with an interval of 0.05). Therefore, the proposed method in this paper can determine the angle of attack distribution of each height under a given design reference wind speed, which can provide an important reference for the wind resistance design of the bridge.

Conclusions
(1) Based on the actual topographic conditions and measured data, the influence effects of wind direction on the wind profile are clarified by classifying and discussing the incoming wind speed profiles with different wind directions. (2) According to the actual wind speed profile shape and mechanism analysis, a mixed wind speed profile model (MWSPM) suitable for V-shaped gorge terrain is proposed, and the profile functions of wind speed ratio under different wind directions and different wind speed thresholds are obtained. (3) GEV distribution and Logistic distribution can describe the distribution of wind speed and angle of attack corresponding to different heights and wind directions in mountainous gorge areas. (4) Based on the correlation of average wind parameters, the joint probability model of wind speed and angle of attack at different heights along the gorge direction is constructed, and a combined wind parameter profile model is developed under different exceedance probabilities based on the inverse first-order reliability method (IFORM).
It is worth noting that the sample wind speed used in this paper is not large enough due to the influence of samples, but this paper aims to provide a method for constructing the joint probability model of wind parameters. It will be further explored when there are enough samples to supplement in the future.