Composite high-temperature drought index construction method based on AI technology

Abstract

The invention belongs to the technical field of disaster remote sensing monitoring, and relates to an AI technology-based composite high-temperature drought index construction method, which comprises the following steps of: collecting multi-source data, preprocessing the data, and constructing a high-temperature drought factor and an evaluation index; then, a drought monitoring model and a high-temperature monitoring model are respectively constructed by using multiple AI algorithms including machine learning and deep learning, finally, a composite high-temperature drought index is constructed, the constructed composite high-temperature drought index is used for monitoring and evaluating a composite high-temperature drought event, multi-source data are fused, and the accuracy of the composite high-temperature drought event is improved. According to the method, factors influencing high temperature and drought disasters are comprehensively considered, so that the recognition precision of the composite high temperature and drought disasters is remarkably improved, the AI technology is applied, internal association between data is fully explored, and the method has higher generalization ability, can be suitable for different climate regions and time scales, has good expansibility and is suitable for popularization and application. The method can be popularized and applied to construction of other composite disaster indexes, and provides support for monitoring and prediction of various extreme climate events.

Description

Technical field:

[0001] The present invention belongs to the technical field of disaster remote sensing monitoring, and specifically is a method for constructing a composite high temperature drought index based on AI technology. The constructed index can be used to monitor and evaluate composite high temperature drought events. Background technology:

[0002] Against the backdrop of global warming, the frequency, duration, and spatial range of extreme events such as heat and drought have increased. These two extreme events can have serious impacts on agricultural production and the ecological environment, including reduced agricultural production, vegetation degradation, water shortages, and reduced biodiversity, which in turn endanger food security and socioeconomic stability, and pose a serious threat to human health. When these two extreme climate events, heat and drought, occur simultaneously or subsequently, a compound heat and drought event will form. Since the impacts of the two are superimposed, the destructiveness of this compound event will be far greater than that of a single heat or drought event. In recent years, compound heat and drought events have occurred frequently in many regions around the world, causing significant socioeconomic losses. Therefore, it is particularly urgent to effectively monitor and evaluate compound heat and drought events.

[0003] The monitoring and evaluation of disaster events usually rely on scientific and reasonable indices, which can accurately reflect the degree of disaster occurrence and its spatiotemporal variation characteristics. However, for complex disaster events, it is difficult to fully reveal its complexity by relying on a single variable, so it is necessary to integrate multiple related variables. At present, the commonly used composite high temperature drought index is usually constructed by combining drought and high temperature indicators, and the calculation of these two indicators is mainly based on meteorological interpolation data or reanalysis data. However, the spatial resolution of such data is usually above 0.1°, which limits the spatial resolution of the composite high temperature drought index. In regional-scale monitoring, this rough resolution makes it difficult to capture subtle differences in geographical and environmental characteristics, thereby affecting the accuracy of the monitoring results. To solve this problem, the present invention starts with improving the resolution of high temperature and drought indicators, constructs high temperature and drought indicators with high spatial resolution, and constructs a composite high temperature drought index based on this, thereby providing an effective indicator for accurate monitoring at a regional scale. Summary of the invention:

[0004] The purpose of this invention is to solve the problems of insufficient monitoring range and accuracy, and to design a method for constructing a composite high temperature and drought index based on AI technology to better meet the needs of practical applications. The composite high temperature and drought index constructed by this method can effectively characterize the occurrence and intensity of composite high temperature and drought events, thereby providing a reliable scientific basis for the quantitative analysis and monitoring of composite extreme climate events.

[0005] In order to achieve the above-mentioned purpose, the method for constructing a composite high temperature drought index based on AI technology according to the present invention specifically comprises the following steps:

[0006] S1. Data collection:

[0007] Collect multi-source data, including ground meteorological data, remote sensing data and reanalysis data;

[0008] S2. Data preprocessing:

[0009] The small amount of missing data in the ground meteorological data was interpolated, the storage format and coordinate system of remote sensing data and reanalysis data were unified, the spatial resolution was adjusted to 1000 m using bilinear resampling, and the temporal resolution was synthesized to a monthly scale, and finally the data were cropped to the study area;

[0010] S3. High temperature and drought factors and evaluation index construction:

[0011] The key factors of precipitation, soil moisture, surface temperature, vegetation conditions, surface water conditions, air dryness, evapotranspiration, temperature and altitude are comprehensively considered. Among them, the altitude and 2m height temperature data can be directly used as independent variables in the monitoring model training data set, and other meteorological and remote sensing data are included in the data set as independent variables after calculating the relevant indexes; at the same time, the standardized precipitation evapotranspiration index (SPEI) and standardized temperature index (STI) are calculated based on ground meteorological data, which are used to measure drought and high temperature respectively, as dependent variables in the training data set;

[0012] S4. Construction of high temperature and drought monitoring model:

[0013] We used a variety of AI algorithms including machine learning and deep learning to build drought monitoring models and high temperature monitoring models, respectively, and used the determination coefficient R 2 The model fitting accuracy is evaluated by using the root mean square error (RMSE) and the best performance model to output the SPEI index data set and the STI index data set. 2 The formula for RMSE is: Where n is the number of samples; y i is the true value; is the model prediction value; is the average of all true values.

[0014] S5. Construction of composite high temperature drought index:

[0015] The joint distribution method based on Copula function was used to construct the joint distribution of SPEI and STI variables, and the standardized compound dry and hot index (SCDHI) was established. The constructed compound dry and hot index was used to monitor and evaluate compound high temperature and drought events.

[0016] As a further technical solution of the present invention, the ground meteorological data in step S1 include precipitation, daily average temperature, daily maximum temperature, daily minimum temperature, relative humidity, station air pressure, wind speed and sunshine hours; the remote sensing data include MODIS and SRTM, wherein the vegetation status, surface temperature, evapotranspiration, potential evapotranspiration and surface reflectivity data are all from MODIS satellite sensors; the altitude data is from the Shuttle Radar Topography Mission (SRTM); the reanalysis data sets include ERA5-Land, GLDAS, CHIRPS and TerraClimate, wherein the temperature data is from ERA5-Land, the soil moisture data is from GLDAS, the precipitation data is from CHIRPS, and the saturated vapor pressure difference data is from TerraClimate.

[0017] As a further technical solution of the present invention, step S3 is a process of calculating the relevant index based on precipitation, soil moisture, surface temperature, vegetation conditions, surface water conditions, air dryness, and evapotranspiration data as follows:

[0018] Calculate the precipitation state index PCI based on the precipitation data Pre:

[0019] The soil moisture state index SMCI is calculated based on the soil moisture data SM:

[0020] The temperature state index TCI is calculated based on the land surface temperature data LST:

[0021] Calculate the Enhanced Vegetation Index (EVI) based on the band reflectance data: Where ρ NIR , RED and ρ BLUE Represent the reflectance of near-infrared, red and blue light bands respectively;

[0022] The normalized water index NDWI6 is calculated using surface reflectance data: Where b02 and b06 represent bands 2 and 6 of the surface reflectivity, respectively;

[0023] The normalized saturated vapor pressure difference index SVPD is calculated by the saturated vapor pressure difference VPD:

[0024] The normalized evapotranspiration index SET is calculated using the evapotranspiration data ET:

[0025] The normalized potential evapotranspiration index SPET is calculated using the potential evapotranspiration data PET:

[0026] In the above formula, X i represents the X value of the i-th month of a certain year; X max and X min Indicates the maximum and minimum values ​​of the corresponding grid of X in the i-th month of a certain year.

[0027] As a further technical solution of the present invention, the specific process of calculating the standardized precipitation evapotranspiration index (SPEI) in step S3 is: firstly, the potential evapotranspiration (PET) is calculated using the Penman-Monteith formula, then the difference between precipitation and potential evapotranspiration is calculated, and then the data sequence is fitted using the Log-logistic probability distribution, and finally normalization is performed to obtain the SPEI value.

[0028] As a further technical solution of the present invention, when calculating the standardized temperature index (STI) in step S3, the Gamma distribution probability of the temperature in a certain period of time is first calculated, and then normalized to obtain the STI value.

[0029] As a further technical solution of the present invention, the AI ​​algorithm in step S4 includes XGBoost, SVR and ANN algorithms.

[0030] As a further technical solution of the present invention, the specific process of step S5 is:

[0031] S51. Calculate the joint probability p of a composite high temperature drought event. The two random variables X and Y are SPEI and STI respectively. When the variable X is less than or equal to the threshold value x and the variable Y is greater than or equal to the threshold value y, a composite high temperature drought event occurs. The formula of the joint probability p is: p = P(X≤x, Y≥y) = uC(u,v), where u = P(X≤x), v = P(Y≥y), which are the marginal distributions of the random variables X and Y respectively; C(u,v) = P(X≤x, Y≤y), which is the joint distribution formed by combining the two marginal distributions by the Copula function;

[0032] S51. Perform an inverse normal distribution transformation on the joint probability p, thereby obtaining an index characterizing the composite high temperature drought event, namely, the composite high temperature drought index SCDHI: SCDHI = φ -1 (F(p)), where φ -1 is the inverse cumulative distribution function of the standard normal distribution, and F is the marginal cumulative distribution function, which remaps p to a uniform distribution.

[0033] Compared with the prior art, the present invention has the following beneficial effects:

[0034] (1) The present invention integrates multi-source ground meteorological, remote sensing and reanalysis data, comprehensively considers the factors affecting high temperature and drought disasters, and thus significantly improves the recognition accuracy of compound high temperature and drought disasters.

[0035] (2) By improving the spatial resolution, the present invention can accurately capture the extreme climate change in local areas at a finer scale, thereby significantly enhancing the monitoring accuracy of the composite high temperature drought index.

[0036] (3) The present invention uses AI technology to fully explore the intrinsic relationships between data, has stronger generalization capabilities, and can be applied to different climate regions and time scales.

[0037] (4) The present invention has good scalability and can be applied to the construction of other composite disaster indices to provide support for the monitoring and prediction of various extreme climate events. Description of the drawings:

[0038] Figure 1 This is a flow chart of the method for constructing a composite high temperature drought index based on AI technology involved in the present invention.

[0039] Figure 2 It is a schematic diagram of the Pearson correlation analysis between various high temperature and drought related factors involved in the present invention and SPEI and STI.

[0040] Figure 3 The present invention relates to a long-term series diagram of SPEI, STI and SCDHI indexes in Ningxia Hui Autonomous Region from 2001 to 2022.

[0041] Figure 4 The present invention relates to a spatial distribution map of typical composite high temperature drought events identified using the SCDHI index. Specific implementation method:

[0042] The present invention will be further described below by way of embodiments in conjunction with the accompanying drawings.

[0043] Embodiment 1:

[0044] This embodiment constructs a composite high temperature drought index based on AI technology and is used to monitor and evaluate composite high temperature drought events in a certain area. Specifically, the following steps are included:

[0045] S1. Data collection:

[0046] Multi-source data are collected, including ground meteorological data, remote sensing data and reanalysis data; the ground meteorological data include precipitation, daily average temperature, daily maximum temperature, daily minimum temperature, relative humidity, station air pressure, wind speed and sunshine hours; remote sensing data include MODIS and SRTM, among which vegetation status, surface temperature, evapotranspiration, potential evapotranspiration and surface reflectivity data are all from MODIS satellite sensors; altitude data are from the Shuttle Radar Topography Mission (SRTM); reanalysis data sets include ERA5-Land, GLDAS, CHIRPS and TerraClimate, temperature data are from ERA5-Land, soil moisture data are from GLDAS, precipitation data are from CHIRPS, and saturated vapor pressure difference data are from TerraClimate.

[0047] S2. Data preprocessing:

[0048] The surface meteorological data have been quality controlled, and a small amount of missing data has been interpolated; the storage format and coordinate system of remote sensing and reanalysis data have been unified, the spatial resolution has been adjusted to 1000 m using bilinear resampling, and the temporal resolution has been synthesized to a monthly scale, and finally the data have been cropped to the study area;

[0049] S3. High temperature and drought factors and evaluation index construction:

[0050] The causes of drought and high temperature are complex. This embodiment comprehensively considers factors related to drought and high temperature, such as precipitation, soil moisture, surface temperature, vegetation status, surface water status, air dryness, evapotranspiration, temperature and altitude. Among them, the altitude and 2m height temperature data can be directly used as independent variables in the monitoring model training data set, while other data need to calculate the relevant index first and then be included in the data set as independent variables. The specific calculation formula is shown in Table 1 below; the precipitation state index (PCI) is calculated based on precipitation data and is used to evaluate regional precipitation conditions. The soil moisture state index (SMCI) is calculated based on soil moisture data and is used to evaluate changes in soil moisture levels. The temperature state index (TCI) is calculated based on land surface temperature (LST) data to reflect the degree of heat stress of vegetation and soil. The enhanced vegetation index (EVI) is a remote sensing index used to measure vegetation density and health status. Compared with the traditional normalized difference vegetation index (NDVI), EVI can effectively reduce the impact of atmospheric conditions and soil background on vegetation measurement. The Normalized Difference Water Index (NDWI) is a vegetation index calculated using surface reflectance data, and is mainly used to assess the distribution and moisture content of surface water bodies. Evapotranspiration (ET) is the total amount of surface water transferred to the atmosphere through evaporation and plant transpiration, while potential evapotranspiration (PET) is the maximum amount of water that plants can release assuming sufficient water conditions. Saturated vapor pressure deficit (VPD) refers to the difference between the actual water vapor pressure and the saturated water vapor pressure in the air at a given temperature, and is the driving factor for the intensification of drought. The three indices, the Normalized Evapotranspiration Index (SET), the Normalized Potential Evaporation Index (SPET) and the Normalized Saturated Vapor Pressure Deficit Index (SVPD), are obtained by normalizing the ET, PET and VPD data, respectively.

[0051] Table 1: Summary of selected high temperature and drought related factors

[0052]

[0053] In the table, X i represents the X value of the i-th month of a certain year; X max and X min Indicates the maximum and minimum values ​​of the corresponding grid of X in the i-th month of a certain year.

[0054] The surface meteorological data are then used to calculate the standardized precipitation evapotranspiration index (SPEI) and the standardized temperature index (STI), which are used as drought and high temperature evaluation indicators, respectively, and as the dependent variables of the monitoring model. The specific calculation method is as follows:

[0055] (1) Calculation of Standardized Precipitation Evapotranspiration Index (SPEI): First, the Penman-Monteith formula is used to calculate the potential evapotranspiration (PET), then the difference between precipitation and potential evapotranspiration is calculated, and then the Log-logistic probability distribution is used to fit the data sequence, and finally normalization is performed to obtain the SPEI value;

[0056] (2) Calculation method of standardized temperature index (STI): First calculate the gamma distribution probability of temperature in a certain period of time, and then perform normalization processing on it to obtain the STI value.

[0057] This embodiment also evaluates the selection and availability of drought and high temperature factors. The reasonable selection of factors plays a key role in model construction. The drought factors selected when simulating the site standardized precipitation evapotranspiration index SPEI are PCI, SMCI, TCI, EVI, NDWI, SET, SVPD, SPET, and elevation. Elevation is a static variable, so no correlation analysis is performed. Multiple factors affect high temperature and drought at the same time. This study selected some drought-related factors as well as average temperature, maximum temperature, and minimum temperature as high temperature-related factors to simulate the site standardized temperature index STI. The Pearson correlation coefficient R is used to perform correlation analysis on the independent variables and dependent variables in the training data set. The calculation formula is as follows:

[0058]

[0059] Where x i and i is the observed value of variables X and Y corresponding to point i; and is the mean of variables X and Y. The analysis results are as follows: Figure 2 As shown in the figure, each high temperature and drought factor has a good correlation with STI and SPEI, and all of them have passed the 0.01 significance test.

[0060] S4. Construction of high temperature and drought monitoring model:

[0061] This implementation case takes the extreme gradient boosting algorithm (XGBoost) as an example to build high temperature and drought monitoring models respectively. XGBoost performs well in fitting on the training set and the test set, and its effects are shown in Table 2. Subsequently, the two models are applied to generate SPEI and STI data sets respectively to provide data support for the construction of the composite high temperature drought index.

[0062] Table 2: XGBoost fitting accuracy comparison

[0063]

[0064] S5. Construction and application of composite high temperature drought index:

[0065] The SCDHI was further constructed by using the precipitation evapotranspiration index (SPEI) and standardized temperature index (STI) datasets constructed by the S4 process. The specific process is as follows:

[0066] S51. Calculate the joint probability p of a composite high temperature drought event. The two random variables X and Y are SPEI and STI respectively. When the variable X is less than or equal to the threshold value x and the variable Y is greater than or equal to the threshold value y, a composite high temperature drought event occurs. The formula of the joint probability p is: p = P(X≤x, Y≥y) = uC(u,v), where u = P(X≤x), v = P(Y≥y), which are the marginal distributions of the random variables X and Y respectively; C(u,v) = P(X≤x, Y≤y), which is the joint distribution formed by combining the two marginal distributions by the Copula function;

[0067] S51. Perform an inverse normal distribution transformation on the joint probability p, thereby obtaining an index characterizing the composite high temperature drought event, namely, the composite high temperature drought index SCDHI: SCDHI = φ -1 (F(p)), where φ -1 is the inverse cumulative distribution function of the standard normal distribution, and F is the marginal cumulative distribution function, which remaps p to a uniform distribution;

[0068] The composite heat and drought index (SCDHI) is used to monitor and evaluate composite heat and drought events. Summer drought and extreme high temperatures have a significant impact on human life and agriculture. Therefore, June to August are selected as the research months. The research time range is from 2001 to 2022. The SPEI, STI and SCDHI index values ​​of each grid point are spatially averaged to obtain long-term series data, as shown in Figure 2. Figure 3 The classification standard of composite high temperature and drought events based on the SCDHI index is shown in Table 3. When the SCDHI value is ≤-0.5, it indicates that a composite high temperature and drought event has occurred; when the SCDHI value is ≤-1.5, it indicates that a composite high temperature and drought event of severe or above grade has occurred. The results show that during the study period, especially in June 2005, June 2006, June 2009, August 2013, August 2015, July and August 2021, and June 2022, composite high temperature and drought events occurred frequently, and some periods were severe or above.

[0069] Table 3: SCDHI composite high temperature and drought level classification table

[0070]

[0071] June 2009, July and August 2021 were selected as the periods of occurrence of typical composite high temperature and drought events. According to the grade classification criteria, the spatial distribution maps of the above events were drawn. The results are as follows: Figure 4 As shown, combined with the spatial distribution map and the "China Meteorological Disaster Yearbook", it was found that the precipitation in June 2009 was extremely small and the temperature was abnormally high, which further aggravated the spread of drought. Except for the areas along the Yellow River and its tributaries, most areas were in severe drought. Ningxia experienced a wide-ranging and severe compound high-temperature drought event; in the summer of 2021, Ningxia's climate characteristics of high temperature and low rainfall were very significant. The compound high-temperature drought events that occurred in July and August showed the significant characteristics of long duration, high intensity and wide impact range.

[0072] Compared with the "China Meteorological Disaster Yearbook", it is found that the SPEI and STI output by the monitoring model can accurately reflect the temporal and spatial changes of meteorological drought and high temperature events in Ningxia. The constructed composite high temperature drought index (SCDHI) has significant effectiveness in monitoring composite high temperature drought events and can reflect the actual situation. The composite high temperature drought index constructed by this embodiment can realize accurate monitoring of composite high temperature drought events in summer.

Claims

1. A method for constructing a composite high temperature drought index based on AI technology, characterized in that: The specific steps include: S1. Data collection: Collect multi-source data, including ground meteorological data, remote sensing data and reanalysis data; S2. Data preprocessing: interpolate the small amount of missing data in the ground meteorological data, unify the storage format and coordinate system of remote sensing data and reanalysis data, use bilinear resampling to adjust the spatial resolution to 1000 meters, and synthesize the temporal resolution to the monthly scale, and finally crop the data to the study area; S3. Construction of high temperature and drought factors and evaluation indexes: Comprehensively consider the key factors of precipitation, soil moisture, surface temperature, vegetation conditions, surface water conditions, air dryness, evapotranspiration, temperature and altitude. Among them, the altitude and 2m height temperature data can be directly used as independent variables in the monitoring model training data set. Other meteorological and remote sensing data are included in the data set as independent variables after calculating the relevant indexes. At the same time, the standardized precipitation evapotranspiration index SPEI and the standardized temperature index STI are calculated based on ground meteorological data to measure drought and high temperature, respectively, as dependent variables in the training data set; S4. Construction of high temperature and drought monitoring models: Use a variety of AI algorithms including machine learning and deep learning to build drought monitoring models and high temperature monitoring models respectively. 2 The model fitting accuracy is evaluated by using the root mean square error (RMSE) and the best performance model to output the SPEI index data set and the STI index data set. 2 The formula for RMSE is: Where n is the number of samples; y i is the true value; is the model prediction value; is the average of all true values; S5. Construction of composite high temperature and drought index: Use the joint distribution method based on the Copula function to construct the joint distribution of the two variables SPEI and STI, establish the standardized composite high temperature and drought index SCDHI, and use the constructed composite high temperature and drought index to monitor and evaluate composite high temperature and drought events.

2. The method for constructing a composite high temperature drought index based on AI technology according to claim 1, characterized in that: The ground meteorological data described in step S1 include precipitation, daily average temperature, daily maximum temperature, daily minimum temperature, relative humidity, station air pressure, wind speed and sunshine hours; the remote sensing data include MODIS and SRTM, among which the vegetation status, surface temperature, evapotranspiration, potential evapotranspiration and surface reflectivity data are all from MODIS satellite sensors; the altitude data are from the Space Shuttle Radar Topography Mission; the reanalysis data sets include ERA5-Land, GLDAS, CHIRPS and TerraClimate, the temperature data are from ERA5-Land, the soil moisture data are from GLDAS, the precipitation data are from CHIRPS, and the saturated vapor pressure difference data are from TerraClimate.

3. The method for constructing a composite high temperature drought index based on AI technology according to claim 2, characterized in that: Step S3 is a process of calculating the relevant index based on precipitation, soil moisture, surface temperature, vegetation conditions, surface water conditions, air dryness, and evapotranspiration data as follows: Calculate the precipitation state index PCI based on the precipitation data Pre: The soil moisture state index SMCI is calculated based on the soil moisture data SM: The temperature state index TCI is calculated based on the land surface temperature data LST: Calculate the Enhanced Vegetation Index (EVI) based on the band reflectance data: Where ρ NIR , RED and ρ BLUE Represent the reflectance of near-infrared, red and blue light bands respectively; The normalized water index NDWI6 is calculated using surface reflectance data: Where b02 and b06 represent bands 2 and 6 of the surface reflectivity, respectively; The normalized saturated vapor pressure difference index SVPD is calculated by the saturated vapor pressure difference VPD: The normalized evapotranspiration index SET is calculated using the evapotranspiration data ET: The normalized potential evapotranspiration index SPET is calculated using the potential evapotranspiration data PET: In the above formula, X i represents the X value of the i-th month of a certain year; X max and X min Indicates the maximum and minimum values ​​of the corresponding grid of X in the i-th month of a certain year.

4. The method for constructing a composite high temperature drought index based on AI technology according to claim 3, characterized in that: The specific process of calculating the standardized precipitation evapotranspiration index SPEI in step S3 is as follows: first, the potential evapotranspiration PET is calculated using the Penman-Monteith formula, then the difference between precipitation and potential evapotranspiration is calculated, then the data sequence is fitted using the Log-logistic probability distribution, and finally normalization is performed to obtain the SPEI value.

5. The method for constructing a composite high temperature drought index based on AI technology according to claim 4, characterized in that: When calculating the standardized temperature index STI in step S3, the Gamma distribution probability of the temperature in a certain period of time is first calculated, and then normalized to obtain the STI value.

6. The method for constructing a composite high temperature drought index based on AI technology according to claim 5, characterized in that: The AI ​​algorithms described in step S4 include XGBoost, SVR and ANN algorithms.

7. The method for constructing a composite high temperature drought index based on AI technology according to claim 6, characterized in that: The specific process of step S5 is: S51. Calculate the joint probability p of a composite high temperature drought event. The two random variables X and Y are SPEI and STI respectively. When the variable X is less than or equal to the threshold value x and the variable Y is greater than or equal to the threshold value y, a composite high temperature drought event occurs. The formula of the joint probability p is: p = P(X≤x, Y≥y) = uC(u,v), where u = P(X≤x), v = P(Y≥y), which are the marginal distributions of the random variables X and Y respectively; C(u,v) = P(X≤x, Y≤y), which is the joint distribution formed by combining the two marginal distributions by the Copula function; S51. Perform an inverse normal distribution transformation on the joint probability p, thereby obtaining an index characterizing the composite high temperature drought event, namely, the composite high temperature drought index SCDHI: SCDHI = φ -1 (F(p)), where φ -1 is the inverse cumulative distribution function of the standard normal distribution, and F is the marginal cumulative distribution function, which remaps p to a uniform distribution.

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