Introduction

Continental red beds are a distinctive kind of sedimentary stratum in the continental basin which regards well development of red color as an identification mark (He et al. 2022; Lai et al. 2019; Turner 1981). Continental red beds are not entirely red and often comprise a variety of other colors (Fu et al. 2023; Jiang et al. 2021; Lai et al. 2019; Singh et al. 2021; Turner 1981). A wide range of sedimentary facies develops in the continental red beds (Adardor et al. 2021; Taral et al. 2022; Turner 1981). The color, grain size and structure of continental red beds vary abruptly in time and space (Adardor et al. 2021; Turner 1981). The distinctive features of continental red beds have attracted attention to the origin of continental red beds that generates particular color and sedimentary characteristics. Doubts have always existed between the sedimentary origin and diagenesis origin (Jiang et al. 2015; Lai et al. 2019; Sheldon 2005; Singh et al. 2021, 2009; Turner 1981; Xie et al. 2021). Early research indicates that arid and hot climates are the key to constructing red bed sedimentation (Bai et al. 2020; Gerhard 2000). Arid and semi-arid climate with seasonal rainfall gives rise to extreme variations in the water table and sediment discharge (Dam and Andreasen 1990; Hofmann et al. 2000; Knight and Evans 2017; Rubio Cisneros and Holbrook 2021a; Tunbridge 1981; Zhou et al. 2019). Oxidation–reduction environment and grain size evolve rapidly in time and space. The ephemeral fluvial and shallow water delta develop with frequently alternating sandstone and mudstone, as well as red and gray colors (Dam and Andreasen 1990; Deluca and Eriksson 1989; Hofmann et al. 2000; Horn et al. 2018; Knight and Evans 2017; Mader 1985; Rubio Cisneros and Holbrook 2021b; Taral et al. 2022; Tunbridge 1981; Zhou et al. 2019). In contrast, some researchers doubt the climatic origin of red beds since red bed sedimentation has also been recorded in other different climate settings depending on various conditions (Gerhard 2000; Lai et al. 2019; Sheldon 2005; Singh et al. 2021, 2009).

The original discussion of red bed sediments results in more investigations of red bed sediments in different areas. Varied researches are applied to reveal the origin of red bed sedimentation, including climate reconstruction, mineralogy, petrology, provenance and diagenetic simulation experiment (Aehnelt et al. 2021; Jiang et al. 2015; Mongelli et al. 2006; Sheldon 2005; Singh et al. 2009). However, the actual origin becomes more confusing. Diagenesis, nature of source rocks and drainage conditions represent strong correlations to red bed sedimentation (Aehnelt et al. 2021; Jiang et al. 2015; Sheldon 2005; Singh et al. 2009; Xie et al. 2021). During burial diagenesis, the form of hematite and illite increases the red coloration to construct red bed sedimentation, especially at the high diagenetic temperature (Jiang et al. 2015; Singh et al. 2009). On the other hand, red bed sedimentation may inherit the red coloration from old red bed sedimentation, and the color is preserved with well drainage conditions (Mongelli et al. 2006; Singh et al. 2009). Nevertheless, paleoclimate is still first investigated to exclude its controls on the color of sediments. In addition, since the paleoclimate strongly affects the sedimentary composition, type and distribution of continental red beds, paleoclimate reconstruction is still the key to fully understanding continental red beds.

A series of methods can reconstruct paleoclimate, such as mineralogy, geochemistry, paleobiology and paleomagnetism geophysics (Bai et al. 2020; Onoue et al. 2022; Pang and Yun 2013; Wang et al. 2020; Zhu et al. 2020). Qualitative reconstruction of paleoclimate dominates the research system. Quantitative methods are less reported. Numerous empirical equations have been established to reveal paleoclimate based on the sedimentary records of fine-grained clastic rocks and paleosols (Rasmussen et al. 2011; Zhang and Yang 2021). Paleosols have received more attention as they can indicate the regional paleoclimate. Mean annual temperature (MAT) and mean annual precipitation (MAP) can be calculated based on magnetization rate, element characteristics, Fe content of Fe–Mn nodules and isotopes of carbon, oxygen and hydrogen (Chen et al. 2022; Gillot et al. 2022; Jordanova et al. 2022; Michel et al. 2022; Wang et al. 2023; Zhang and Yang 2021). In contrast, fewer empirical equations have developed based on fine-grained clastic rocks. Only MAT can be calculated based on the index of sodium depletion fraction and CIA (Deng et al. 2022; Zhang and Yang 2021). Moreover, the quantitative reconstructions of fine-grained clastic rocks only reveal the paleoclimate of source areas. Although previous researchers have developed the foundation of paleoclimate reconstructions to a certain extent, actual paleoclimate reconstruction is still in doubt based on the geochemistry as there are varied background and database application conditions. For the database, applying geochemistry in the ancient sedimentary basin is challenging to get paleosols and sufficient clastic samples from cores. For background conditions, the reconstructions of geochemistry will fall into significant errors without correlating influences of structure and geomorphic evolution (slop and relief), sedimentations (sedimentary mixing, recycling and sorting) and diageneses (Deng et al. 2022; Guo et al. 2021; Zhang and Yang 2021).

Compared to the geochemistry methods, such as the chemical weathering index, index of sodium depletion fraction, element characteristics, paleobiology is a promising tool for quantitative paleoclimate reconstruction since paleobotany and paleozoology record the actual paleoclimate without affection of structure, geomorphic evolution, sedimentations and diageneses. Hence, abundant methods have been established to reconstruct the paleoclimate based on paleobiology data, including Detrended Correspondence Analysis, Pollen–Climate Response Surface, Climate Factor Transfer Function, Modern Analogue Technique, Coexistence Approach (CA), etc. (Birks and Seppä, 2004; Da et al. 2023; Dhakal et al. 2022; Dugerdil et al. 2021; Gibson et al. 2022; Klemm et al. 2013; Markgraf et al. 2002; Mosbrugger and Utescher 1997; Mu et al. 2015; Overpeck et al. 1985; Poole et al. 2005; Shen et al. 2006; Utescher et al. 2014). To utilize these methods, a considerable amount of paleobiology data is required, specifically sporopollen data. However, sporopollen fossils are difficult to survive in the red bed sedimentation due to the unique conditions of original vegetation, transportation, burying and water medium. In the continental red beds, quantitative paleoclimate reconstruction remains challenging.

Objectives and geological background

The paper aims to introduce the normal distribution constrained method (NDCM) to reconstruct paleoclimate quantitatively for continental bed sedimentation, which is crucial to reveal the origin of red bed sedimentation. Previous methods always address reconstructing the meaning value of varied climate parameters with large mount data during a certain geological period (Birks and Seppä, 2004; Klemm et al. 2013; Markgraf et al. 2002; Mosbrugger and Utescher 1997; Mu et al. 2015; Shen et al. 2006; Utescher et al. 2014). The insufficient and discontinuity of data improve uncertainties and reduce vertical revolution in the long time-scale paleoclimate analysis. Ephemeral change and fluctuation of paleoclimate are not presented. The ephemeral changes and fluctuations have a certain influence on the color and sedimentary characteristics. The origin of continental red beds has remained a long-standing issue due to ignoring ephemeral changes and fluctuations in paleoclimate analysis. The NDCM focuses on compensating for errors and missing resulting from insufficient sporopollen data in the continental red beds. Ephemeral changes and fluctuations of paleoclimate, which can be inferred from NDCM, also improve the understanding of paleoclimate evolution and help to reveal the origin of continental red bed sedimentation. To show the advantage and applicability of NDCM in insufficient data background, this paper chooses the lower fourth sub-member of Shahejie Formation (Es4x) in the Bonan Sag as the research objective (Fig. 1).

Fig. 1
figure 1

Overview map of the Bonan Sage; a the location of Bohai Bay Basin in China; b the location of Bonan Sag in the Bohai Bay Basin; c the structural top view of the lower fourth sub-member of Shahejie Formation in the Bonan Sag; d the color distribution of mudstone during the lower fourth sub-member of Shahejie Formation in the Bonan Sag

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The Bonan sag, located in the mid-west of the Zhanhua sag of the Jiyang depression, is the largest secondary negative structural unit in Zhanhua sag, a half graben-like depression that has boundaries which are steep in the northwest and gentle to the south-east (Fig. 1c). During the period of Es4x, the continental red beds developed widely in the Bonan Sag (Fig. 1d). The color and sedimentary structure varied in a wide range (Fig. 2). The continental red beds developed red-gray color. The ‘red’ category, which comprised yellow to orange, brown, maroon, deep purple and red, mainly developed in the mudstone. The ‘gray’ category included dark to light gray, greenish-, brownish, bluish-gray, green and dark gray. The gray color developed in both sandstone and mudstone. The dark gray color could only be seen in the sag’s center. The other gray color is distributed sparsely in the widely developed red color. The grain size varied from gravel to mud (Fig. 2). The sorting and rounding also varied in a wide range (Fig. 2). The grain size decreased with the sorting and rounding increased from edge to central in the Bonan Sag (Xu et al. 2021). The varied color and sedimentary structure indicated the development of different sedimentary facies, including alluvial, fluvial, delta and lacustrine facies. The variation in sedimentary characteristics infers the paleoclimate, water discharge and provenance were all in fluctuation background.

Fig. 2
figure 2

Sedimentary characteristics of continental red bed during the lower fourth sub-member of Shahejie Formation; a steel-gray pebble conglomerate with poor sorting and rounding, L10, 2479.73 m; b steel-gray granule-pebble conglomerate in medium sorting and subcircular–subangular shape, L10, 2489.05 m; c gray pebbled coarse sandstone with well sorting and rounding, LX153, 3852 m, d gray fine sandstone with fuzzy trough cross-bedding, K76, 2512.61 m, e brown-gray mudstone, XYS9, 792.87 m; f steel-gray mudstone, XBS1, 3559.28 m; g dark gray mudstone gypsum intercalation, Y186, 4181.44 m; h brown–red mudstone, XYS9, 3613.42 m

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Methodology

The ‘nearest living relative’ (NLR) philosophy provides the fundamental methodology to describe climatic parameters based on fossil sporopollen data (Dhakal et al. 2022; Gibson et al. 2022; Mosbrugger and Utescher 1997; Utescher et al. 2014). The climatic parameters of NLR quantify the climatic requirement of the corresponding fossil taxon with the consistency of climatic requirement between a fossil taxon and its NLR. Revealing possible paleoclimate conditions is viable when the taxon and NLR data have been determined. Potential conditions of paleoclimate are disorderly and variable in the long time-scale geological period. Re-evaluation and data reconstruction are needed to present the concentration and fluctuation of paleoclimate. The content of sporopollen taxon indicates the probability of potential paleoclimate conditions. Since climatic variation follows the normal distribution in a certain period, the paleoclimate reconstruction can be established with constraints of normal distribution and taxon content (Katz and Brown 1992; Meehl et al. 2000; WGI 2021).

Sample data

Key input parameters for the method only need the taxon of sporopollen and the content of sporopollen taxon. Several foundations are also required to provide qualitative background for the method, including qualitative paleoclimate reconstruction, stratigraphic units and NLRs (nearest living relatives). The qualitative reconstruction of paleoclimate establishes a first-order constraint for the quantitative paleoclimate reconstruction. The frequency and range of fluctuation in paleoclimate are more comprehensive and accurate with qualitative and quantitative reconstructions. In addition, the qualitative reconstruction assists in defining internal periods of sequence units. Stratigraphic units and their interval periods provide isochronal constraints for this method. This paper applies a sequence stratigraphic approach to determine stratigraphic units. The other lithostratigraphic or chronostratigraphic framework is also suitable. The NLRs lay the foundation of quantitative parameters of paleoclimate based on the taxon of sporopollen.

Since few wells get complete and continuity records of Es4x in the Bonan Sag, this paper applies comprehensive data to establish fundamental work. Comprehensive indications of logging and sporopollen data in paleoclimate are used to reconstruct the paleoclimate evolution qualitatively. Both logging and seismic data support the determination of sequence stratigraphic framework with the constraint of paleoclimate evolution. Logging data define units in detail. Seismic data establish the macroscale framework and outline boundaries in areas where no logging data can be used. For quantitative paleoclimate reconstruction, only one well (XBS1) of sporopollen data and its NLRs are selected to import the method since insufficient sporopollen data are found with significant discontinuities and low vertical resolution in other wells (Tables 1 and 2).

Table 1 Nearest living relatives and its climatic parameter indicators of each sporopollen taxon in the transgressive system tract stage
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Table 2 Nearest living relatives and its climatic parameter indicators of each sporopollen taxon in the regressive system tract stage
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Workflow

The NDCM can be done by following the steps (Fig. 3).

Fig. 3
figure 3

Workflow of normal distribution constrained method

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Qualitative reconstruction of paleoclimate evolution

The paleoclimate evolution can be inferred from the ratio of feature elements and pollen assemblages. Due to the deficiency of cores and element logging, this paper applies natural gamma-ray spectrometry logging as a replacement. The Th/K and Th/U ratios of mudstone provide continuous paleoclimate information (Fig. 4).

Fig. 4
figure 4

Paleoclimate evolution and sequence architecture division of the lower fourth sub-member of Shahejie Formation from XYS9

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When applying Th/K and Th/U ratios to invert paleoclimate evolution, distinguishing different indications of the varied sedimentary environments is the key to reconstructing actual paleoclimate evolution (Xu et al. 2021). The Th/K ratio is higher in the exposure environment. In contrast, the higher ratio indicates a cold and arid climate in a submerged environment. The dual character also develops in the Th/U ratio. The low Th/U ratio indicates increasing water depth in a submerged environment. In an exposure environment, the low Th/U ratio indicates long-term exposure. Moreover, the increasing Th/U ratio presents enhancing surface runoff (Xu et al. 2021). The previous research suggests that most of the research region is in an exposure environment during the period of the Es4x (Xu et al. 2021). Hence, the high ratios of Th/K and Th/U present a hot-humid climate and strong surface runoff, respectively. The Th/K and Th/U ratios help to establish the continuous paleoclimate evolution and provide a superficial understanding of the paleoclimate fluctuations (Fig. 4).

The paleoclimatic indications from sporopollen fossils help to reduce uncertainty in the paleoclimate inversion from Th/K and Th/U ratios since various factors can give rise to errors (Fig. 5). According to NLRs and its climatic parameters, different paleoclimate indications from sporopollen fossils can be determined (Tables 1, 2 and 3). The NLRs data originate from the Palaeoflora Database and previous related research. Meanwhile, climatic parameters of the NLRs are achieved, including mean annual temperature (MAT), mean temperature of the coldest month (CMT), mean temperature of the warmest month (WMT), mean annual precipitation (MAP), mean precipitation of the wettest month (HMP) and mean precipitation of the driest month (LMP) (Tables 1 and 2).

Fig. 5
figure 5

Paleoclimate evolution and sequence architecture division of the lower fourth sub-member of Shahejie Formation from XBS

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Table 3 Climatic indications of different sporopollen taxon
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Determining paleoclimate indications of sporopollen taxon used the quantitative determination method (Lei et al. 2018; Xu et al. 2021). The content of sporopollen subsequently reveals the temperature and humidity evolution of paleoclimate (Fig. 5). Then, the paleoclimate evolution is reconstructed based on the comprehensive inversion of Th/K and Th/U ratios and constraints of sporopollen content (Figs. 4 and 5).

Establishing sequence stratigraphic framework

The sequence stratigraphic framework is established with a comprehensive analysis of paleoclimate evolution, stratigraphic stacking patterns and seismic reflection termination types (Figs. 4 and 5). The definition of sequence boundaries follows the sequence stratigraphic approach with stratigraphic stacking patterns and seismic reflection termination types. The definition of sequence architecture introduces paleoclimate evolution on the basic analysis of stratigraphic stacking patterns and seismic reflection termination types. The variations in paleoclimate fluctuations and attributes provide a profound sight for identifying the sequence architecture. The details are presented in the published paper by Xu et al. 2021.

Initial reconstruction of paleoclimate distribution model

Reconstructing normal distribution is the key to quantifying paleoclimate since climate varied in the constrain of normal distribution during a certain period. During a geological time scale, paleoclimatic environments varied widely. The widely paleoclimatic environments give rise to the diversity of sporopollen taxon (Tables 1 and 2). There are wide and multiple overlapping intervals developing in the paleoclimate environments (Tables 1 and 2). Based on the developing frequency of sporopollen taxon, the uniform distribution model is applied to present the concentration of overlapping intervals and construct the initial distribution to normal distribution in the maximum extent (Figs. 6a–c, 7a–c, 8a–c, 9a–c).

Fig. 6
figure 6

Initial and normalized distribution model of atmospheric temperature parameters during the transgressive system tract stage. a Initial distribution model of mean annual temperature (MAT ℃); b initial distribution model of mean temperature of the coldest month (CMT ℃); c initial distribution model of mean temperature of the warmest month (WMT ℃); d normalized distribution model of mean annual temperature (MAT ℃); e normalized distribution model of mean temperature of the coldest month (CMT ℃); f normalized distribution model of mean temperature of the warmest month (WMT ℃)

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Fig. 7
figure 7

Initial and normalized distribution model of precipitation parameters during the transgressive system tract stage. a Initial distribution model of mean annual precipitation (MAP mm); b initial distribution model of mean precipitation of the wettest month (HMP mm); c initial distribution model of mean precipitation of the driest month (LMP mm); d normalized distribution model of mean annual precipitation (MAP mm); e normalized distribution model of mean precipitation of the wettest month (HMP mm); f normalized distribution model of mean precipitation of the driest month (LMP mm)

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Fig. 8
figure 8

Initial and normalized distribution model of atmospheric temperature parameters during the regressive system tract stage. a Initial distribution model of mean annual temperature (MAT ℃); b initial distribution model of mean temperature of the coldest month (CMT ℃); c initial distribution model of mean temperature of the warmest month (WMT ℃); d normalized distribution model of mean annual temperature (MAT ℃); e normalized distribution model of mean temperature of the coldest month (CMT ℃); f normalized distribution model of mean temperature of the warmest month (WMT ℃)

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Fig. 9
figure 9

Initial and normalized distribution model of precipitation parameters during the regressive system tract stage. a Initial distribution model of mean annual precipitation (MAP mm); b initial distribution model of mean precipitation of the wettest month (HMP mm); c initial distribution model of mean precipitation of the driest month (LMP mm); d normalized distribution model of mean annual precipitation (MAP mm); e normalized distribution model of mean precipitation of the wettest month (HMP mm); f normalized distribution model of mean precipitation of the driest month (LMP mm)

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Taking the MAT interval of Ulmipollentes sp in the RST as an example, the uniform distribution of the interval is described in detail. First, determine the step length of the interval. The MAT of Ulmipollentes sp varies between − 1.2 °C and 24.3 °C (Tables 1 and 2). The step size of 0.1℃ can cover the potential value and variation in the MAT. In addition, the step size can also be defined based on the high-resolution evolution of paleoclimate. The high-resolution evolution inverts the average magnitude of variation in paleoclimate. For instance, if the 2% of average magnitude in variation is determined from the high-resolution feature element ratio or other paleoclimatic indicators, the step size of 0.51 °C is chosen. Different step sizes result in disparate distribution models. If no available data help to determine step size, the step size of 0.1 °C provides a nearly accurate and complete constraint. In this paper, although the average Th/K and Th/U ratios of mudstone intervals provide information on the magnitude of variation, the resolution is insufficient since some mudstone intervals are widely separated in the vertical direction (Fig. 4). Hence, the step size of 0.1 °C is applied.

Secondly, define the value of the MAT interval based on the step size. 254 temperature values are defined in the constraint of step size. Finally, the MAT interval (67) counter is assigned to every value to reconstruct the uniform distribution. When the workflow above is applied to every interval of the same climatic parameter, the uniform distribution model of the climatic parameter is established (Fig. 7a).

Morphological examination of the initial distribution model

The previous step applies a uniform distribution model to approximate the normal distribution. However, insufficient sporopollen data cause errors due to the limited paleoclimate information from the low sample size. Skew distributions develop in the initial distribution (Figs. 6a–c, 7a–c, 8a–c, 9a–c). Consequently, the morphological examination is applied to determine the necessity of further normalizations. Moreover, different kinds of skew distribution suit disparate normalized methods. If the initial distribution model fits well with the normal distribution, the workflow skips to step 6 (Fig. 3).

Both graphical and parameterization methods can examine the initial distribution. The graphical examination verifies the initial distribution model based on frequency distribution histograms and probability density curves (Figs. 6a–c, 7a–c, 8a–c, 9a–c). In the positive skewness distribution, the highest point of the curve shifts to the left of the X-axis (Figs. 7a–c and 9a–c). The curve of the right half is gentler and has a longer tail line extending infinitely until it approaches the X-axis (Figs. 7a–c and 9a–c). The negative skewness develops opposite characteristics to the positive skewness (Figs. 6a–c and 8a–c). The parameterization examination uses kurtosis (K), skewness (S) and standard errors (SE) to verify the initial distribution model (Table 4). The positive and negative values of S indicate the positive and negative skewness, respectively. The positive and negative values of K present leptokurtic and platykurtic, respectively. In addition, the ratios between K or S and SE indicate the extent of skewness and kurtosis. When 1.96< \(\frac{\left|K\right|}{SE}\) or \(\frac{\left|S\right|}{SE}\) < 2, obviously skewness and kurtosis develop in the initial distribution model. 2 < \(\frac{\left|K\right|}{SE}\) or \(\frac{\left|S\right|}{SE}\) < 3 indicates the medium degree of skewness and kurtosis. \(\frac{\left|K\right|}{SE}\) or \(\frac{\left|S\right|}{SE}\) > 3 marks high degree of skewness and kurtosis.

Table 4 Discriminant parameters of initial and normalized distribution model
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Normalization processing of distribution model

According to the morphological examination of the initial distribution model, different normalized methods are applied to reconstruct the paleoclimate distribution model, which conforms to the normal distribution furthest (Table 4). The square root transformation suits to correct weak-medium degree skewness and kurtosis. The normalization distribution of paleoclimate can be reconstructed using Eq. 1:

$$y^{\prime} = \sqrt y$$
(1)

The logarithmic transformation suits to correct the high degree skewness and kurtosis. The calculation uses Eq. 2:

$$y^{\prime} = \ln y\,or\,y^{\prime} = \lg y$$
(2)

The inverted transformation suits to correct the distribution that fluctuates at both ends of the distribution. The calculation uses Eq. 3:

$$y^{\prime} = 1/y$$
(3)

In Eqs. 1–3, y′ is the normalized data, and y is the initial data in the initial distribution model. Different normalized methods also have application prerequisites. The square root and inverted transformation are only available for the positive skewness. When using the square root transformation, the initial data need to be greater than or equal to 0. Similarly, the inverted transformation requires a dataset that is greater than 0. Hence, a pre-processing of initial data is also needed with a linear or reverse conversion. In the square root transformation, reverse conversion transforms the negative values into satisfactory ones. The negative skewness can convert to positive skewness. The calculation uses Eq. 4:

$$y={x}_{max}-x$$
(4)

where \({x}_{max}\) is the max value in the initial data and \(x\) is each value in the initial data. If only negative values develop in the initial data, the simple linear conversion can fix the data with Eq. 5:

$$y=x+\left|{x}_{min}\right|$$
(5)

where \(\left|{x}_{min}\right|\) is the absolute value of minimum value in the initial data. For the inverted transformation, the negative values, 0 value and negative skewness can be correlated using Eq. 6:

$$y={x}_{max}-x+1$$
(6)

The correlation for only negative and 0 values uses Eq. 7:

$$y=x+\left|{x}_{min}\right|+1$$
(7)

When the normalization process finishes, the initial normalized distribution model needs to be exanimated again to determine the distribution status (Table 4). If the skewness and kurtosis fall off and approach a normal distribution, the next step can conduct with the initial normalized distribution model (Table 4; Figs. 6d–f, 7d–f, 8d–f, 9d–f). In contrast, the initial normalized distribution model needs iteration of the normalization process before approaching normal distribution. Noteworthy, it is challenging to correlate the distribution model with normal distribution completely. The distribution model in slight skewness and kurtosis can provide available paleoclimate information (Table 4; Figs. 6d–f, 7d–f, 8d–f, 9d–f).

Quantitative reconstruction of paleoclimate

The normalized distribution model gives well constrain on quantitative reconstruction. The normalized average value (μ), standard deviation (Std) and confidence interval (CI) of 90% describe the distribution characteristics of the normalized distribution model in different climatic parameters. The calculations of μ and Std have various tools, including Statistical Product and Service Solutions (SPSS), Origin, Python, etc. However, the calculated results of the normalized distribution model only present the normalized parameters instead of the actual parameters. The reverse calculations are needed to restore the normalized parameters to the original parameters. The equations of reverse calculations derive from the equations of normalized methods (Eqs. 1–7). Subsequently, the μ, Std and CI in the original dataset reveal the true paleoclimate (Table 5). It is important to note that the restored Stds do not indicate the degree of dispersion since simple reverse calculations can't restore accurate Std of paleoclimate. Although the Std of the normalized distribution model may be greater or less than the accurate Std, the Std of normalized distribution can evaluate the dispersion evolution tendency of different intervals when the same normalized methods apply to the initial distribution model. Based on the results of different paleoclimate parameters, the potential evapotranspiration rate (PER) is applied to evaluate the arid degree. The PER can be calculated by Eqs. 8–9:

$$PER=PET/P$$
(8)
$$PET=58.93\times BT$$
(9)

where P is the MAP, PET is the possible evaporation amount, BT is the annual average biological temperature, equal to MAT (Table 6) (Holdridge 1947; Mao et al. 2011).

Table 5 Quantitative reconstruction of paleoclimate from normal distribution constrained method and coexistence approach
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Table 6 Evaporative, precipitation, temperature parameters and results in the calculation of potential evapotranspiration rate during the lower fourth sub-member of Shahejie Formation
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Results

Paleoclimate evolution and sequence stratigraphic framework

Both paleoclimate evolution and sequence stratigraphic methods are used to establish a sequence stratigraphic framework. The three-order sequence of Es4x (SQ Es4x) is defined with the routine analysis of sequence stratigraphy, including seismic reflection termination types and stratigraphic stacking patterns (Xu et al. 2021). The abrupt changes of paleoclimate also prove the development of three-order sequence boundaries at the top and bottom of Es4x (Fig. 4). Moreover, the frequency and amplitude evolution of paleoclimate fluctuations identify the sequence architecture in the SQ Es4x (Fig. 4). In the early SQ Es4x, the Th/K and Th/U ratio indicated the increase in humidity. Meanwhile, the frequency and amplitude enhanced in the fluctuations. In the late SQ Es4x, the paleoclimate transformed into a stable humid and cold climate with more hygrophytic sporopollen fossils and fewer fluctuations of Th/K and Th/U ratios (Figs. 4 and 5). The different paleoclimate constructed the transgressive system tract (TST) and regressive system tract (RST) in the early and late SQ Es4x, respectively. The different paleoclimate also resulted in varied responses in sedimentary records presented on the stacking patterns and seismic reflection termination types. More detailed descriptions of paleoclimate evolution and sequence stratigraphy are displayed in the published research (Xu et al. 2021).

Initial distribution model of paleoclimate

Based on the NLRs of sporopollen taxon, the initial distribution models of TST and RST have been established in the constraint of uniform distribution (Figs. 6a–c, 7a–c, 8a–c, 9a–c). The initial distribution models of atmospheric temperature develop negative skewness and platykurtic kurtosis in both the TST and RST stages (Table 4; Figs. 6a–c and 8a–c). The negative skewness and platykurtic kurtosis are in different degrees. The negative skewness is mainly in a medium degree. The degrees of most platykurtic kurtosis were higher than the negative skewness. In the precipitation parameters, different characteristics develop (Table 4; Figs. 7a–c and 9a–c). The distributions were mainly positive and platykurtic. Weak skewness and kurtosis develop excepting the LMP. The distributions of LMP develop medium skewness with weak kurtosis during the TST and RST stage.

Normalized distribution model of paleoclimate

As most distributions are in the weak-medium skewness and kurtosis, normalized processing applies square root transformation (Table 4; Figs. 6a–c, 7a–c, 8a–c, 9a–c). The square root transformation calibrates the initial distribution model to approximate normal distribution with the massive sample size (Table 4; Figs. 6d–f, 7d–f, 8d–f, 9d–f). The same normalization processing of square root transformation also establishes the comparability between different stages. As negative skewness and negative values develop in the atmospheric temperature parameters (Figs. 6a–c and 8a–c), the reverse conversion transforms are applied using Eq. (4) before the normalization processing. The K and S of most parameters decrease (Table 4). That indicates the normalization processing fits the initial distribution approaching normal distribution. However, the \(\frac{\left|S\right|}{SE}\) and \(\frac{\left|K\right|}{SE}\) reached high value in all distributions (Table 4). The massive sample size results in the errors of discriminant parameters. The increase in sample size can raise the SE. When the sample size reaches a certain order of magnitude, the minimal SE results in high discriminant parameters. The high discriminant parameters lose the ability to distinguish the degrees of skewness and kurtosis. In this setting, graphic examinations are applied. The frequency distribution histograms and probability density curves of most distributions approximate normal distribution. Few normalized distributions still develop slight skewness and kurtosis with weak dual modality characteristics (Figs. 7d, f, 9f). It is a challenge to fix the distributions to standard normal distribution completely. Significantly, almost no methods can fix dual or multiple modalities to a single modality in distributions. However, the slight skewness, kurtosis and multiple modality characteristics affect the actual data weakly in the massive sample size. The normalized distribution model revealed the paleoclimate (Figs. 6d–f, 7d–f, 8d–f, 9d–f).

Quantitative characteristics of paleoclimate

The quantitative characteristics are reconstructed with μ and CI, calculated inversely from the normalized distribution model (Table 5). The Stds of the normalized distributions model indicate the dispersion of paleoclimate, namely the amplitude of paleoclimate fluctuations (Table 5).

The atmospheric temperature and precipitation varied from the TST to the RST stage (Table 5). The μ of MAT, CMT and WMT decreased from the TST to the RST stage. The CI of 90% in MAT, CMT and WMT presented lower minimum interval values during the RST stage. In contrast, the maximum interval values of CI fluctuated slightly. The Std increased from the TST and RST stages. For the precipitation, the higher μ of MAP, HMP and LMP developed in the RST stage (Table 5). The 90% CI of HMP and LMP presented the same increasing tendency from the TST to the RST stage. However, the 90% CI of MAP developed a higher minimum interval value and a lower maximum interval value in the RST stage (Table 5). The change range was small in all the precipitation parameters from the TST to the RST stage. The PER was high and decreased slightly from the TST to the RST stage (TST: 6.28, RST: 6.04). The abnormal evolution tendency of various parameters indicated the complexity of the precipitation (Table 5). The fluctuation might give rise to the complexity and uniqueness of the precipitation. The characteristics of fluctuation will be discussed in the discussion part.

Discussion

Comparison between NDCM and CA

The consistence approach provides an operable way to reconstruct paleoclimate quantitatively with certain accuracy (Mosbrugger and Utescher 1997; Poole et al. 2005; Utescher et al. 2014). In order to evaluate the normal distribution constrained method, we also apply the coexistence approach (Table 5; Figs. 10 and 11). The red line of Figs. 10 and 11 marks the coexistence interval of the TST and the RST stage which indicates the mainly variation range of paleoclimate (Figs. 10 and 11). The comparison between the results of NDCM and CA reveals the comprehensiveness and accuracy of NDCM (Table 5; Figs. 10 and 11). For the atmospheric temperature parameters, results from NDCM fall in or approach the reconstructed interval of CA, except the CMT. The CMT intervals are mainly defined by the CMT endmember of Cyathidites and Lycopodiumsporites (Figs. 10 and 11). However, the Cyathidites and Lycopodiumsporites develop rarely (Tables 1 and 2). The CA reconstructed the CMT without the constraint of frequency. Positive deviation develops and leads to the oversized interval of CMT. Compared with the CA, the NDCM takes the frequency of different sporopollen taxon into account. Strong constraints are provided by the well-developed sporopollen taxon, like the Ulmipollenites, Quercoidites, Ephedriptes and Taxodioideae (Tables 1 and 2). As most sporopollens survive in the colder CMT interval, the NDCM restores smaller values than the CA in the CMT. Differing from the atmospheric temperature parameters, the precipitation parameters present significant contrasts between the results of NDCM and CA (Table 5; Figs. 10 and 11). The contrasts were also caused by the frequency of different sporopollen taxa (Tables 1 and 2). The huger order of magnitude expands the contrasts in the precipitation parameters. In addition, the strong constraint of NDCM corrects the CA’s dual interval of LMP into a single interval. Similarly, the NDCM also calibrates the errors of MAT, CMT and WMT in the CA, presenting the higher atmospheric temperature of the RST stage. In summary, the NDCM reconstructs more available and comprehensive results than the CA.

Fig. 10
figure 10

Coexistence analysis of transgressive system tract stage. The sporopollen taxon which is presented on the Y axis in the formation of series number is shown in Table 1. The red line marks the end members of coexistence intervals

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Fig. 11
figure 11

Coexistence analysis of regressive system tract stage. The sporopollen taxon which is presented on the Y axis in the formation of series number is shown in Table 2. The red line marks the end members of coexistence intervals

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Fluctuation and evolution of paleoclimate

Several abnormalities develop between the μ and CI in different paleoclimate parameters (Table 5). A comprehensive analysis of μ, CI, Std and qualitative paleoclimate evolution provides a significant sight to reveal the fluctuation characteristics and understand the abnormalities (Table 5). Most CIs of MAT, CMT and WMT developed lower minimum and stable maximum interval values during the RST stage, indicating decreasing atmospheric temperature in the hot background. Moreover, the abnormalities indicated that the paleoclimate didn't evolve in the linear model from the TST to the RST stage. The fluctuation controlled the evolution of atmospheric temperature. The higher Stds of MAT, WMT, CMT presented stronger amplitude of fluctuations toward the colder climates during the RST stage. The contrary evolution tendency developed in the qualitative paleoclimate evolution reconstructed by the Th/K ratio (Fig. 4). The Th/K curves fluctuated in huge amplitude during the TST stage. The high frequency caused the contrary characteristics of amplitude in the TST stage. As the Th/K curves were in a long time scale, the single fluctuation of the Th/K ratio presented the sum of frequent fluctuations in the short time scale. Without the exhibition of frequency, the amplitudes of Th/K curves were enhanced in the form of a single fluctuation during the TST stage. In the RST stage, more straight sections developed in the Th/K curves. The fluctuation frequency decreased in the atmospheric temperature. The increasing fluctuation amplitude compensated for the decreasing fluctuation frequency. Hence, the fluctuation amplitude of Th/K curves decreased slightly during the RST stage. Moreover, the compensation effects between amplitude and frequency of fluctuations made the μ, Std and CI of MAT, CMT and WMT vary in a small degree (Table 5). On the whole, the atmospheric temperature fluctuated with decreasing frequency and increasing amplitude from the TST to the RST stage.

Precipitation parameters were more sensitive to fluctuations during the TST and RST stage. The same increasing tendency developed in μ, Std and CI of HMP and LMP from the TST to the RST stage (Table 5). The increasing tendency indicated increasing precipitation and fluctuation amplitude in the short time scale. However, the actual precipitation evolution is complex and nonlinear, which can be inferred from the inconsistencies between the monthly (Std and CI of HMP and LMP) and annual parameters (Std and CI of MAP). The Th/U curves fluctuated more frequently in the TST stage (Fig. 4). In the similar effects of atmospheric temperature, the high frequency of short time scale compensated for the low amplitude of long time scale, which could be shown in the scale of Th/U curves (Fig. 4). The compensation effects even raised the Std of the MAP and made the Th/U curves fluctuate in the high amplitude during the TST stage (Table 5 and Fig. 4).

Moreover, different frequencies and amplitudes of the fluctuations made the CI end members of MAT vary in the contrary tendency (Table 5). The increasing amplitude of fluctuations raised the minimum CI value of MAT from the TST to the RST stage. The MAT developed a higher low limit in the RST stage. The low value of MAT was raised in the distribution. The μ of MAT increased responding to the variation in fluctuation amplitude. However, the μ of MAT varied in a very small scale due to the compensation effect of increasing fluctuation frequency. The increasing frequency of fluctuations raised the maximum CI value of MAT in the TST stage. The high upper limit developed and improved the high value of MAT. The mode value of MAT decreased from 1035 to 897 mm from the TST to the RST stage. All the above characteristics give reasons to believe the total precipitation was higher in the TST stage. Although the TST stage developed a higher total precipitation, more arid weather can be inferred from the paleoclimatic indication of sporopollen fossils and PER (Table 6). As humidity is the comprehensive effect of precipitation and atmospheric temperature, the correlation between precipitation and atmospheric temperature gives well understanding of the special characteristics. The higher atmospheric temperature evaporated a large amount of precipitation. The higher total precipitation has no significant improvement effects on the humidity during the TST stage. In contrast, the higher low limit of MAT and decreasing atmospheric temperature improved humidity effectively in the RST stage.

The Es4x developed after the ETM2 and ECCO (Zhou 2018). The continental red beds of Es4x recorded the paleoclimate characteristics of the post-period of the extreme hyperthermal event. They developed significant differences compared to the period of the extreme hyperthermal event. According to the PER > 6, the paleoclimate was still in the hot and arid background during the Es4x (Table 6) (Holdridge 1947; Mao et al. 2011). However, the increasing humidity and the decreasing atmospheric temperature developed in the post-period of the extreme hyperthermal event instead of the long-term stable hot and arid weather in the extreme hyperthermal event. The characteristics of fluctuations distinguished the post-period of extreme hyperthermal events from the period of extreme hyperthermal events. The decreasing frequency developed with the increasing amplitude in the post-period. In contrast, the fluctuations always developed strong amplitude with low frequency during extreme hyperthermal events (Eide et al. 2018; Milliman and Farnsworth 2011).

Origin of red beds in the post-period of extreme hyperthermal events

The Es4x developed both red and gray sediments at the same time (Figs. 1 and 2). The hot and arid background still played a key role in the origin of red sediments. The red mudstone developed abundantly in the vertical direction and covered most research regions in the space. However, other phenomena also indicated the other origin of red sediments. First, the red color mainly developed in the mudstones (Fig. 2). Although the fluvial system constructed most sandstone of the research region in the water saturated environment, some fluvial sandstones might transform into red color when the hot and arid background made the channel exposed in the dry season. The deficiency of red sandstones indicated the other origin of the red color in the post-period of extreme hyperthermal events. Secondly, the gray mudstone didn’t correspond to the high Th/U and Th/K ratios (Figs. 4 and 5). As the high Th/U and Th/K ratios indicated the high precipitation and increasing humidity, the gray mudstone should develop correspondingly. Finally, the red color developed in a disorganized distribution (Fig. 12). If only the paleoclimate controlled the development of red beds, red mudstone should distribute in a relatively high position. Actually, the red mudstone developed in both low and high positions. The gray mudstones also distributed irregularly in the same way (Fig. 12). Combined with the previous research on origins, the differences in drainage conditions and parent rocks perhaps play a certain degree of control in the development of red coloration (Sheldon 2005; Singh et al. 2021, 2009). Different drainage conditions might cause the red color deficiency of sandstones. The disorganized distribution of red mudstone might be affected by local provenance and drainage conditions. However, the confirmation of origins needs further research in provenance and hydrology. As it is not the key concern of this paper, we just propose the possibility of other origins.

Fig. 12
figure 12

Cross-well section of regressive system tract stage. The location is shown in Fig. 1

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Contrary to the origin of red coloration, the paleoclimate significantly controlled the sedimentary characteristics of red beds (Xu et al. 2021). The frequent precipitation fluctuation increased the total precipitation in the TST stage. The stronger water discharge enhanced the total sediment supply of the TST stage. The different enrichment of the sand body developed with the sand body concentration the of TST stage. Moreover, flood events develop frequently with the frequent precipitation fluctuation of the TST stage. Frequent flood events resulted in the diversion of channels. The lateral migration developed in the TST stage (Fig. 5).

Conclusions

The NDCM provides a quantitative way to reconstruct paleoclimate with low data requirement. The quantitative paleoclimate reconstruction helps to understand the paleoclimate characteristics in the post-period of the extreme hyperthermal events. Based on the paleoclimate characteristics, a new perspective is established to evaluate the origin of continental red beds.

  1. 1.

    Compared to the CA, the NDCM uses the development frequency constraints of sporopollen to calibrate errors caused by low sample size and different development frequencies of sporpollen.

  2. 2.

    The hot and arid weather still plays a key role during the Es4x. However, the increasing humidity and decreasing atmospheric temperature indicates the paleoclimate is in the processes of breaking away from the hot and arid background.

  3. 3.

    The paleoclimate fluctuation develops decreasing frequency and increasing amplitude in the post-period of extreme hyperthermal events during the Es4x.

  4. 4.

    In the post-period of extreme hyperthermal events, the arid and hot weather still control the development of continental red beds. The provenance and drainage condition may make a contribution to the origin of the red color.