Calibration of satellite typhoon data based on attitude modified buoy

Abstract

To address the issue of poor accuracy in typhoon wind data, this paper presents a methodology for the calibration of typhoon wind data while conducting an analysis and evaluation of the associated uncertainties. The approach commences with the introduction of two wind field data calculation models. The first model facilitates the correction of buoy attitude, thereby transforming inaccurate buoy wind speed data into real-speed data. In parallel, the second model enables the conversion of buoy-observed true velocity into neutral stable stratified wind parameters, leveraging satellite-derived data for precise calculations. Subsequently, the paper undertakes the task of spatio-temporal alignment between buoy data and satellite observations. Ultimately, a comprehensive comparative analysis is conducted by juxtaposing the ERA5 database with data collected from a moored buoy equipped with the R.M. YOUNG wind monitor. A new simulation method for satellite wind speed data inversion is proposed, and the experimental results demonstrate the effectiveness of the proposed calibration method in enhancing the accuracy of typhoon wind field data. In particular, the maximum wind speeds recorded were 20.15 m s⁻¹ and 13.22 m s⁻¹ during Typhoon "Jangmi (202005)" and "Maysak (202009)," respectively. Furthermore, the mean square errors (MSE) for our method were measured at 0.31 m s⁻¹ and 0.28 m s⁻¹, outperforming the satellite-derived indicators. The expanded uncertainty of measurement results for the two typhoons was calculated at 0.39 m s⁻¹ and 0.34 m s⁻¹, closely aligning with the MSE values. These computational models present a valuable means of enhancing the precision and reducing uncertainty in satellite-derived data. The findings presented in this paper hold great promise for applications in typhoon forecasting, investigations of air-sea interactions, and disaster prevention and mitigation efforts.

1 Introduction

Typhoon is a type of high-intensity tropical cyclone characterized by its ability to transport energy and precipitation across significant distances. It consistently accompanies strong winds, heavy rainfall, abrupt shifts in barometric pressure, rapid fluctuations in temperature and humidity, and other intense convective weather patterns. In recent years, there has been a growing awareness among researchers about the significant role that typhoons play as a key variable influencing global meteorological and climate changes.

An increasing number of meteorologists are recognizing the paramount importance of ensuring the accuracy and reliability of typhoon data. The presence of inaccuracies and a high degree of uncertainty in satellite remote sensing of typhoons severely constrains our capacity to predict and analyze these extreme oceanic weather events (Fangli et al. 2017; Xiang et al. 2021).

Two primary methods are employed for gathering typhoon data: in-situ measurement and satellite remote sensing.

Single point measurements involve the installation of wind sensors on a stationary platform, and these platforms can be categorized into at least three main types. The first category comprises wind sensors placed on offshore oil platforms (Haines et al. 2021; Jiang et al. 2019; Xu et al. 2019). The second category involves wind sensors installed on Volunteer Observing Ships (VOS) (Pei et al. 2019; Han 2003; Liu 2009^). The third category utilizes meteorological observation stations located on remote and isolated islands, far from the mainland (Xie et al. 2010; Chen et al. 2011; Xu et al. 2013; Chen 2022; Haines et al. 2021; Liang et al. 1991; Yao et al. 2015). In general, the accuracy of wind data on the mainland is better than 0.3 m s⁻¹. However, single point measurements suffer from a significant limitation, as the geographic location of the measurement is essentially fixed. This approach cannot capture wind field values in remote oceanic areas, which are crucial for large-scale wind field data required in scientific research.

The satellite observation mode, known as large area measurement (Fangli et al. 2017), has garnered significant attention in recent years. Researchers have increasingly focused on the applicability of satellite-derived wind field data, leading to numerous related studies (Xu 2022; Zhang et al. 2020; Mingsen et al. 2019; Wenming et al. 2021; Shasha et al. 2017; Guanbo et al. 2021; Wang and Dong 2020). As a result, research into satellite remote sensing of wind fields is expanding.

Currently, the accuracy of typhoon wind field measurements obtained through satellite remote sensing is approximately 1.6 m s⁻¹ (Zhang et al. 2022). Regrettably, when oceanographic satellites are tasked with observing high wind speeds (> 25 m s⁻¹), the accuracy of satellite data declines. Errors increase, leading to a significant decrease in performance."

At present, there is no established calibration station for recording ocean high wind speeds, especially during events like typhoons or hurricanes. An increasing number of oceanographers are recognizing the necessity of direct in-situ wind data observations during typhoons. Buoys have emerged as a valuable tool to fulfill this objective. Over the past few years, some oceanographers have conducted research on buoy-based observations. Notable examples include TAO (Tropical Atmosphere Ocean)/TRITON (Triangle Trans-Ocean Buoy Network) and NDBC (National Data Buoy Center) (Zhu et al. 2019; Gong et al. 2019; Wang et al. 2020; Jiang et al. 2016; Saha and Zhang 2022; Hu et al. 2001; Gu et al. 2021; Zhanget al. 2021; Ando et al. 2017).

Unlike measurements taken at stationary land-based stations, buoys are constantly subject to tilting and swaying, resulting in significant discrepancies between the wind speed values recorded by sensors on the buoys and the actual wind speed. Despite significant progress in typhoon measurements using buoys over the years, a fundamental challenge remains unresolved: the distortion of wind speed values is still a prominent issue, making them unsuitable for direct use.

To address this critical challenge and enhance the precision of buoy wind speed measurements, particularly in typhoon conditions, researchers have introduced a novel attitude calculation model for the calibration of buoy-derived wind speed values. This innovation holds the potential to significantly enhance the accuracy, quality, and reliability of wind speed measurements from buoys (Young et al. 2017).

To validate the new attitude calculation method, researchers designed a series of air-sea interface buoys, which were deployed in the Northwest Pacific Ocean. These buoys were equipped with two specialized ocean wind measuring instruments (Young et al. 2017). They served as calibration stations, providing real-time wind data during typhoon passages. This data was analyzed to enhance the accuracy and reliability of satellite wind data while also improving the overall quality of satellite-based observations.

To assess the reliability of the research findings, a mathematical model has been developed. This model comprehensively analyzes all the factors that influence the calibration of typhoon data. It quantitatively evaluates the research outcomes, ultimately enhancing the scientific rigor of the research methodology.

This paper begins by obtaining wind field data from two typhoons in the Northwest Pacific Ocean in 2020 using satellite remote sensing instruments. Additionally, a set of mooring buoys, equipped with anemometers, was strategically placed near the typhoon paths to acquire in-situ typhoon data from very close proximity, ensuring data synchronization. Subsequently, two calculation models were introduced: one for converting buoy wind speed error data into true speed data and another for transforming buoy true speed data into neutral and stable stratified wind that could be compared with satellite data. The satellite remote sensing data, collected from the same geographical locations as the buoys, were selected to ensure spatial synchronization. Following this, numerical calculations for the two typhoon fields were conducted, and the Quantile–Quantile (QQ) plots were generated, along with the calculation of correlation coefficients. To address technical challenges and evaluate the process of measuring typhoon wind field data, uncertainty analysis and assessment of the measurement results are conducted, aiming to provide a quantitative evaluation of the research's technical proficiency.

2 Study on buoy data correction

2.1 Buoy raw data acquisition

The mooring buoy is strategically positioned in the typhoon-prone Northwest Pacific Ocean at a geographical coordinate of 125.96° E longitude and 21.06° N latitude, in an area characterized by a water depth of 5699 m. Figure 1 provides an illustration of the geographical location and on-site photographs.

Fig. 1

Location and layout of mooring buoy

The mooring buoy is equipped with two sets of anemometers, both of which are of the same model and manufactured by R.M. Young Company, USA. The specifications for these anemometers are detailed in Table 1. Positioned in the geometric center of the buoy body, these anemometers are separated by a horizontal distance of 1.0 m, ensuring symmetrical placement. Despite the buoy's swaying motion, the anemometers are consistently exposed to the same wind field. Their heights from the sea surface are 3.0 m and 3.2 m, respectively.

Table 1 Information of the special ocean anemometer instrument

To begin, the wind raw data from the two anemometers was collected. These anemometers, integrated into the mooring buoy, continuously recorded real-time wind speed measurements. Following the guidelines specified in the 'Specification for Typhoon Cyclone Observational Wind Data Differentiation' (Administration and for Market Regulation State Administration for Standardization GB, T 36745-2018 2018), the wind speed data from the two anemometers were denoted as wb1_ij,z and wb2_ij,z respectively. This data was transmitted back via the Iridium satellite system every 10 min, with the respective series numbers assigned as 180681 and 180682 for the two anemometers. The recorded daily data was in a 24-h format, with 'i' representing the hour (ranging from 0 to 23), 'j' representing ten-minute intervals (ranging from 0 to 5), and 'z' indicating the sea level height specific to the buoy's anemometer.

As previously stated, the two instruments are positioned with a horizontal separation of 1.0m and are symmetrically placed in the geometric center of the buoy body. Due to the buoy's swaying motion, the wind field can be reasonably considered to be uniform. In typical conditions, the measurement error of wind speed is expected to be minimal.

The error of the two anemometers is computed using Formula (1)

$$wab_{k} = wb{1}_{k} { - }wb{2}_{k}$$

(1)

A detailed analysis of the accuracy values for the two anemometers is required.

2.2 Buoy data preprocessing

2.2.1 Error data identification and rejection processing

This section conducts data quality control processing to identify and rectify any potential gross errors in the data.

The criterion for identifying gross errors is as follows: Calculate the mean value ($\overline{{\text{wab} }_{k}}$), residual error (ν_wabk), and standard deviation (σ_wab) of the observed data. If ν_wabk > 3σ_wab, the data point is considered a gross error and should be excluded. Subsequently, recalculate the mean value, residual error, and standard deviation for the remaining dataset, and repeat the judgment. Continue this process until all gross error data points have been removed.

To identify and eliminate gross errors, methods like the Pauta criterion, Grubbs criterion, and Dixon criterion can be employed. Grubbs and Dixon criteria are well-suited for situations with a limited number of measurements. For larger sample sizes (exceeding 100 data points), the Pauta criterion is recommended due to its enhanced scientific rigor and accuracy. In this study, to ensure scientific credibility, the Pauta criterion is employed to remove gross errors, and the calculation process is detailed in Formula (2).

$$\left\{\begin{array}{c}\overline{{wab }_{k}}=\frac{1}{1512}\sum_{k=1}^{1512}{wab}_{k}\\ {\nu }_{wabk}=\left|{wab}_{k}-\overline{{wab }_{k}}\right|\\ {\sigma }_{wab}=\frac{1}{k-1}\sqrt{\sum_{k=1}^{1512}{\left({wab}_{k}-\overline{{wab }_{k}}\right)}^{2}}\end{array}\right.$$

(2)

In formula (2), $\overline{{wab }_{k}}$、${\nu }_{wabk}$ and ${\sigma }_{wab}$ are respectively mean value, residual value and standard deviation value.

Following the elimination process, the maximum instrument error remaining is 0.30 m s⁻¹. Therefore, any data with errors less than 0.30 m s⁻¹ are considered reliable for mathematical statistical analysis. These reliable data points can be used for subsequent comparison with satellite remote sensing data in further calculations and calibration.

Before and after the removal process, the standard deviations were 0.36 m s⁻¹ and 0.21 m s⁻¹, respectively. This represents a substantial improvement of 42% in data quality.

Following the removal of gross errors, a data correlation analysis is conducted. The correlation coefficient between the two anemometers is calculated using Formula (3).

$$\uprho 12=\frac{\sum_{k=1}^{1478}\left({wa1}_{k}-\overline{{wa1 }_{k}}\right)\left({wa2}_{k}-\overline{{wa2 }_{k}}\right)}{\sqrt{\sum_{k=1}^{1478}{\left({wa1}_{k}-\overline{{wa1 }_{k}}\right)}^{2}}\sqrt{\sum_{k=1}^{1478}{\left({wa2}_{k}-\overline{{wa2 }_{k}}\right)}^{2}}}$$

(3)

In Formula (3), $\overline{{wa1 }_{k}}$ and $\overline{{wa2 }_{k}}$ represent the mathematical expectations (or means) of the data from the two anemometers. The correlation coefficient between the two anemometers during Typhoon 'Jangmi' is calculated using Formula (4).

$${\uprho 12}_{\text{Jangmi}}=0.987\cong 1.$$

(4)

And the correlation coefficient of the two anemometers in the process of Typhoon “Maysak” is the Formula (5)

$${\uprho 12}_{\text{Maysak}}=0.995\cong 1.$$

(5)

Data reliability analysis was conducted using marine-specific anemometers from R.M. Young Company in the United States. These anemometers are globally recognized in the field of marine meteorology. Prior to deployment, all anemometers were meticulously measured and calibrated by the National Meteorological Station of China. Their performance adheres to the technical requirements outlined in the 'Specification for Surface Meteorological Observation Wind Direction and Wind Speed' (Administration and for Market Regulation State Administration for Standardization GB, T 35227-2017 2018).

After the removal of gross errors, the correlation coefficient indicates a strong linear relationship with a probability of 1. Additionally, simple arithmetic operations do not alter the inherent properties of the wind speed data. Formula (6) is employed to calculate the average value of the two anemometers, which is referred to as the buoy wind speed.

$${wb}_{k}=\frac{{wb1}_{k}+{wb2}_{k}}{2}$$

(6)

2.2.2 Data quality evaluation

During the passage of Typhoon 'Jangmi', an analysis of the data distribution histograms for the two anemometers was conducted, as depicted in Fig. 2. Notably, the wind speed data from both anemometers exhibited a consistent trend. After normalizing the frequency of numerical occurrences, it became evident that the probability distribution of the data was strikingly similar.

Fig. 2

Histogram of data distribution of two anemometers when "Jangmi" passed by

The partial data of wind speed point is shown in the Table 2.

Table 2 Instrument observation data of two typhoons during the passage of "Jangmi" (Part) m s⁻¹

In the process of Typhoon “Maysak” passed by,The data distribution histogram of the two anemometers is shown in Fig. 3.

Fig. 3

Data distribution histograms of the two anemometers when "Maysak" passed by

Just like the Typhoon "Jangmi", and part of the “Maysak “data is shown in Table 3.

Table 3 Instrument observation data of two typhoons during the passage of “Maysak “m⋅s⁻¹

2.2.3 Confirm the statistical distribution of the buoy data

To analyze the statistical distribution of data from the two anemometers, also referred to as the buoy data, QQ plots were drawn for the two typhoons, as depicted in Fig. 4. In the plot, the blue '×' symbols represent matching data points from the two anemometers, while the red line represents the centerline of the first quadrant of the coordinate system, indicating a correlation coefficient of 1.

Fig. 4

QQ bitmaps of the two Anemometers during two typhoons

From Fig. 4, we observe that when the wind speed is below 13 m s⁻¹, the wind speed data from the two anemometers align perfectly with the red line. When the wind speed exceeds 13 m s⁻¹, the wind speed data from the two anemometers exhibit a regular distribution around the red line, indicating that both anemometers conform to the same probability distribution. This observation further reinforces the high reliability of the data from Formula (4) and Formula (5).

2.2.4 Equivalent stress wind conversion processing of buoy anemometer

As mentioned earlier, two sets of anemometers collect wind speed data denoted as wb1_ij,z, wb2_ij,z, with 'z' representing the height of the respective anemometer above the sea surface. Specifically, the sea level heights of the two anemometers on the buoy are 3.0 m and 3.2 m, respectively.

It is worth noting that satellite remote sensing provides equivalent stress wind data at a 10 m sea surface level, with the removal of atmospheric knot effect. Given this difference in data representation, direct calculations between satellite data and anemometer data are not feasible.

To bridge the gap between anemometer data and satellite data, it is essential to transform anemometer data into neutral and stable stratified wind data. To achieve this, we refer to the LKB parameter group jointly published by Liu et al. (1979), as well as the logarithmic wind profile formula proposed by Young (Watson et al. 2003). Additionally, we incorporate national typhoon vortex norms. We employ Formula (7) to adjust the height of wind speed data, resolving the height disparity between satellite and anemometer data.

$${wb1}_{ij,10}={wb1}_{ij,z}\sqrt{\frac{{\kappa }^{2}}{{C}_{d}}}\frac{1}{\text{ln}\left(\frac{z}{{z}_{0}}\right)}$$

(7)

In formula (7), wb1_ij,10 refers to the wind speed at 10m equivalent of the special anemometer.

$\kappa$ is von Kalman constant,

$$\kappa =0.4$$

(8)

C_d is sea surface wind resistance coefficient (Guan and Xie 2004), and the common value range is [1.0 × 10⁻³, 2.0 × 10⁻³], analyze the test and research results of Donelan in 1982,

$$Cd = {1}.{2} \times {1}0^{{ - {3}}}$$

(9)

z is the height from the sea surface of the special anemometer for the buoy (factory number 180681),

$$z = {3}.0\,{\text{m}}$$

(10)

z₀ is the roughness length,

$$z_{0} = {9}.{7} \times {1}0^{{ - {5}}}$$

(11)

wb1_ij,10 is denoted as wb1_ij. And the wb2_ij,10 is denoted as wb2_ij,which is the equivalent wind speed of the other anemometer (factory number 180682) at 10 m, the treatment method of the anemometer is the same.

The satellite data is collected at hourly intervals, with the data at the 'i' time point represented as ws_i. Meanwhile, anemometer data is recorded every 10 min. To address the issue of temporal mismatch, the anemometer data needs to be transformed into hourly intervals. This conversion is achieved using Formulas (12a) and (12b).

$${wb1}_{i}=\sum_{j=0}^{5}{wb1}_{ij}$$

(12a)

$${wb2}_{i}=\sum_{j=0}^{5}{wb2}_{ij}$$

(12b)

The anemometer records six data points within each hour, which can be aggregated and calculated at a 1-h interval, aligning with the satellite data.

2.3 Buoy attitude correction data processing

There are two primary sources of wind speed data: buoy data and satellite data. Sea surface wind, as widely understood, pertains to the horizontal wind parallel to the sea's plane. When a buoy is deployed on the sea surface, it often experiences tilting or swaying motion, causing the wind sensor on the buoy to be unable to maintain a stable horizontal orientation. Consequently, this results in inaccuracies in the recorded wind data.

As illustrated in Fig. 5, to address this challenge, an MTI-300 attitude sensor from XSENS, a Dutch company, is incorporated into the anchor buoy. This sensor facilitates the establishment of a spatial coordinate system, mitigating the limitations of Kalman filtering. It produces data on three-axis acceleration and three-axis angular velocity, enabling precise measurement of parameters such as buoy pitch, roll, and heading angle. Leveraging this wealth of attitude data, a novel attitude correction algorithm has been devised to rectify the erroneous wind data, resulting in the acquisition of high-quality wind field data.

Fig. 5

The attitude correction algorithm of mooring buoy

Figure 5 illustrates the wind sensor as a rigid body, with the buoy's centroid serving as the origin to establish two rectangular coordinate systems. When either the sensor or the buoy is inclined due to wave motion, it is referred to as the buoy's missile coordinate system, denoted as E_A (O, X', Y', Z'). When both the sensor and the buoy are at rest, it is considered the inertial coordinate system, designated as E_B (O, X, Y, Z). We propose a calculation model for converting buoy wind speed error data into accurate wind speed data.

In the context of two rectangular coordinate systems, the six degrees of freedom information of the wind sensor can be encapsulated within a matrix. This matrix is represented in Formula (13a) and Formula (13b).

$${M}_{A}=\left[\begin{array}{cc}{x}_{A}^{/}& \alpha \\ {y}_{A}^{/}& \beta \\ {z}_{A}^{/}& \gamma \end{array}\right]$$

(13a)

$${M}_{B}=\left[\begin{array}{cc}{x}_{B}& {\theta }_{x}\\ {y}_{B}& {\theta }_{y}\\ {z}_{B}& {\theta }_{z}\end{array}\right]$$

(13b)

During the measurement of typhoon wind data, the values in one coordinate system need to be dynamically converted into corresponding values in another coordinate system in real-time. To achieve this, we calculate the dynamic equation around the center of mass, as demonstrated in Formula (14).

$${M}_{A}={R}_{AB}*{M}_{B}+k*\text{E}$$

(14)

where ${\text{R}}_{\text{AB}}$ is the transformation matrix. After derivation and calculation, yaw (${\uptheta }_{{\text{z}}} - {\upgamma }$), pitch (${\uptheta }_{{\text{y}}} - {\upbeta }$$)$ and roll (${\uptheta }_{{\text{x}}} - {\upalpha }$) can be completed in the order of Z, Y and X. The transformation matrix used at this time is in Formula (15).

$${R}_{AB}=\left[\begin{array}{lll}\text{cos}\left({\theta }_{y}-\beta \right)\text{cos}\left({\theta }_{z}-\gamma \right)& \text{cos}\left({\theta }_{y}-\beta \right)\text{sin}\left({\theta }_{z}-\gamma \right)& -\text{sin}\left({\theta }_{y}-\beta \right)\\ \text{sin}\left({\theta }_{x}-\alpha \right)\text{sin}\left({\theta }_{y}-\beta \right)\text{cos}\left({\theta }_{z}-\gamma \right)-\text{cos}\left({\theta }_{x}-\alpha \right)\text{sin}\left({\theta }_{z}-\gamma \right)& \text{sin}\left({\theta }_{x}-\alpha \right)\text{sin}\left({\theta }_{y}-\beta \right)\text{sin}\left({\theta }_{z}-\gamma \right)+\text{cos}\left({\theta }_{x}-\alpha \right)\text{cos}\left({\theta }_{z}-\gamma \right)& \text{sin}\left({\theta }_{x}-\alpha \right)\text{cos}\left({\theta }_{y}-\beta \right)\\ \text{cos}\left({\theta }_{x}-\alpha \right)\text{sin}\left({\theta }_{y}-\beta \right)\text{cos}\left({\theta }_{z}-\gamma \right)+\text{sin}\left({\theta }_{x}-\alpha \right)\text{sin}\left({\theta }_{z}-\gamma \right)& \text{cos}\left({\theta }_{x}-\alpha \right)\text{sin}\left({\theta }_{y}-\beta \right)\text{sin}\left({\theta }_{z}-\gamma \right)-\text{sin}\left({\theta }_{x}-\alpha \right)\text{cos}\left({\theta }_{z}-\gamma \right)& \text{cos}\left({\theta }_{x}-\alpha \right)\text{cos}\left({\theta }_{y}-\beta \right)\end{array}\right]$$

(15)

$\text{E}$ is the inherent error between two sets of rectangular coordinate systems, and $k$ is the factor of proportion.

In previous research, scientific predecessors selected the WHOI buoys and the Salinity Processes in the Upper Ocean Regional Study (SPURS) buoys. Notably, the relative discrepancy in anemometer wind speeds on these buoys was found to reach 5%, with the maximum wind speed error associated with the incident angle being 0.2 m s⁻¹ (Schlundt et al. 2020).

3 Experimental study

3.1 Two examples of typhoon paths

The information regarding the typhoon's path and the evolution of strong wind regions is sourced from the 'Tropical Cyclone Yearbook 2020' (China Meteorological Press 2020). On August 8th and August 9th, 2020, Typhoon 'Jangmi' closely passed by the buoy's location, with the typhoon center at a mere 35 km distance from the buoy site. The trajectory of 'Jangmi' is depicted in Fig. 6 (http://tf.121.com.cn/en/web/tyquery/index.html).

Fig. 6

Moving path of Typhoon "Jangmi"

Between August 29th and August 31st, the super Typhoon 'Maysak' traversed near the buoy's location, with a distance of 302 km between the buoy and the typhoon center. The path of 'Maysak' is illustrated in Fig. 7 (http://tf.121.com.cn/en/web/tyquery/index.html).

Fig. 7

Moving path of Super Typhoon "Maysak"

Wind speed data from both the buoy and the satellite were obtained. These datasets share the same space and timestamp, rendering the measurement and comparative research scientifically sound and theoretically valid.

3.2 Satellite data acquisition and preprocessing

As mentioned earlier, the satellite remote sensing data used in this study is the ERA5 dataset, released by the European Centre for Medium-Range Weather Forecasts (ECMWF). ERA5 is a reanalysis dataset, available in GRIB and NC formats. We selected the specific weather events 'Tropical Storm Jangmi (No. 202005)' and 'Super Typhoon Maysak (No. 202009),' both of which closely passed by the buoy station.

We obtained the ERA5 satellite wind speed data for the two typhoons, 'Tropical Storm Jangmi' and 'Super Typhoon Maysak,' from the ECMWF website. The satellite data represents geospatial statistical averages, covering a geographical coordinate range from 100°E to 150°E and from 0°N to 50°N. We selected equal latitude and longitude sampling with a resolution of 0.25° × 0.25°. The time resolution of the stored data is 1 h. The satellite data is organized as a 160 × 200 matrix, as depicted in Formula (16).

$${Wsi}_{160\times 200}=\left[\begin{array}{lll}{wsi}_{\text{1,1}}& \cdots & {wsi}_{\text{1,200}}\\ \vdots & {wsi}_{n,m}& \vdots \\ {wsi}_{\text{160,1}}& \cdots & {wsi}_{\text{160,200}}\end{array}\right]$$

(16)

In Formula (16), it represents the satellite wind speed data at a specific position, with geographical coordinates relative to the starting point at (100°E, 0°N). Specifically, it represents the satellite wind speed data at a location where the longitude is (100° + 0.25n)E and the latitude is (0 + 0.25 m)N.

The buoy's position is at 125.96°E and 21.06°N. Therefore, the row and column numbers of the matrix can be calculated using Formula (17).

$$\left\{\begin{array}{c}n=\frac{\left(125.96-100\right)}{0.25}=103.8\approx 104\\ m=\frac{\left(21.06-0\right)}{0.25}=84.2\approx 84\end{array}\right.$$

(17)

To enhance the reliability of our analysis results, minimize misjudgments, and reduce coarse errors, we've chosen a satellite data analysis area with a longitude and latitude interval of 0.75° × 0.75°. The buoy's coordinates serve as the center of this selected area. Specifically, the data within rows 83 to 85 and columns 103 to 105 in the corresponding matrix are more accurately aligned with the geographical position of the buoy.

Given that the buoy is constrained by the anchor system, its range of movement on the sea surface forms a circular area with a radius of less than 200 m. Under these conditions, we can consider the geographical position to remain essentially unchanged. During this time, the typhoon wind field can be regarded as a stable random process, with wind field data exhibiting ergodicity across various states. This means that the statistical mean of geospatial data is equivalent to the arithmetic mean over time. Consequently, the wind speed data from the satellite represents the statistical mean of geospatial data, calculated using Formula (18).

$${ws}_{i}=\frac{\sum_{n=83,m=103}^{n=85,m=105}{wsi}_{n,m}}{9}$$

(18)

The wind speed of the buoy, ${wb}_{i}$, represents the arithmetic mean of data collected at 0 min, 10 min, 20 min, 30 min, 40 min, and 50 min intervals, which is a measure of the arithmetic mean over time. At the 'i-th' hour, it can be deduced using Formula (19).

$${ws}_{i}={wb}_{i}$$

(19)

3.3 Measurement comparison analysis process and result analysis

3.3.1 Analysis of wind data of Typhoon "Jangmi"

Typhoon 'Jangmi' passed through the buoy station from 00:00 on August 8, 2020, to 24:00 on August 9, 2020. The satellite data time is in UTC (Coordinated Universal Time) format, while the buoy data time follows Beijing time format. There is a fixed 8-h time difference between the two time systems, and the calculation procedure is outlined in Formula (20).

$${t}_{Beijing}={t}_{\text{satellite}}+8h$$

(20)

After aligning the starting points of the satellite data and the buoy data, professional software was employed to analyze the satellite data. This analysis resulted in the creation of charts depicting the changes in the typhoon's wind field and its path, as illustrated in Figs. 8, 9, 10, 11. A close examination of these figures clearly shows that the buoy and Typhoon 'Jangmi' had a direct encounter, with the closest distance between the two measuring just 36 km.

Fig. 8

Typhoon "Jangmi" wind field (2020-08-08 20:00)

Fig. 9

Typhoon "Jangmi" Wind Field (2020-08-09 02:00)

Fig. 10

Typhoon "Jangmi" Wind Field (2020-08-09 06:00)

Fig. 11

Typhoon "Jangmi" Wind Field (2020-08-09 12:00)

The maximum wind speed recorded by the satellite was 19.88 m s⁻¹, while the buoy measured a maximum wind speed of 20.01 m s⁻¹. Wind speed error is calculated using Formula (21).

$${\Delta we}_{i}={ws}_{i}-{wb}_{i}$$

(21)

The maximum error between buoy data and satellite data is 0.27 m s⁻¹. In this paper, an unbiased estimation processing method is employed, and Formula (22) is utilized to calculate the sample variance.

$${\upsigma }_{Jangmi}=\sqrt{\frac{\sum_{i=1}^{n}{\left({ws}_{i}-\overline{{ws}_{i}}\right)}^{2}}{n-1}}$$

(22)

$\overline{{ws}_{i}}$ is the mathematical expectation of the ${ws}_{i}$.

The mean square error (MSE) of 0.31 m s⁻¹ is obtained, which is better than the satellite design index of 2.0 m s⁻¹. Some main data are shown in Table 4.

Table 4 Measurement comparison data of Typhoon "Jangmi" (part)

3.3.2 Analysis of wind data of Typhoon Maysak

From 00:00 on August 29, 2020, to 24:00 on September 1, 2020, Typhoon 'Maysak' brushed past the buoy, and the typhoon eye was distinctly visible. The closest distance from the buoy to the typhoon center was 243 km. The trajectory of the typhoon and its associated wind field are depicted in Figs. 12, 13, 14, 15.

Fig. 12

Typhoon "Misaku" wind field (2020-08-29 08:00)

Fig. 13

Typhoon Maysak Wind Field (2020-08-30 08:00)

Fig. 14

Typhoon Maysak Wind Field (2020-08-30 22:00)

Fig. 15

Typhoon Maysak Wind Field (2020-08-31 04:00)

The calculation process is consistent with that of Typhoon 'Maysak'. The maximum wind speed recorded by the satellite and the buoy was 12.98 m s⁻¹ and 13.22 m s⁻¹, respectively, resulting in a maximum error of 0.24 m s⁻¹. The Mean Squared Error (MSE) of satellite wind speed is 0.28 m s⁻¹, which is well within the satellite's design specification of 2.0 m s⁻¹. Key data points are summarized in Table 5.

Table 5 Measurement comparison data of Typhoon Maysak (Part)

3.3.3 Confirm the statistical distribution of the buoy data

We generated QQ sub-plots for the buoy during the two typhoons, as depicted in Fig. 16. In this figure, the blue '×' symbols represent matching data points from the two anemometers, and the red line serves as the centerline of the first quadrant of the coordinate system. Data points that align with the red line indicate a correlation coefficient of 1. It is evident from Fig. 16 that when the wind speed is below 11 m s⁻¹, the wind speed datasets from the two anemometers closely align with the red line. When the wind speed exceeds 11 m s⁻¹, the datasets are consistently distributed around the red line. This observation highlights the congruence of the two anemometers in following the same probability distribution, reaffirming the high reliability of the data.

Fig. 16

QQ bitmap of buoy and satellite data during two typhoons

3.4 Analysis and evaluation of uncertainty of measurement comparison results

3.4.1 Uncertainty analysis of measurement results

Since uncertainty is an inherent aspect of any measurement, the metrological comparison of measurement results necessitates an accompanying statement of uncertainty. This statement aims to evaluate the quality and reliability of the measurement results by quantifying the uncertainty introduced by each relevant factor. In our analysis of measurement uncertainty, we adhere to the guidelines provided in the VIM (International Vocabulary of Metrology), which is a collaborative publication by ISO/IEC (ISO, IEC GUIDE 2007). These guidelines help us analyze and quantify the measurement uncertainty associated with our research results.

The steps involved can be summarized as follows. First, we identify the sources of uncertainty and establish a mathematical model. Next, we evaluate the sources of uncertainty by determining the standard uncertainty, which can be categorized as either Class A or Class B evaluation, along with the associated degrees of freedom. We carefully consider whether each source of uncertainty is independent or interrelated, and calculate the combined standard uncertainty and corresponding degrees of freedom. Subsequently, we determine the coverage factor to derive the expanded uncertainty. In the final step, we compile an uncertainty report, summarizing our findings.

3.4.2 Analysis of the standard uncertainty of Typhoon "Jangmi" measurement results

In this paper, our primary focus is on measuring sea wind speed, utilizing both satellite and buoy-based measurement devices. The resulting measurement outcomes are indicated errors in satellite wind speed and buoy wind speed. Since these indicated errors, denoted as $\Delta \text{V}$, cannot be directly measured, we obtain them through a functional relationship between the satellite wind speed variable ${V}_{S}$ and the buoy wind speed variable ${V}_{B}$. This mathematical model is represented in Formula (23).

$$\Delta \text{V}=f\left({V}_{s},{V}_{B}\right)={V}_{S}-{V}_{B}$$

(23)

In practical wind speed measurements, as the absolute true wind speed value remains unknown, we evaluate the uncertainty of the measurement results by selecting estimated values for the satellite wind speed variable and buoy wind speed variable. These estimated values correspond to the measured values obtained from the measuring devices, denoted as ${v}_{Si}$ and ${v}_{Bi}$, respectively. We represent these estimated values as $\Delta {\text{v}}_{i}$. The calculation process is detailed in Formula (24).

$$\Delta {\text{v}}_{i}={v}_{Si}-{v}_{Bi}={ws}_{i}-{wb}_{i}\sqrt{\frac{{\kappa }^{2}}{{C}_{d}}}\frac{1}{ln\left(z/{z}_{0}\right)}$$

(24)

In Formula (24),

$${\text{v}}_{\text{Si}}={\text{ws}}_{\text{i}}$$

(25)

$${\text{v}}_{\text{Bi}}={\text{wb}}_{\text{i}}\sqrt{\frac{{\upkappa }^{2}}{{\text{C}}_{\text{d}}}}\frac{1}{\text{ln}\left(\text{z}/{\text{z}}_{0}\right)}$$

(26)

$\upkappa =0.4, {\text{C}}_{\text{d}}=1.2\times {10}^{-3}, {\text{z}}_{0}=9.7\times {10}^{-5}$ m, and z is the sea level of the anemometer.

According to Formula (24), the propagation law of measurement uncertainty can be obtained, as shown in Formula (27)

$$\begin{aligned}&{u}^{2}\left(\Delta {\text{v}}_{i}\right)={\left(\frac{\partial \Delta {\text{v}}_{i}}{\partial {ws}_{i}}\right)}^{2}\cdot {u}^{2}\left({ws}_{i}\right)+{\left(\frac{\partial \Delta {\text{v}}_{i}}{\partial {wb}_{i}}\right)}^{2}\cdot{u}^{2}\left({wb}_{i}\right)+{\left(\frac{\partial \Delta {\text{v}}_{i}}{\partial \text{z}}\right)}^{2}\cdot {u}^{2}\left(\text{z}\right)\\&-2\left(\frac{\partial \Delta {\text{v}}_{i}}{\partial {ws}_{i}}\right)\left(\frac{\partial \Delta {\text{v}}_{i}}{\partial {wb}_{i}}\right)r\left({ws}_{i},{wb}_{i}\right)u\left({ws}_{i}\right)u\left({wb}_{i}\right)-2\left(\frac{\partial \Delta {\text{v}}_{i}}{\partial {ws}_{i}}\right)\left(\frac{\partial \Delta {\text{v}}_{i}}{\partial \text{z}}\right)r\left({ws}_{i},\text{z}\right)u\left({ws}_{i}\right)u\left(\text{z}\right)\\&+2\left(\frac{\partial \Delta {\text{v}}_{i}}{\partial {wb}_{i}}\right)\left(\frac{\partial \Delta {\text{v}}_{i}}{\partial z}\right)r\left({wb}_{i},z\right)u\left({wb}_{i}\right)u\left(\text{z}\right)={{c}_{1}}^{2}\cdot {u}^{2}\left({ws}_{i}\right)+{{c}_{2}}^{2}\cdot {u}^{2}\left({wb}_{i}\right)+{{c}_{3}}^{2}\cdot {u}^{2}\left(\text{z}\right)\\&-2{c}_{1}{c}_{2}r\left({ws}_{i},{wb}_{i}\right)u\left({ws}_{i}\right)u\left({wb}_{i}\right)-2{c}_{1}{c}_{3}r\left({ws}_{i},\text{z}\right)u\left({ws}_{i}\right)u\left(\text{z}\right)+2{c}_{2}{c}_{3}r\left({wb}_{i},z\right)u\left({wb}_{i}\right)u\left(\text{z}\right)\end{aligned}$$

(27)

In Formula (27), the terms $u\left({ws}_{i}\right)$、$u\left({wb}_{i}\right)$ $\text{and}$ $u\left(\text{z}\right)$ represent the standard uncertainties introduced by the wind measurement results of the satellite, the wind measurement results of the buoy, and the sea-level change of the anemometer. The sensitivity coefficients are defined in Formula (28)-(30).

$${{c}_{1}}={}^{\partial \Delta {{\text{v}}_{i}}}/{}_{\partial w{{s}_{i}}}$$

(28)

$${{c}_{2}}={}^{\partial \Delta {{\text{v}}_{i}}}/{}_{\partial w{{b}_{i}}}=\sqrt{\frac{{{\kappa }^{2}}}{{{C}_{d}}}}\frac{1}{ln\left( z/{{z}_{0}} \right)}$$

(29)

$${{c}_{3}}={}^{\partial \Delta {{\text{v}}_{i}}}/{}_{\partial \text{z}}=w{{b}_{i}}\cdot \sqrt{\frac{{{\kappa }^{2}}}{{{C}_{d}}}}\frac{{{z}_{0}}}{z}=3.7\times {{10}^{-4}}*w{{b}_{i}}$$

(30)

$r\left({ws}_{i},{wb}_{i}\right)$, $r\left({ws}_{i},\text{z}\right)$, $r\left({wb}_{i},z\right)$ are the respectively correlation coefficients ${ws}_{i}$, ${wb}_{i}$ and $\text{z}$.

Since the wind measurement results of the satellite and the buoy are opposite to each other, so it can be obtained in Formula (31)-(32)

$$r\left({ws}_{i},{wb}_{i}\right)=0$$

(31)

$$r\left({ws}_{i},\text{z}\right)=0$$

(32)

By simplifying Formula (27),${u}^{2}\left(\Delta {\text{v}}_{i}\right)$ can be obtained in Formula (33)

$${u}^{2}\left(\Delta {\text{v}}_{i}\right)={u}^{2}\left({ws}_{i}\right)+{{c}_{2}}^{2}\cdot {u}^{2}\left({wb}_{i}\right)+{{c}_{3}}^{2}\cdot {u}^{2}\left(\text{z}\right)+2{c}_{2}{c}_{3}r\left({wb}_{i},z\right)u\left({wb}_{i}\right)u\left(\text{z}\right)$$

(33)

u(wbi) is the standard uncertainty, which is assessed as follows.

The standard for buoy measurements utilizes ocean-specific anemometers from R.M. Young Company in the United States. These anemometers have been meticulously calibrated by meteorological stations and hold certificate numbers GQJ(C)LS2019-0644 and GQJ(C)LS2019-0646. The extended uncertainties for both anemometers are identical, totaling 0.20 m s⁻¹. Through a class B assessment, we calculate the standard uncertainty using Formula (34).

$$u\left( w{{b}_{i}} \right)={}^{{{U}_{\text{RM}}}}/{}_{k}$$

(34)

Considered that the certificate issued by China's National Meteorological Station is reliable, so can get the degree of freedom in Formula (35),

$${\nu }_{{wb}_{i}}=\infty$$

(35)

u(wsi) is the standard uncertainty, which is assessed as follows.

As previously mentioned, satellite data pertains to the information in rows 83–85 and columns 103–105 of the matrix, with each group containing 9 data points. Throughout the history of the typhoons, the density distribution of matched data is depicted in Fig. 17. The red line within Fig. 17 serves as the dividing line between satellite and buoy data, distinguishing positive and negative matches.

Fig. 17

density distribution of satellite and buoy wind speed matching data

As observed in Fig. 4, the data distribution exhibits regional aggregation and complies with the repeatability condition. Consequently, we opt for a Class A evaluation and employ the Bessel formula for calculations, as outlined in Formula (36).

$$u\left({ws}_{i}\right)=\sqrt{\frac{\sum_{i=1}^{n}\left({ws}_{i}-\overline{{ws}_{i}}\right)}{n-1}}$$

(36)

The degrees of freedom are shown in Formula (37)

$${\nu }_{{ws}_{i}}=n-1$$

(37)

u(z) is the standard uncertainty, which is assessed as follows.

The range of variation for sea-level height of the anemometer falls within the interval [0.7, 3.9]. We employ a Class B assessment and assume a uniform distribution. This assessment leads to a half-width of the range, that is ${a}_{z}=\frac{\left(3.9-0.7\right)}{2}m=1.6m$. Substituting this value into Formula (27), we obtain a standard uncertainty of 0.11 m s⁻¹. We calculate the degrees of freedom using Formula (38).

$${\nu }_{z}=\frac{1}{2{\left\{\frac{\sigma \left(u\left(\text{z}\right)\right)}{u\left(\text{z}\right)}\right\}}^{2}}$$

(38)

In Formula (38), $\frac{\sigma \left(u\left(\text{z}\right)\right)}{u\left(\text{z}\right)}$ is the relative standard deviation of $u\left(\text{z}\right)$, which can be determined $\frac{\sigma \left(u\left(\text{z}\right)\right)}{u\left(\text{z}\right)}=0.25$, so the${\nu }_{z}=8$.

At this point, the standard uncertainty analysis has been completed, and the summary is shown in Table 6.

Table 6 Summary of sources of standard uncertainty

In the calculation of synthetic standard uncertainty, through discrete data operation, we can be obtained ${c}_{2}=1.12$, $r\left({wb}_{i},z\right)=0.08$, and can be obtained ${u}_{c}=0.19\text{m}\cdot {\text{s}}^{-1}$ by using Formula (27), we can be obtained ${\nu }_{c}=7$ by using Formula (39)

$${\nu }_{c}=\frac{{u}_{c}^{4}}{\frac{{u\left({ws}_{i}\right)}^{4}}{{\nu }_{{ws}_{i}}}+\frac{{u\left({wb}_{i}\right)}^{4}}{{\nu }_{{wb}_{i}}}+\frac{{u\left(\text{z}\right)}^{4}}{{\nu }_{z}}}$$

(39)

The inclusion factors are determined as follows.

Because the three sources of standard uncertainty belong to the normal distribution and are basically independent from each other, so the extended uncertainty can be calculated by the t distribution, the confidence probability $\text{p}=95\text{\%}$, and the degree of freedom ${\upnu }_{\text{c}}=7$, and the inclusion factor $\text{k}={\text{t}}_{\text{p}}\left({\upnu }_{\text{c}}\right)=2.36$. At this time, the extended uncertainty can be obtained Formula (40).

$${{U}_{Jangmi}=k\cdot u}_{c}$$

(40)

Then, we have given the uncertainty report. Typhoon "Jangmi" measurement comparison measurement results extended uncertainty is in Formula (41)-(42)

$${U}_{Jangmi}=0.39\text{m}\cdot {\text{s}}^{-1}$$

(41)

$${k}_{Jangmi}=2.36.$$

(42)

The buoy data was considered the true value of wind speed, and the analysis of the error in the indicated values of satellite data was conducted from August 4, 2020, to August 14, 2020. The temporal consistency between the two datasets was found to be excellent, as depicted in Fig. 12. The specific focus of the analysis was the period from August 8, 20:00, to August 9, 8:00, represented by the vertical green line in Fig. 12. The maximum allowable error was set at 0.39 m s⁻¹, and the extended uncertainty range is illustrated in Fig. 18.

Fig. 18

Uncertainty analyses of Typhoon "Jangmi" measurement results

3.4.3 Standard uncertainty analysis of Typhoon "Maysak" Measurement results

The uncertainty analysis process of Typhoon "Maysak" is the same as "Jangmi", and the extended uncertainty can be obtained is in Formula (43)-(44).

$${U}_{\text{Maysak}}=0.34\text{m}\cdot {\text{s}}^{-1}$$

(43)

$${k}_{\text{Maysak}}=2.23$$

(44)

Satellite and buoy wind speed data, indication errors, and extended uncertainty are shown in Fig. 19.

Fig. 19

Uncertainty analysis diagram of Typhoon "Maysak" measurement results

In Sects. 3.3.1 and 3.3.2, it was established that the Mean Squared Error (MSE) of the measurement results for Typhoons "Jangmi" and "Maysak" stands at 0.31 m s⁻¹ and 0.28 m s⁻¹, respectively. The extended uncertainties for the two typhoon measurement results are 0.39 m s⁻¹ and 0.34 m s⁻¹. The close proximity of these values indirectly validates the scientific and credible nature of the MSE analysis process, affirming the study's foundation on sound methodology and high-quality results.

4 Summary and conclusion

The Mean Squared Error (MSE) was found to be 0.31 m s⁻¹ during the observation of Typhoon "Jangmi (202005)" and 0.28 m s⁻¹ during the observation of "Maysak (202009)." Both of these values surpass the satellite remote sensing design index of 2.0 m s⁻¹ and also exceed the current common technical index value of 1.6 m s⁻¹. The calculation models employed in this study have the remarkable potential to enhance the accuracy of satellite data by up to 40%, especially in challenging environments such as typhoons.

In this paper, a new simulation method for satellite wind speed data inversion is proposed, and the inversion effect is accurate and reliable. And an analysis was conducted on 108 pairs of buoy and satellite data. Among these, 95 pairs exhibited errors of less than 0.30 m s⁻¹, accounting for 88% of the dataset. This indicates a high level of data consistency. The approach of utilizing buoys to calibrate satellite remote sensing data is deemed acceptable, and the calibration process has the potential to enhance the quality of marine satellite data.

Uncertainty analysis and the evaluation of measurement results have been conducted to provide a quantitative assessment of result quality. A new uncertainty analysis and evaluation method for satellite wind speed measurement is proposed, and the practical results show that it meets the application requirements. The measurement extended uncertainty for "Jangmi" is 0.39 m s⁻¹, while for "Maysak," it stands at 0.34 m s⁻¹. Notably, these values closely align with the Mean Squared Error (MSE), which signifies a rigorous and trustworthy analytical process.

In typhoon-prone regions like the Western Pacific Ocean and the South China Sea, it is possible to strategically select monitoring stations and deploy multiple sets of mooring buoys to establish a real-time typhoon monitoring network. This network can effectively gather in-situ meteorological data, including the formation, evolution, and trajectory of typhoons. Such data are invaluable for conducting in-depth research into the underlying mechanisms of typhoons.

Moreover, the multi-point data obtained from this typhoon monitoring array can be harnessed to enhance the precision of satellite remote sensing data over extensive areas and wide-ranging scales, transforming it into a valuable point-to-area resource for more accurate insights.

The findings of this study fulfill the technical specifications required for typhoon wind speed analysis and exhibit robust applicability to a broad range of scenarios and conditions.

Typhoon is one of the key directions of the comprehensive research of oceanography and meteorology, and the importance of measuring accurate and reliable typhoon data is self-evident to the development of human society. In the next step, we will continue to track and carry out typhoon field calibration studies to obtain more data and strive to obtain more accurate and reliable calibration methods.