A proposal of a semi-empirical method for modifying the atmospheric pressure and wind fields of tropical cyclones

ABSTRACT The actions of wind and atmospheric pressure associated with tropical cyclones (e.g. typhoons) are considered the primary factors behind the generation of storm surges, though the fields used in meteorological models can sometimes deviate from observations. To improve these, the direct modification method (DMM) has been previously proposed, though this only modifies the wind field of a typhoon, and further development is necessary for applying it to storm surge hindcasts. The present work describes the development of a semi-empirical gradient wind balance-based method (GWB-M) for modifying both the wind and pressure fields in meteorological models, based on the dynamic relationship between the wind and pressure in typhoons (i.e. gradient wind balance). The applicability of GWB-M was assessed through a storm surge hindcast based on Typhoon Faxai in 2019, which generated powerful waves and a storm surge at Tokyo Bay. GWB-M improved the time series of 10 m wind speed and sea level pressure, with their spatial distributions being more realistic than those in DMM and blending parametric typhoon models (BM), which cannot take into account the influence of the complex topography around Tokyo Bay. Further, the maximum sea level anomalies after the typhoon made landfall were also captured by GWB-M with a higher accuracy than DMM.


Introduction
Storm surges are abnormal rises in sea level (Gill 1982) that can cause inundation and are associated with economy loses and casualties in coastal areas (e.g. Typhoon Haiyan/Yolanda in 2013 in the Philippines; Esteban et al. 2015;National Disaster Risk Reduction and Management Council 2014). They are mainly generated by the combined action of the atmospheric pressure and wind stress caused by a tropical cyclone (e.g. typhoons in the western North Pacific), during its passage. For enclosed, shallow water areas, such as Tokyo Bay in Japan, the effects of the wind stress can be predominant, resulting in pronounced storm surges. Therefore, the accuracy of estimating these two meteorological forcings, particularly that of the wind, is crucial for simulating the storm surges that can be generated by a tropical cyclone (hereafter, typhoon).
To obtain such forcings used in storm surge hindcasts, the outputs from meteorological models provided by meteorological organizations have been widely used. ERA-Interim (Schenkel and Hart 2012) and the more recent ERA5 (Xiong et al. 2022) by the European Center for Medium-Range Weather Forecasts are typical examples of recent studies (e.g. Dullaart et al. 2020;Muis et al. 2016). If the area of interest is limited, data with relatively high spatial resolution can also be available. The output of the Meso-Scale Model (MSM) provided by the Japan Meteorological Agency (JMA) is one example (Heidarzadeh et al. 2021;Iwamoto and Takagawa 2017;Nakamura, Mäll, and Shibayama 2019): it is limited to the western North Pacific, though provides hourly sea level pressure and 10 m wind fields with a spatial resolution of 5 km. Additionally, dynamic downscaling using meteorological models such as WRF (Skamarock et al. 2008) can be performed to obtain the high spatiotemporal resolution data of the meteorological forcings. Using such data, notable storm surges that occurred in the past have been successfully simulated (e.g. Mori et al. 2014). However, global reanalysis datasets having a coarse spatial or temporal resolution, such as ERA-Interim and ERA5, are not always capable of adequately capturing the actual meteorological fields of a typhoon. Additionally, the output of MSM (or similar data product) and its downscaled results through WRF can also deviate from the actual typhoon parameters, such as intensity and track, among others (Iwamoto and Kawai 2020;Takagi and Takahashi 2022).
To address this issue using a simplified approach, roughly two methods for modifying the meteorological fields of a typhoon can be considered. Following the definition of Pan et al. (2016), one is a blending method (BM) and the other is a direct modification method (DMM). In BM, the meteorological fields near the typhoon center (simulated by meteorological models) are replaced by those of parametric typhoon models (e.g. Holland 1980;Jelesnianski 1965;Mitsuta and Fujii 1987;Myers 1954;Schloemer 1954). Recently, this has been used to improve both the meteorological fields of typhoons and storm surge hindcasts in coastal areas of the East China Sea (Oey and Chou 2016), Tokyo Bay (Liu and Sasaki 2019;Takagi and Takahashi 2022), Bay of Bengal (Murty et al. 2020;Shashank, Sriram, and Sannasiraj 2022), and the Gulf of Mexico (Vijayan et al. 2021). In DMM, the wind field of a typhoon from a meteorological model is multiplied by a correction factor to achieve levels of wind speed maxima consistent with the best track data. Compared to BM, the wind speeds modified by DMM have shown better performance (Pan et al. 2016) and improved the accuracy of wave hindcasts (Li et al. 2020), though they are less frequently used in storm surge hindcasts. One possible reason is that the current DMM approach only modifies the wind field of a typhoon, not its pressure field. Thus, the discrepancies between the observed and simulated central pressures cannot be resolved, and sea level anomalies simulated in storm surge hindcasts can be underestimated. This indicates that further development of DMM is necessary for these models to be able to accurately hindcasts storm surges.
Therefore, to extend on the method proposed by Pan et al. (2016), the authors developed a method that modifies both the typhoon wind and pressure fields while satisfying the well-known correlation between the two (Atkinson and Holliday 1977) near the center of a typhoon. Also, its applicability to storm surge hindcasts was assessed through the case of Typhoon Faxai in 2019. According to the JMA best track (Japan Meteorological Agency 2022), Typhoon Faxai had a particularly low central pressure (960 hPa) when it crossed over the Miura Peninsula and made landfall in Chiba prefecture on September 9, 2019 (see Figure 1). It passed directly over Tokyo Bay and had a small radius of maximum wind compared to that of other major typhoons that have recently made landfall in Japan (Suzuki et al. 2020). The maximum sustained winds exceeded 40 m/s even as it passed through Tokyo Bay, generating powerful waves (Tamura et al. 2021) and a storm surge. This paper is organized as follows. Section 2 describes the data and methodology used for the storm surge hindcasts. Section 3 presents the results and discussion. Finally, the conclusions are presented in section 4. Figure 2 summarizes the framework used in this study. Typhoon Faxai was rather unusual due to its relatively small size (Suzuki et al. 2020), thus requiring a high-resolution spatiotemporal meteorological field to be employed to accurately hindcast its storm surge. With this in mind, the dynamically downscaled results from MSM were used (Iwamoto and Kawai 2020). The downscaling was performed using the Weather Research and Forecasting model (WRF V3.9, Skamarock et al. 2008). The detailed configuration is summarized in Appendix A (see Table A1). Note that the downscaling results of WRF accurately simulated the synoptic scale fields of 10 m wind and sea level pressure, although there were some discrepancies from observed values near the typhoon center (Iwamoto and Kawai 2020, see also Figure A1).

Modelling framework
During post-processing, the WRF-simulated 10 m wind and sea level pressure fields of Typhoon Faxai were separately modified by using BM, DMM, and the newly proposed method. The meteorological fields of a parametric typhoon model were also simulated when applying BM. The details of each method are described in section 2.2.
Finally, the Regional Oceanic Modelling System (ROMS) (Shchepetkin and McWilliams 2005) was used to conduct the storm surge hindcasts using the five simulated wind stress and sea level pressure fields (see section 2.3).

Blending method
Following Pan et al. (2016) and Liu and Sasaki (2019), the meteorological fields of WRF F WRF (i.e. 10 m wind and sea level pressure fields) were modified to become the synthetic fields F BM : where F PTM is the meteorological fields of a parametric typhoon model (PTM), r is the distance from the typhoon center (m), and r 0 is the radius of maximum sustained winds (m). The meteorological fields near the typhoon center in WRF were completely replaced by those of a parametric typhoon model and smoothly connected to that outside the region occupied by the low pressure system.
The tangential wind speed of a parametric typhoon model V PTM was simulated as follows (Mitsuta and Fujii 1987): where r is the distance from the typhoon center (m), f is the Coriolis parameter, V T is the moving speed of a typhoon (m/s), θ is the angle between the moving direction of a typhoon and the radial vector, ρ a is the air density (¼ 1:1 kg/m 3 ), and P is the atmospheric pressure (Pa). Since Eq. 3 represents the wind speed neglecting the ground surface friction, V PTM was corrected by multiplying the wind speed by the reduction factor C X ð Þ (Fujii and Mitsuta 1986): where X is the nondimensional distance from the typhoon center normalized by the radius of maximum sustained winds (¼ r=r 0 ), X p (¼ 0:5), C 1 ð Þ (¼ 2=3), C X p À � (¼ 1:2), and k (¼ 2:5) are parameters in the model. Additionally, to account for changes in wind direction due to ground surface friction, the inflow angle of wind was set as 30 � (Fujii and Mitsuta 1986).
The pressure field of a parametric typhoon model P PTM was calculated using the classical radial profile (Myers 1954;Schloemer 1954): where P c is the central pressure (Pa), and ΔP (¼ P e À P c ) is the difference between the central pressure P c and the environmental pressure outside the typhoon region P e (Pa). Eq. 5 was also used when calculating the pressure gradient term @P=@r in V PTM . Note that the model parameters P c , V T , and the location of the center of Typhoon Faxai were given by the JMA best track, r 0 and ΔP were given by the estimated result when Eq. 5 was fitted to the sea level pressure observed around Tokyo Bay (Iwamoto and Kawai 2020).

Direct modification method
The modification of the wind speeds of WRF V WRF was performed using the observed maximum sustained wind speed V m Obs: in the JMA best track, following Pan et al. (2016), where V m WRF is the maximum wind speed in V WRF , and r m is the distance from the typhoon center to be modified. The value of r m (¼ 6r 0 ) was set according to Pan et al. (2016). As previously mentioned, Pan et al. (2016) did not consider the modification in the pressure field of the typhoons; thus, when using DMM in this study, the pressure field in WRF was not modified.

Modification based on gradient wind balance in typhoons
A semi-empirical, gradient wind balance-based modification method (GWB-M) was developed for modifying the typhoon fields simulated in WRF, which improves on the existing DMM (Li et al. 2020;Pan et al. 2016). The key features include a modification of the pressure field and the associated wind speed being corrected as a result. First, the modified sea level pressure field P GWB-M was derived using the original pressure field P WRF as follows: where w is the Gaussian weight (Oey and Chou 2016), P PTM is the pressure field of the parametric typhoon model mentioned before, and r c (¼ r 0 = ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 2 log 2 p ) is the characteristic distance from the typhoon center. As the distance between the grid to be corrected and the typhoon center increases, the weight decays exponentially from one to zero. The e −1 -decay radius of the weight to the modified forcings was set to r 0 (same as in BM and DMM), that is, w ¼ 0:5 when r ¼ r 0 .
Then, a correction of the wind speed associated with the pressure field change was derived by assuming that the wind field of a typhoon at 10 m height is in a state of gradient wind balance, defined by Eq. 9 (Gill 1982), where V is the tangential wind speed (m/s). Substituting pressure and wind fields before (P b and V b ) and after the correction (P a and V a ) into Eq. 9 and subtracting the former from the latter, Re-arranging Eq. 10 according to the pressure difference P 0 ;P a À P b and the associated change of wind speed V 0 ;V a À V b , a relationship between V 0 and V b can be derived: If P 0 ¼ 0 at all locations, meaning that both pressure fields are equivalent, there should be no change from The change in wind speed V 0 can be both positive or negative, depending on @P 0 =@r.
Note that V 0 was derived by neglecting the effect of ground and sea surface friction, and this may result in an overestimation of its value. Thus, V 0 was corrected by multiplying by a wind speed reduction factor (0.65) (Vickery et al. 2009). Also, the background wind speed V B , which is independent of the gradient wind balance, should be removed from the initial wind speed V WRF for use as V b in Eq. 12. This was done following the typhoon removal procedure implemented in WRF (Davis and Low-Nam 2001). At first, the Poisson equations of the stream function φ and the velocity potential χ are solved: where � is the gradient operator in the threedimensional space, � is the vertical component of relative vorticity (m −1 ), and δ is the divergence (m −1 ). The bold font indicates a vector, and the non-bold font denotes a scalar. � and δ were calculated by using the wind field in WRF V WRF ; u WRF ; v WRF ð Þ, where u and v denote the zonal and meridional wind components, respectively. φ and χ are two-dimensional functions on a plane; thus, the vertical components of �φ and �χ are zero. Eq. 13 was solved by the Dirichlet boundary condition, and both � and δ were set to 0 outside the radius (300 km) from the typhoon center, following Davis and Low-Nam (2001), and considering the small radius of the maximum sustained wind in Typhoon Faxai ( < 50 km, see Suzuki et al. 2020). Note that, when the typhoon center was near the lateral boundaries of WRF, it was necessary to provide appropriate boundary values so that the solution reflected the asymmetric structure of the typhoon. Thus, when the center was a certain distance (200 km) from the northern or southern boundaries, Eq .13 was solved iteratively by updating the boundary values using the previous solution.
Then, the spatial gradients of φ and χ were taken for yielding the non-divergent (rotational) wind V φ and divergent (irrotational) wind V χ (Holton and Hakim 2012), where k is a unit vector orthogonal to the ground surface. Ideally, the speeds of V φ and V χ are equivalent to the tangential wind speed and inflowing wind speed into the typhoon center near the ground. Thus, the residual of the wind fields can be regarded as the background wind (i.e.
. As a result, the composite wind field was obtained by includes the effect of wind speed reduction due to the friction. Therefore, this effect should be removed before using V comp as V b in Eq. 12, which neglects the effect of the friction. Consequently, V comp was divided by the wind speed reduction factor (i.e. V b ¼ V comp =0:65). Then, V 0 was calculated by substituting the above-mentioned values into Eq. 12.
To modify V WRF , the zonal and meridional components of the correction amount Δu and Δv were calculated using V 0 and the wind speed reduction factor, where θ comp is the angle (positive counterclockwise) between the wind vector of V comp and a horizontal axis, that is, where u comp and v comp are the zonal and meridional wind components of V comp , respectively. Note that V χ can be regarded as the inflowing wind; thus, V comp and θ comp already included the change in wind direction due to the friction. At last, V WRF was modified by adding Δu and Δv to u WRF and v WRF .

Storm surge modeling
ROMS is a free-surface and terrain-following coordinate ocean model that uses a mode-splitting method to solve primitive equations (Shchepetkin and McWilliams 2005). The computational domain was composed of two nested grid systems (see Figure 1), using a two-way nesting approach to treat the water mass budget between parent and child grids. The spatial resolutions of Domains 1 and 2 were 0.01 (ffi where ρ a is the air density (¼ 1:1kg/m 3 ), C D is the surface drag coefficient, and V is the wind speed at 10 m height (m/s). C D was estimated from Eq. 17 (Mitsuyasu and Honda 1982;Nakamura, Mäll, and Shibayama 2019), C D was assumed to be constant if V 10 exceeds 30 m/s, for limiting the momentum transfer under strong wind conditions (Powell, Vickery, and Reinhold 2003). Note that for simplification purposes, astronomical tides and the effect of waves on storm surges (i.e. wave setup) were excluded from the hindcasts. Also, given that at the time the simulation was started a part of the target typhoon was already inside the parent grid (Domain 1), the meteorological forcings were linearly ramped from a calm state to the initial forcing fields at 0900 JST, September 7, for 12 hours.

Statistical analysis
The Mean Error (ME), the Root Mean Square Error (RMSE), and the ratio of the maxima of the observed and simulated sea level anomalies (Peak ratio) were used for the statistical analysis of the storm surge hindcasts: where x s is the time series of simulated sea level anomalies, x o is the time series of observed sea level anomalies, and N is the total number of time series. Note that these values were calculated from 2100 JST, September 8 to 1200 JST, September 9, and the observed sea level anomalies were derived at the six tide stations (see Figure 1), using raw tidal observations (see Appendix B). Figure 3 shows an example of applying GWB-M to the case when Typhoon Faxai made landfall at the head of the bay (0500 JST, September 9). By solving Eqs. 13 and 14, the background wind field V B was removed from the initial wind field V WRF ; speed V B was less than 10 m/ s, generally consistent with the observations (23 km/h ffi 6.4 m/s, Takagi and Takahashi 2022). The typhoon center in WRF was slightly positioned westward compared to the JMA best track data, as seen in P 0 . This resulted in the relatively higher wind speed change of V WRF (i.e. V 0 ) on the right-hand side of the typhoon center in WRF (see the middle center panel in Figure 3). Figure 4a shows the simulated 10 m wind fields at 0500 JST, September 9. The wind field in PTM shows a symmetric spatial pattern near the typhoon center. The wind field in WRF had asymmetric variability due to topographic effects (i.e. deformation of the wind field and reduction of wind speed), and its speed was generally higher over sea than on land. Additionally, the wind direction in WRF at that time in the inner side of Tokyo Bay was toward the northeast, and that in PTM was east to southeast. The wind field modified using BM was almost symmetrical near the typhoon center, as expected from the wind field in PTM. In contrast, the clear asymmetry around the center was maintained when using DMM and GWB-M, as seen in WRF. Using BM, the wind speed over the ocean intensified up to 35 m/s at the innermost part of the bay, with DMM and GWB-M enhancing the wind further over a wider area (reaching to about 40 m/s; Figure 4a). The wind speeds on the A-B cross-section show that both were comparable, except near the head of the bay (GWB-M was about 5 m/s larger than DMM) (Figure 4b).

Modification of the wind and pressure fields
The pressure field in WRF did not show a clear pressure drop at the typhoon's center ( 990 hPa), although that of PTM decreased to nearly 960 hPa (Figures 4a and 4c). At this time, Typhoon Faxai had already made landfall and the central pressure according to the JMA best track data was 960 hPa (see Figure  A1b). For the modified pressure fields, BM and GWB-M yielded central pressures of nearly 960 hPa, indicating that they resulted in a more accurate simulation of the pressure field of the typhoon (although the associated changes in wind speed were different). However, DMM did not modify the pressure field in WRF, overestimating it by over 20 hPa near the typhoon center ( Figure 4c).
The modifications using BM and GWB-M improved the time series of sea level pressure at all locations ( Figure 5a). The maximum discrepancies occurred at Tateyama, relatively far from the typhoon track, and even these were still less than 10 hPa. For other sites located near the track, the sharp pressure falls were well captured, and the discrepancies were less than 5 hPa. The time series of the 10 m wind speed modified by BM and GWB-M were similar (their discrepancies around the maxima were up to 5 m/s; Figure 5c) despite the noticeable difference in the wind fields ( Figure 4a). DMM modified the wind speed from 2200 JST, September 8 to 0900 JST, September 9; the wind speed of DMM otherwise remained largely unchanged from that of WRF. It is noteworthy that GWB-M showed a slightly larger discrepancy with observed values, particularly after the passage of the typhoon (0400 JST to 0600 JST, September 9). Since Typhoon Faxai had a considerably smaller radius of maximum sustained wind compared to other typhoons (Suzuki et al. 2020), the wind speed near the center can be highly sensitive to its track, and the small track error in WRF (see Figure  A1a) could have caused such discrepancies. Overall, the general wind direction was well estimated throughout all locations (Figure 5d). Table 2 summarizes the mean, minimum, and maximum values of the Mean Error (ME), the Root Mean Square Error (RMSE), and the ratio of the maxima in the observed and simulated sea level anomalies (i.e. peak ratio). When applying BM and GWB-M, the ME values indicate very small errors. The RMSE did not show an obvious improvement when applying BM, though it generally decreased when applying DMM and GWB-M (the mean value in GWB-M was the smallest among the five meteorological forcings). Additionally, the peak ratios in BM and GWB-M were closest to 1.0, whereas those in DMM showed only limited improvement.

Hindcast results of storm surge induced by Typhoon Faxai
As shown in Figure 6, two local peaks were observed at all tide stations, with the second peaks being regarded as the seiche (Suzuki et al. 2020). The second peak in Yokohama occurred 1 hour later than that in Chiba, indicating that the sea level anomalies at the innermost bay propagated to its mouth as a free wave (Konishi, Kamihira, and Segawa 1986). The hindcast using WRF underestimated the observed time series of sea level anomalies. The first and second peaks were underestimated by up to 0.4 m and 0.5 m, respectively. The simulated sea level anomalies when applying BM improved, especially for the first peaks (increased by up to 0.4 m compared to that with WRF), except for the case of Tokyo. The improvement in the second peaks was not significant. It should be mentioned that the difference between the sea level anomalies when using PTM and using the meteorological fields modified by BM was, at most, about 0.15 m for both peaks. Given that these peaks occurred when the typhoon made landfall, it indicates that the typhoon fields near the center of both were nearly identical, which caused this slight difference. In addition, DMM partially improved the hindcast, particularly around the second peaks at Tokyo and Chiba (by about up to 0.3 m), where the wind setup tends to be large. However, no obvious effect was confirmed for the first peaks (e.g. Yokosuka), which occurred when the typhoon center approached the tide stations. This appears to be due to the pressure field in DMM, being identical to that in WRF, indicating some limitations of DMM when applying it to storm surge hindcasts.
The results using the meteorological fields modified by GWB-M outperformed those of DMM, and they also show the same degree of improvement around the first peaks, and larger increments around the second peaks (above 0.3 m) than when applying BM. Since the pressure field near the center of the typhoon modified using GWB-M was essentially the same as that of BM (see Eq. 7 and Figure 4), the sea level anomalies induced by both pressure fields (i.e. inverted barometer effect) were almost equivalent (the difference was less than 5 hPa, see Figure 4c). Thus, the difference in the sea level anomalies between BM and GWB-M in Figure 6 primarily occurred due to wind field differences near the typhoon center, as shown in Figure 4a. Such differences are caused by topographic effects, which BM  Figure 3. Note that the locations appeared the maximum winds in PTM and BM differed from that in r 0 due to the property of the parametric typhoon model used (Mitsuta and Fujii 1987). The bold black line indicates the cross-section used in the wind speed and sea level pressure comparisons. (b) and (c) Wind speeds and sea level pressures on the A-B cross-section in each case. cannot physically consider. In fact, there are several low mountains around Tokyo Bay, which are a few hundred meters in elevation (see Figure 5b), and south-westerly winds (i.e. parallel to the bay axis), influenced by the topography, were prevalent (see Figure 4a). However, BM replaced the asymmetric wind field in WRF with a nearly symmetric, westerly, or north-westerly wind field. This wind field would result in a more minor increase in sea levels than would be the case with southwesterly winds, due to the shorter fetch. Hence, when the topography heavily affects the meteorological fields of a typhoon, as in the case of Typhoon Faxai, the improvement in storm surge hindcasts using BM may be insufficient. This highlights one of the limitations of BM and why GWB-M would be preferable in such situations.
Note that an appropriate value for the wind speed threshold when calculating the surface drag coefficient C D is still a matter of debate, though a wind speed of 30 m/s has been recognized as one possible criteria (Sroka and Emanuel 2021). However, several formulae for calculating C D have been proposed since the existence of a wind speed threshold was reported (Powell, Figure 5. (a), (c), and (d) Time series of observed and simulated sea level pressure, 10 m wind speed, and wind direction. The heights in 5c represent the altitude from the ground at which the anemometers are installed, and the observed wind speeds were converted to those at a 10 m height using the 1/7 th power law. (b) Locations of observation sites around the track of the typhoon. The topography data was obtained from GSI maps (Geospatial Information Authority of Japan, 2022). Table 2. Mean values of ME, RMSE, and the ratio of the maxima in the observed and simulated sea level anomalies at the six tide stations. BM, DMM, and GWB-M indicate the meteorological forcings modified by the methods corresponding to their names. Note that the minimum and maximum values of these statistics are also presented in the brackets. Vickery, and Reinhold 2003); C D can vary from 1:5 � 10 À 3 up to 3:0 � 10 À 3 when the wind speed is nearly 30 m/s, depending on the formulae (Bryant and Akbar 2016). Also, according to Lin and Chavas (2012), the use of two formulae widely used in storm surge simulations (Garratt 1977;Large and Pond 1981) can result in a 10 % difference in the average simulation of storm surge heights at New York City or Tampa Bay.
Given that the C D calculated by Eq. 17 were within the range of values calculated by the aforementioned formulae, it is thus possible that the hindcast results may have a margin of error similar to that indicated by Lin and Chavas (2012).

Applicability and limitations of the proposed GWB-M
Although GWB-M shows a better performance than the methods previously proposed (BM and DMM) when performing storm surge hindcasts, several things should be noted. First, the assumption that the atmosphere is in a state of gradient wind balance, expressed through Eq. 9, is not necessarily valid for all typhoons. In the case of Typhoon Faxai, which had a small radius of maximum wind speed and a sharp pressure gradient around its center even after making landfall, a strong horizontal vortex was maintained and thus was almost in a state of gradient wind balance, as Miyamoto, Fudeyasu, and Wada (2022) suggested. However, a typhoon transitioning into an extra-tropical cyclone can deviate from this balanced state. Such a typhoon typically has a lower intensity and the threat of storm surges is comparatively lower, a significant storm surge may still occur if it intensifies rapidly (Nakamura, Mäll, and Shibayama 2019). Nevertheless, GWB-M cannot treat such typhoons, and this is one of the limitations of the method. Second, GWB-M has not been validated for meteorological grid data having coarse spatial resolution (,O 10 ð Þ km), such as ERA-Interim and ERA5. Indeed, the downscaled result from WRF was used in this study, having a spatial resolution of almost 1 km in Domain 2, which satisfied the empirical threshold for simulating a typhoon around its center ( < 2 km, see Gentry and Lackmann 2010). Thus, examining whether GWB-M is also effective in such coarse data appears necessary.
Finally, GWB-M still has room for further improvement in terms of the method for modifying the meteorological fields of a typhoon. One example is the decomposition of the typhoon vortex and background winds. If the output from meteorological models covers a sufficiently large area (i.e. the western North Pacific), a simpler method (Lin and Chavas 2012), which considers the spatial average of the wind field within the typhoon region (,O 1000 ð Þ km), could be used. This study used the output from WRF, where the spatial coverage was limited to the seas around Japan, and did not consider larger areas. Additionally, the wind  Table 2. The sampling intervals of the observations are described in Appendix B (see Table B1).
speed reduction factor used in this study (0.65) is rather arbitrary, though within the ranges reported elsewhere (Vickery et al. 2009). This value is almost consistent with that in the Japanese technical standard in coastal engineering (The Overseas Coastal Area Development Institute of Japan 2020), while higher values have also been used for assessing its impact on storm surge hindcasts (e.g. Lin and Chavas 2012). Therefore, the applicability of GWB-M should be evaluated more extensively in the future.

Conclusion
This study developed a semi-empirical, gradient wind balance-based method (GWB-M) to modify the wind and pressure fields of a typhoon to improve on the existing DMM (Pan et al. 2016), which only modifies the wind field. This method was applied to 2019 Typhoon Faxai, which was downscaled by WRF, and its applicability was assessed through a storm surge hindcast in Tokyo Bay.
The proposed GWB-M improved the time series of 10 m wind speed and sea level pressure around the typhoon track. Similar improvements were confirmed with DMM and the method used by blending parametric typhoon models (BM). However, the wind fields modified by these methods show a contrasting feature. DMM and GWB-M directly modified the southwesterly wind field in WRF (i.e. parallel to the bay axis), which was influenced by the topography around the bay. In contrast, BM replaced the wind field in WRF with a nearly symmetrical westerly or north-westerly wind field, generally orthogonal to the bay axis, resulting in a shorter fetch. As a result, the second peaks of the sea level anomalies after the typhoon made landfall were well captured when using GWB-M, and less so with BM (or DMM due to the unmodified pressure field): the peak ratio (i.e. the ratio of the maxima of the observed and simulated sea level anomalies) was, on average, 0.84 at the six tide stations (0.65 in DMM and 0.84 in BM). These results indicate that the usage of GWB-M can outperform DMM, and it may also be more suitable than BM for cases where the topographic effects on the wind field are significant, as in Tokyo Bay.
However, GWB-M was only applied to the highresolution typhoon fields, not to reanalysis data (e.g. ERA-5). In addition, GWB-M still has room for improvements: the method of decomposing the typhoon vortex and background winds can be further simplified for meteorological grid data that covers a larger domain area, and the optimized values of parameters (such as the wind speed reduction factor) may vary compared to those used in this study. Therefore, further evaluation of the range of applicability and validity of the method is recommended in future research.

Acknowledgments
This work was financially supported by Ministry of Land, Infrastructure, Transport and Tourism in Japan. The observation data were provided by the Japan Meteorological Agency, Yokosuka city, the Japan Coast Guard, the Geospatial Information Authority of Japan and the Maritime Information group of the Port and Airport Research Institute. A part of the present work was performed as a part of activities of Research Institute of Sustainable Future Society, Waseda Research Institute for Science and Engineering, Waseda University. The authors would like to appreciate the contributions of three anonymous reviewers, who provided many constructive comments throughout the review process.

Disclosure statement
No potential conflict of interest was reported by the authors.

Appendix B. Sea level anomalies around Tokyo Bay
Raw tidal observations at Tokyo Bay were gathered in 2019. As shown in Figure 1 and Table B1, data from a total of six tide stations were collected. To derive the anomalies, the astronomical tides were simulated using 60 harmonic constants with tide tables provided by the Japan Coast Guard and the Japan Meteorological Agency. For Kurihama, the astronomical tide was directly estimated by the least square method from the yearlong tidal records. Finally, to calculate the anomalies, the derived time series were subtracted from the raw tidal records. Since the raw data included signals that were not the subject of this study, the Lanczos-cosine filter (Thomson and Emery 2014) was applied to filter out signals longer than 3 days (the filter length was 21 days and the cutoff period was 5 days). The filtering process could successfully remove signals longer than a week in which the amplitude was of the order of 0.1 m ( Figure B1). Figure B1. Estimated sea level anomalies at Yokohama tide station. The inset shows the frequency characteristic of the Lanczoscosine filter for each periodic band.