Statistical Optimization and Analysis on the Spatial Distributions of Ice Ridge Keel in the Northwestern Weddell Sea, Antarctica

Abstract

Statistical optimization methods serve as fundamental tools for studying sea-ice-related phenomena in the polar regions. To comprehensively analyze the spatial distributions of ice ridge keels, including the draft and spacing distributions, a statistical optimization model was developed with the aim of determining the optimal keel cutoff draft, which differentiates ice ridge keels from sea ice bottom roughness. By treating the keel cutoff draft as the identified variable and minimizing the relative errors between the theoretical and measured keel spatial distributions, the developed model aimed to seek the optimal keel cutoff draft and provide a precise method for this differentiation and to explore the impact of the ridging intensity, defined as the ratio of the mean ridge sail height to spacing, on the spatial distributions of the ice ridge keels. The utilized data were obtained from observations of sea ice bottom undulations in the Northwestern Weddell Sea during the winter of 2006; these observations were conducted using helicopter-borne electromagnetic induction (EM-bird). Through rigorous analysis, the optimal keel cutoff draft was determined to be 3.8 m, and this value was subsequently employed to effectively differentiate ridge keels from other roughness features on the sea ice bottom. Then, building upon our previous research that clustered measured profiles into three distinct regimes (Region 1, Region 2, and Region 3, respectively), a detailed statistical analysis was carried out to evaluate the influence of the ridging intensity on the spatial distributions of the ice ridge keels for all three regimes. Notably, the results closely matched the predictions of the statistical optimization model: Wadhams’80 function (a negative exponential function) exhibited an excellent fit with the measured distributions of the keel draft, and a lognormal function proved to effectively describe the keel spacing distributions in all three regimes. Furthermore, it was discovered that the relationship between the mean ridge keel draft and frequency (number of keels per kilometer) could be accurately modeled by a logarithmic function with a correlation coefficient of 0.698, despite considerable data scatter. This study yields several significant results with far-reaching implications. The determination of the optimal keel cutoff draft and the successful modeling of the relationship between the keel draft and frequency represent key achievements. These findings provide a solid theoretical foundation for analyzing the correlations between the morphologies of the sea ice surface and bottom. Such theoretical insights are crucial for improving remote sensing algorithms for ice thickness inversion from satellite elevation data, enhancing the accuracy of sea ice thickness estimations.

1. Introduction

Climate warming has triggered a series of profound environmental changes in the polar regions. The declining polar sea ice and consequent rising sea levels are not only reshaping polar ecosystems and natural environments but also redrawing the geostrategic landscape of the Arctic and Antarctic, which have attracted increasing global attention [1,2,3,4,5,6].

The complex seasonal and interannual variations in sea ice represent defining characteristics of polar seas and are closely intertwined with the global climate system and ecosystems. They significantly modulate the momentum, heat, and mass exchanges among the atmosphere, sea ice, and ocean, contributing to phenomena such as rising sea levels, accelerated climate warming, and more frequent extreme weather events [7,8,9,10,11]. Understanding the morphology and distribution of sea ice is essential for improving parameterization schemes for sea ice thermodynamics and optimizing remote sensing algorithms for ice thickness estimation by combining them with sea ice roughness data, which are the direct thermodynamic outcomes of interactions among the atmosphere, sea ice, and ocean [12,13,14,15].

Ice ridges, characterized by their sails (above-surface protrusions) and keels (below-bottom extensions) and formed by dynamic deformation induced by wind, waves, and ocean currents, as well as the crushing and overlapping of collisions within brash ice at the edges of ice floes, are prominent morphological features of sea ice and key elements in the sea ice structure [14]. A predefined cutoff height or draft is commonly used to distinguish ice ridge sails/keels from other sea ice surface/bottom undulations: features with height/draft values below this cutoff are categorized as non-ridged ice surface/bottom undulations, while those above are identified as ice ridge sails/keels [16,17]. This cutoff parameter is crucial for estimating the volume and thickness of sea ice, understanding the exchanges of heat, energy, and momentum between the atmosphere and ocean, and developing large-scale sea ice thermodynamic models [18,19]. Consequently, the study of ice ridge sail and keel morphologies and distributions has become an increasing area of research [13,18,19,20,21,22,23].

Despite its importance, large-scale, continuous, and high-precision field observations and quantitative analyses of sea ice bottom draft remain scarce, owing to the significant challenges posed by harsh and complex polar environments, the random and intricate spatial distribution of sea ice, and the limitations of observational equipment in terms of adaptability and resolution. The current methods for inferring sea ice bottom morphology and spatial distribution from ice surface data obtained via satellite altimetry (e.g., IceBridge, CryoSat-2, and SMOS) predominantly rely on empirical approaches, which often employ preassumed parameters regarding snow and ice morphology and distribution [14,24,25]. However, unrealistic theoretical assumptions and the omission of key parameters compromise the accuracy of representing real-situation conditions and explaining the variations in polar sea ice morphology and distribution. This limitation hinders the refinement of polar marine climate models.

In contrast, mathematical models and methods offer a powerful alternative. They can effectively describe the complex processes of sea ice freezing and melting, its random spatiotemporal distribution, and the interactions between sea ice and external driving forces such as wind, currents, and waves. Additionally, these mathematical models and methods can generate data that are difficult to obtain through field measurements or laboratory simulations, making them invaluable tools in sea ice research [7,17].

Against this backdrop, utilizing a dataset of sea ice bottom undulations collected in the Northwestern Weddell Sea during the winter of 2006 and building on our previous research on ice ridge sail morphology and classification [17,22], this study aims to determine the optimal keel cutoff draft and analyze the morphology and distribution of ice ridge keels by integrating data analysis, mathematical modeling, statistical optimization, and parameter identification techniques. The ultimate goal is to provide a novel and objective theoretical basis and experimental validation for the inversion of polar sea ice thickness using satellite or other altimetry techniques derived from surface elevation data.

This paper is organized as follows. Section 2 details the geographical region, field conditions, and data collection and processing methods. Section 3 develops a nonlinear-constrained statistical optimization model for determining the optimal keel cutoff draft, where the performance function is defined as the sum of relative errors between measured and theoretical keel spatial distributions, and the cutoff draft serves as the parameter to be identified. Section 4 presents a comprehensive analysis of the influence of the ridging intensity (the ratio of mean ridge sail height to spacing) on the keel spatial distributions, integrating previous clustering results [22], and explores the relationship between keel draft and frequency (number of ridge keels per kilometer), as well as that between the sail height and frequency based on earlier research [17]. Finally, the conclusions are presented in the last section.

2. Investigated Region and Data

2.1. Geographical Region and Field Conditions

From 24 August to 29 October 2006, the Alfred Wegener Institute for Polar and Marine Research carried out the Winter Weddell Outflow Study in the northwest Weddell Sea of Antarctica. During this study, a helicopter-borne electromagnetic induction bird (EM-bird) system was employed to collect data on the sea ice roughness.

The measurement area spanned a latitude ranging from 60° S to 66° S and a longitude ranging from 40° W to 60° W. A total of 17 helicopter flights were conducted, comprehensively covering the diverse ice conditions and types encountered throughout the expedition (Figure 1).

Considering the field conditions and the sea ice formation mechanism [19], the investigated area was partitioned into three distinct sub-regions: the northern, central, and southern sub-regions, respectively.

The northern sub-region, located within the marginal ice zone between approximately 60° S and 62° S, was characterized by relatively weak dynamic forces, which led to the minimal deformation of both the sea ice surface and bottom. Consequently, ice ridge sails and keels were scarce and smaller, mainly due to limited ice floe overlap. In fact, some ice bottom profiles in this sub-region showed the complete absence of ice ridge keels.

In contrast, the central sub-region, ranging from around 62° S to 63.5° S, experienced a significant increase in sea ice thickness. This growth was driven by the dynamic interactions between first- and second-year ices. Ice that endured the summer underwent further freezing and development during the subsequent winter, giving rise to a greater abundance of larger ice ridge sails and keels.

The southern sub-region was predominantly dominated by first-year ice formed in the Larsen Polynya. Environmental forces induced icebergs and glacial ice with distinct velocities, generating a remarkable diversity of ice ridge sails and keels. The outflowing branch of the Weddell Gyre, notable for its intense ice deformation processes, featured the most prominent and densely packed ridges along the shelf ice edge. Meanwhile, the Larsen Polynya exhibited an almost complete absence of ridge keels on the ice bottom and a low ridge frequency on the ice surface [17]. Here, ice ridges were smaller in size and fewer in number, and were primarily formed by broken level ice.

2.2. Data Collection and Processing

A total distance of 2988.5 km was surveyed, comprising 94 profiles of sea ice surface and bottom undulations. These profiles varied in length, ranging from 6.3 to 56.8 km.

The EM-bird system used in the survey was trailed 20 m below the helicopter and operated at an altitude of 10–20 m above the sea ice surface [17] (Figure 2). The Riegl LD90 laser altimeter, mounted on the nose of the EM-bird, was responsible for randomly measuring sea ice surface elevations. This altimeter achieved an accuracy of 2.5 cm, emitting infrared pulses at a wavelength of 905 nm with a sampling frequency of 100 Hz.

Regarding the EM-bird itself, it was 3.5 m in length, 0.35 m in diameter, and weighed 105 kg. It collected sea ice bottom undulation data at a sampling frequency of 10 Hz. The system offered an accuracy of more than 0.1 m when measuring level ice; however, this accuracy diminished when measuring rough or deformed ice bottoms.

The helicopter maintained a flight speed of approximately 40 m/s. This speed resulted in distinct spatial sampling distances for the measurement instruments: the laser altimeter had a sampling distance of 0.3–0.4 m between adjacent data points, while the EM-bird’s sampling distance ranged from 3 m to 4 m.

A positioning system (GPS) fixed on the helicopter deck recorded the altitude of the helicopter relative to a reference point at intervals of 0.1 s. Meanwhile, an onboard computer facilitated the real-time storage and processing of all collected data, ensuring efficient data management during the survey.

The accuracy of sea ice surface and bottom undulation measurements is highly susceptible to variations in the flight altitude and speed of the helicopter. To mitigate the influence of the helicopter’s motion state on the elevation data, a three-step automatic filtering method, as described in [17], is indispensable. This method combines three key data sources: the flight altitude of the helicopter relative to the water line (h), the distance from the helicopter to the sea ice surface measured by the laser altimeter (h_s), and the distance from the helicopter to the ice bottom measured by the EM-bird (h_b).

For corresponding measurement points in the two datasets, the sea ice surface elevation and sea ice bottom draft can be accurately calculated as h − h_s and h_b − h, respectively. And the corresponding morphologies of the sea ice profile are illustrated in Figure 3, by assuming the ice ridge keel cross-sections as isosceles triangles.

3. Optimal Model of the Cutoff Draft

The optimal identification of the keel cutoff draft is widely recognized as a pivotal parameter for accurately distinguishing ridge keels from other ice bottom undulations. This parameter significantly influences the correlation between sea ice surface and bottom characteristics, as well as the effective estimation of ice–water dynamic interactions, mirroring the critical impact of the cutoff height on ice ridge sails [17]. Therefore, determining the keel cutoff draft represents a fundamental step in analyzing ridge keel morphology and understanding the relationship between sea ice surface and bottom features.

Traditionally, the keel cutoff draft has been determined by integrating the Rayleigh criterion [26] with field observation experience [27,28]. Nevertheless, owing to the absence of a defined upper limit for the threshold value, the keel cutoff draft selected by the Rayleigh criterion depended generally on the considerate data set and the researcher’s subjective, empirical judgment. An excessively large cutoff value would tend to overlook some smaller ridge keels. These smaller keels, although less prominent, can still play a significant role in the overall dynamics of sea ice, including its interaction with ocean currents, heat exchange, and the mechanical properties of the ice pack. Moreover, the lack of an upper bound constraint for the Rayleigh criterion is associated with an arbitrary cutoff parameter and inconsistent results in different previous studies, undermining the reliability and comparability of studies related to sea ice morphology, highlighting the need for a more refined and data-driven approach to determining the keel cutoff draft.

In this section, we will develop a nonlinear statistical optimization model. Subsequently, the optimal keel cutoff draft obtained from this model will be used to objectively differentiate local undulations and ridge keels on the sea ice bottom.

3.1. Classical Theoretical Distributions of the Keel Draft and Spacing

Let f_d = f_d (d; d_c−k, Θ) and f_s = f_s (s; d_c−k, Ψ) denote the theoretical probability density functions for the keel draft and spacing, respectively. Both functions are Lipschitz continuous. Here, d∈D: = [d_c−k, d_max] denotes the ridge keel draft, where d_max∈R⁺ (R⁺ is the set of positive real numbers) is the maximum keel draft and d_c−k∈U(d_c−k) is the keel cutoff draft, with U(d_c−k) being the admissible parameter set for this parameter. Additionally, s∈S: = [s_min, s_max] denotes the keel spacing, with s_min and s_max being the minimum and maximum spacings, respectively, both s_min and s_max belonging to R⁺ and satisfying s_min < s_max, Θ, and Ψ representing the parameter sets closely associated with the cutoff draft.

Two classical theoretical functions have been proposed to characterize the ridge keel draft distribution, one of which was proposed by Hibler et al. [29]

$$f_d(d;d_{c-k},{\lambda }_1)={\displaystyle \frac{2\sqrt{{\lambda }_1/\pi }\exp (-{\lambda }_1{d}^2)}{erfc(d_{c-k}\sqrt{{\lambda }_1})}},\hspace{1em}d>d_{c-k},$$

(1)

where $erfc(x)={\displaystyle \frac{2}{\pi }}{\displaystyle {\int }_x^{+\infty }{e}^{-t^2}dt}$ and λ₁ is a relevant parameter, with the mean keel draft <d> determined by the following:

$$<d>={\displaystyle \frac{\exp (-{\lambda }_1{d_{c-k}}^2)}{\sqrt{{\lambda }_1\pi }erfc(d_{c-k}\sqrt{{\lambda }_1})}}.$$

(2)

Another theoretical function was proposed by Wadhams [30], who showed that the following negative exponential function can accurately model the distribution of keel drafts and proved it to be a powerful tool for characterizing the statistical behavior of sea ice keel drafts.

$$f_d(d;d_{c-k},{\lambda }_2)={\lambda }_2\exp (-{\lambda }_2(d-d_{c-k})),\hspace{1em}d>d_{c-k}$$

(3)

where λ₂ is a relevant parameter with a mean keel draft <d> determined by the following:

$$<d>=d_{c-k}+{{\lambda }_2}^{-1}$$

(4)

In subsequent sections, Equation (1) will be referred to as the Hiblers’72 function, and Equation (3) as the Wadhams’80 function for simplicity.

Building on the assumption of spatial independence, Hibler et al. [29] employed a homogeneous Poisson process to model the occurrence of ridge keels and derived a negative exponential distribution density function that effectively describes the distribution of ridge keel spacing.

$$f_s(s;d_{c-k},{\lambda }_3)={\lambda }_3\exp (-{\lambda }_{3}s),\hspace{1em}\hspace{1em}d>d_{c-k},$$

(5)

where λ₃ is the distribution shape parameter related to the mean spacing <s> by the following:

$${\lambda }_3=<s{>}^{-1}.$$

(6)

Wadhams and Davy [31] showed that a lognormal function provides a superior fit for the distribution of ridge keel spacing compared to other models.

$$f_s(s;d_{c-k},\theta ,\mu ,\sigma )={\displaystyle \frac{1}{\sqrt{2\pi }\sigma (s-\theta )}}\cdot \exp \left(-{\displaystyle \frac{{\left(\ln (s-\theta )-\mu \right)}^2}{2{\sigma }^2}}\right) s>\theta , d>d_{c-k},$$

(7)

where θ is the shift parameter, and μ and σ are the mean and standard error of ln(s–θ), respectively. They are all related to the mean spacing <s> by the following:

$$<s>=\theta +\exp (\mu +{\displaystyle \frac{{\sigma }^2}{2}}).$$

(8)

3.2. Statistical Optimization Model of the Cutoff Draft

As the following two equations illustrate, let $E_d(d_{c-k})$ and $E_s(d_{c-k})$ represent the relative errors, respectively, between the measured ridge keel draft distribution and its corresponding theoretical function, and between the measured ridge keel spacing distribution and its associated theoretical function.

$$E_d(d_{c-k})={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n\left|f_d(d_i,d_{c-k};{\lambda }_2)-f_{d-i}\right|}}{{\displaystyle {\sum }_{i=1}^n\left|f_{d-i}\right|}}},\hspace{1em}{d}_i>d_{c-k},$$

(9)

$$E_s(d_{c-k})={\displaystyle \frac{{\displaystyle {\sum }_{j=1}^m\left|f_s(s_j,d_{c-k};\theta ,\mu ,\sigma )-f_{s-j}\right|}}{{\displaystyle {\sum }_{j=1}^m\left|f_{s-j}\right|}}},$$

(10)

where d_i is the measured ridge keel draft, s_j is the measured ridge keel spacing, f_d−i is the probability density of the measured ridge keel draft, f_s−j is the probability density of the measured ridge keel spacing, and i = 1, 2, …, n, j = 1, 2, …, m, for the given cutoff draft d_c−k∈U(·).

The admissible parameter set for the keel cutoff draft U(·) = {d_c−k|3.5 m ≤ d_c−k ≤ 4.2 m} is limited by two critical constraints: the Rayleigh criterion and the necessity of adequately encompassing the measured keel drafts. Mathematically, this set possesses distinct properties. Specifically, U(·) is nonempty, bounded, closed, and convex, indicating the existence of valid limited parameter values within it.

To find out the best fits for the ridge keel spatial distributions, set d_c−k,0 = 3.5 m as the initial cutoff draft and Δd_c−k = 0.1 m as the step increment. The analytical results indicate that, for any keel cutoff draft within the admissible parameter set U(·), the Wadhams’80 function provided the optimal fit for the observed keel draft distribution. In contrast, the lognormal function proved to best fit the measured keel spacing distribution. This suggests that these two respective functions effectively capture the statistical characteristics of keel drafts and spacings across the admissible range of the cutoff draft parameter.

We set

$$J_{d-s}(d_{c-k})=E_{d-w}(d_{c-k})+E_{s-l}(d_{c-k}),$$

(11)

$$E_{d-w}(d_{c-k})={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n\left|f_{d-w}(d_i,d_{c-k};{\lambda }_2)-f_{d-i}\right|}}{{\displaystyle {\sum }_{i=1}^n\left|f_{d-i}\right|}}},\hspace{1em}{d}_i>d_{c-k},$$

(12)

$$E_{s-l}(d_{c-k})={\displaystyle \frac{{\displaystyle {\sum }_{j=1}^m\left|f_{s-l}(s_j,d_{c-k};\theta ,\mu ,\sigma )-f_{s-j}\right|}}{{\displaystyle {\sum }_{j=1}^m\left|f_{s-j}\right|}}},$$

(13)

where J_d−s(d_c−k) denotes the performance function, E_d−w(d_c−k) is the relative error between the measured keel draft distribution and Wadhams’80 function, E_s−l(d_c−k) is the relative error between the measured keel spacing distribution and lognormal distribution, f_d−w(d_i, d_c−k; λ₂) denotes the probability density function of the Wadhams’80 function for the keel draft (Equation (3), i = 1, 2, …, n), and f_s−l(s_j, d_c−k; θ, μ, σ) is the lognormal distribution for keel spacing (Equation (7), j = 1, 2, …, m), respectively.

A statistical optimization model of the cutoff draft is then established as follows:

$$\left\{\begin{array}{l}\min J_{d-s}(d_{c-k})\\ s.t. \hspace{1em}{f}_{d-w}(d_i,d_{c-k};{\lambda }_2)\in M(U_{ad}(d_{c-k})),\hspace{1em}i=1,\cdots ,n\\ \hspace{1em} \hspace{1em} f_{s-l}(s_j,d_{c-k};\theta ,\mu ,\sigma )\in N(U_{ad}(d_{c-k})),\hspace{1em} j=1,\cdots ,m\\ \hspace{1em} \hspace{1em} d_{c-k}\in U_{ad}(d_{c-k})\end{array}\right.,$$

(14)

where

M(U(d_c−k)) = {f_d−w (d_i, d_c−k; λ₂)|f_d−w(d_i, d_c−k; λ₂) is the corresponding value of Wadhams’80 probability density function at the keel cutoff draft d_c−k∈U(d_c−k) (refer to Equation (3)), i = 1, 2, …, n}.

N(U(d_c−k)) = {f_s−l (s_j, d_c−k; θ, μ, σ)|f_s−l(s_j, d_c−k; θ, μ, σ) is the corresponding value of the lognormal probability density function at the keel cutoff draft d_c−k∈U(d_c−k) (refer to Equation (7)), j = 1, 2, …, m}.

Figure 4 shows how the performance function J_d−s(d_c−k), the relative errors E_d−w(d_c−k) and E_s−l(d_c−k) vary when the keel cutoff draft increases. Evidently, when the cutoff draft attains a value of 3.8 m, E_d−w(d_c−k), E_s−l(d_c−k), and J_d−s(d_c−k) each reach their respective minimum values, so the value of 3.8 m is judiciously selected as the optimal keel cutoff draft. Consequently, the statistical optimization model (Equation (14)) offers a precise and data-driven criterion for distinguishing ridge keels from other ice bottom features.

4. Results

4.1. Keel Draft and Spacing Distributions in Different Regimes

Considering the influences of geographical location and growth environment, Tan et al. [22] developed an intelligent algorithm that employed ridging intensity R_i as the clustering index to cluster the measured profiles into three distinct regimes based on an increasing ridging intensity, labeled as Region 1 (R_i ≤ 0.01), Region 2 (0.01 < R_i ≤ 0.026), and Region 3 (R_i > 0.026), respectively. The results showed that Region 1 corresponds to the northern and part of the southern sub-regions, Region 2 to the remaining southern sub-region, and Region 3 to the central sub-region.

The clustering results from reference [22] are fundamental for all subsequent statistical analyses and discussions, which aim to explore how the ridging intensity affects the ridge keel spatial distributions, thereby providing a crucial framework for understanding the complex relationships between sea ice morphological characteristics and environmental factors.

4.1.1. Distribution of Keel Draft

To thoroughly compare the observed and theoretical distributions of ridge keel draft, Figure 5 illustrates the probability density functions (PDFs) of keel draft for the three regions (Region 1, Region 2, and Region 3). In Region 1 (Figure 5a), the data exhibit only slightly perceptible scatter, while Regions 2 (Figure 5b) and 3 (Figure 5c) show minor scatter at the higher ends. Notably, Wadhams’80 function demonstrates a strong fit with the measured keel draft distribution for each of the three regions.

In contrast, for all three regions, the Hiblers’72 function systematically overestimates intermediate keel draft values while consistently underestimating the observed data at both the lower and higher tails. Although the deviations at the high tails gradually decrease from Region 1 to Region 3, Figure 5 clearly highlights the poor fits between the Hiblers’72 function and the observed keel draft distributions for all regions.

To compare the fitted results of Hiblers’72 and Wadhams’80 functions to the keel draft distributions for the three regions, linear regression analysis was performed between the theoretical and the real measured values. Simultaneously, to quantify the deviation between the theoretical and measured keel draft distributions, we define the following mean absolute error MAED:

$$MAED={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n\left|f_d(d_i,d_{c-k};\Theta )-f_{d-i}\right|}}{n}},\hspace{1em}{d}_i>d_{c-k},$$

(15)

where the variables and functions are defined in accordance with Equation (9).

The analysis results are presented in

Table 1. Linear correlation coefficients and MAED values between the theoretical and measured keel draft distributions.

Cluster	Hiblers’72 Distribution		Wadhams’80 Distribution
	r	MAED	r	MAED
Region 1	0.872	0.258	0.908	0.194
Region 2	0.976	0.089	0.984	0.076
Region 3	0.970	0.085	0.973	0.070

. For Region 2 (moderate ridging intensity), Wadhams’80 function exhibits the strongest correlation with the measured keel draft distribution, with a linear correlation coefficient of 0.984 and a MAED value of 0.076. Both Region 1 and Region 3 show linear correlation coefficients exceeding 0.900; however, Region 1 has a MAED value of 0.194—an order of magnitude higher than other two, indicating a distinct error. The corresponding Hiblers’72 function demonstrates similar trends in the three regions but with consistently smaller linear correlation coefficients and larger MAED values than those of Wadhams’80 function. These results further confirm that Wadhams’80 function provides a better fit to the observed keel draft distribution compared to Hiblers’72 for each of the three regions.

4.1.2. Distribution of Keel Spacing

To compare the theoretical distributions with the observed keel spacing data, Figure 6 presents the probability density functions (PDFs) of keel spacing for the three regimes. Although there is a slight overestimation at the lower tails of the observed data, the lognormal function effectively describes the keel spacing distribution in each region. In contrast, the exponential function systematically overestimates moderate ridge spacings while underestimating observed keel spacings at both the lower and higher tails in all regions. Notably, the deviation of the measured data tails decreases gradually from Region 1 to Region 3, aligning with a reduction in the range of overestimation.

As shown in

Table 2. Linear correlation coefficients and MAES values between the theoretical and measured keel spacing distributions.

Cluster	Exponential Distribution		Lognormal Distribution
	r	MAES	r	MAES
Region 1	0.723	0.472	0.866	0.279
Region 2	0.904	0.613	0.968	0.183
Region 3	0.888	0.875	0.964	0.196

, which lists the correlation coefficients between the two theoretical functions and the measured data, the linear correlations between the lognormal function and the observed keel spacing distributions are significantly stronger than those of the exponential function for all regions.

Similarly, to quantify the deviation between the measured keel spacing distribution and theoretical function, we conducted a linear regression analysis, and MAES represents the mean absolute error between the measured keel spacing distribution and theoretical function, defined as follows:

$$MAES={\displaystyle \frac{{\displaystyle {\sum }_{j=1}^m\left|f_s(s_j,d_{c-k};\Psi )-f_{s-j}\right|}}{m}}$$

(16)

where the variables and functions are defined in accordance with Equation (10).

The analysis results are shown in Table 2. In Region 1, where ice ridging intensities are the lowest, the lognormal function exhibits the weakest correlation with the measured keel spacing distributions, with a linear correlation coefficient of only 0.866 and a maximum MAES of 0.279. In contrast, Regions 2 and 3 show stronger correlations: the linear correlation coefficients exceed 0.960 and MAES values decrease to below 0.200. Notably, Region 2 (representing moderate ice ridging intensities) achieves the largest linear correlation coefficient (0.968) and the lowest MAES (0.183).

For the exponential function, the results show lower linear correlations and significantly larger MAES values than those of the lognormal function. Additionally, the trend in MAES variation for the exponential function contradicts the correlation coefficient for all three regions. These findings indicate that the lognormal function is more effective for predicting keel spacing distributions, strongly supporting the results presented in Figure 6.

4.2. Correlation Between the Draft and Frequency of Ridge Keel

To comprehensively characterize the morphological and spatial distribution of ice ridge keels, this study analyzes their horizontal and vertical features, quantified by the keel frequency (number of keels per kilometer) and keel draft, respectively. Figure 7a illustrates the relationship between the mean keel draft and frequency, with data in the investigated area considered as a whole.

Notably, despite higher data dispersion compared to the relationship between the mean sail height and frequency (Figure 7b), a significant logarithmic correlation (correlation coefficient r = 0.698) exists between the keel draft and frequency. This correlation reflects the complex, irregular nature of the keel structure and distribution, which exhibit greater morphological variability than the ridge sails. Logarithmic regression models further confirm that both the mean keel draft and sail height increase logarithmically with frequency. However, the increment ratios of the keel draft and sail height to the corresponding frequencies decrease as the frequency increases, indicating the non-linear evolutionary patterns of the ice ridge.

An analysis of the observed data in Figure 7 reveals a region-dependent response: for a given increase in the ridge frequency, there are more pronounced increases in both draft and spacing in the lower ridging intensity regions compared to the higher ridging intensity regions. The logarithmic regression further highlights that the mean keel draft exhibits substantially less variability than the mean keel spacing. These consistent findings validate the use of logarithmic relationships as a robust mathematical framework for modeling the morphological evolution of ice ridge keels.

5. Conclusions

During the 2006 Winter Weddell Outflow Study in the northwest Weddell Sea, Antarctica, a helicopter-borne EM-bird system was utilized to comprehensively measure sea ice bottom roughness, effectively covering the diverse ice types and regimes within the investigated region. A total of 94 sea ice bottom roughness profiles were collected, with lengths ranging from 6.3 to 56.8 km, aggregating to a cumulative length of 2988.5 km. This study conducted a detailed analysis of the field environment, the underlying formation mechanisms, and the distinct characteristics of ice ridges in the three different investigated sub-regions: the northern, central, and southern sub-regions, respectively. By processing the sea ice bottom roughness data, the distributions of ridge keel draft and spacing, as well as the correlation between keel draft and frequency, were systematically analyzed by statistical methods. The key findings are summarized as follows:

• A statistical optimization model with nonlinear constraints was developed to accurately determine the optimal keel cutoff draft by minimizing the relative errors between the theoretical and measured ridge keel spatial distributions. The analysis revealed that, irrespective of the cutoff draft value, Wadhams’80 function (a negative exponential function) consistently provided the best fit for the measured keel draft distribution. Conversely, a lognormal function effectively characterized the corresponding keel spacing distribution. Furthermore, an optimal cutoff draft value of 3.8 m was obtained by combining the statistical optimization model, serving as a definitive discriminant threshold for differentiating ridge keels from other sea ice bottom roughness features in the study region.
• Under the optimal cutoff draft of 3.8 m, the influences of ridging intensity on the ridge special distributions were systematically analyzed based on the clustering results of our previous research [22], which clustered the measured profiles into three distinct regimes according to the increasing ridging intensity: Region 1 (R_i ≤ 0.01), Region 2 (0.01 < R_i ≤ 0.026), and Region 3 (R_i > 0.026), respectively. For any region, Wadhams’80 function and the lognormal function demonstrated the best fit for keel draft and spacing distributions, respectively, achieving larger linear correlation coefficients and smaller mean absolute errors (MAED for keel draft, and MAES for keel spacing). In contrast, the Hiblers’72 model and exponential function exhibited significant deviations from the observed data. Notably, Wadhams’80 and lognormal functions are applicable across a broader spectrum of the ice ridging intensity, enabling more accurate predictions of keel draft and spacing distributions in complex, variable sea ice environments.
• A logarithmic regression analysis, considering both the vertical and horizontal characteristics of the ridge keels, demonstrated a significant logarithmic increase in the mean keel draft with frequency. This relationship was statistically validated by a correlation coefficient of r = 0.698 at a 95% confidence level. Notably, while the increment ratios of keel draft to frequency exhibited a decreasing trend with increasing keel frequency, the mean keel draft generally increased, consistent with the measured distributions of keel morphology and spacing.