Statistical Downscaling Approach Research Articles

Spatial downscaling of GRACE-derived groundwater storage changes across diverse climates and human interventions with Random Forests

The Gravity Recovery and Climate Experiment (GRACE) satellites launched in 2002, which were followed by the GRACE Follow-On mission in 2018, represented a significant advancement in global groundwater storage monitoring. However, the inherent limitation of a relatively low spatial resolution confines its applicability primarily to large river basins and aquifers. This study introduces a novel statistical downscaling approach using Random Forests (RF) to enhance the spatial resolution from 3° × 3° to 0.25° × 0.25° for GRACE-derived groundwater storage (GWS) changes. Focusing on sub-Saharan Africa, the Middle East, and South Asia across diverse climates and human interventions, the analysis spans from February 2003 to December 2022. Predictors encompass hydrological fluxes and their accumulation, including IMERG precipitation, GLEAM evapotranspiration, and GLDAS runoff, alongside GLDAS soil moisture and MODIS NDVI. RF models show robust performance across 114 aquifers, with a Pearson’s correlation coefficient (CC) exceeding 0.92 for training sets and 0.85 for validation sets. Validation against in-situ groundwater level observations from wells in India reveals satisfactory performance, with positive Spearman’s CC values between in-situ groundwater levels and downscaled terrestrial water storage anomaly (TWSA) for a majority of dug and bore wells. This study concludes with the generation of monthly downscaled GWS changes (0.25° × 0.25°) spanning from February 2003 to December 2022. These datasets serve as a foundation for facilitating finer-scale hydrological studies and formulating local groundwater management strategies.

Statistical downscaling of GRACE terrestrial water storage changes based on the Australian Water Outlook model

The coarse spatial resolution of the Gravity Recovery and Climate Experiment (GRACE) dataset has limited its application in local water resource management and accounting. Despite efforts to improve GRACE spatial resolution, achieving high resolution downscaled grids that correspond to local hydrological behaviour and patterns is still limited. To overcome this issue, we propose a novel statistical downscaling approach to improve the spatial resolution of GRACE-terrestrial water storage changes (ΔTWS) using precipitation, evapotranspiration (ET), and runoff data from the Australian Water Outlook. These water budget components drive changes in the GRACE water column in much of the global land area. Here, the GRACE dataset is downscaled from the original resolution of 1.0° × 1.0° to 0.05° × 0.05° over a large hydro-geologic basin in northern Australia (the Cambrian Limestone Aquifer—CLA), capturing sub- grid heterogeneity in ΔTWS of the region. The downscaled results are validated using data from 12 in-situ groundwater monitoring stations and water budget estimates of the CLA’s land water storage changes from April 2002 to June 2017. The change in water storage over time (ds/dt) estimated from the water budget model was weakly correlated (r = 0.34) with the downscaled GRACE ΔTWS. The weak relationship was attributed to the possible uncertainties inherent in the ET datasets used in the water budget, particularly during the summer months. Our proposed methodology provides an opportunity to improve freshwater reporting using GRACE and enhances the feasibility of downscaling efforts for other hydrological data to strengthen local-scale applications.

A model output statistic-based probabilistic approach for statistical downscaling of temperature

A model output statistic-based probabilistic approach for statistical downscaling of temperature

A new statistical downscaling approach for short‐term forecasting of summer air temperatures through a fusion of deep learning and spatial interpolation

AbstractReliable early forecasting of extreme summer air temperatures is essential for effectively managing and mitigating the socioeconomic damage caused by thermal disasters. Numerical weather prediction models have become valuable tools for forecasting air temperature; however, they incur high computational costs, resulting in coarse spatial resolution and systematic bias owing to imperfect parametrization. To address these problems, we developed a novel statistical downscaling and bias correction method (named DeU‐Net) for the maximum and minimum air temperature (Tmax and Tmin respectively) forecasts obtained from the Global Data Assimilation and Prediction System with a spatial resolution of 10 to 1.5 km over South Korea through the fusion of deep learning (i.e., U‐Net) and spatial interpolation. In this study, we used a methodology to decompose statistically downscaled Tmax and Tmin forecasts into temporal dynamics over South Korea and spatial fluctuations by pixels. When comparing the proposed DeU‐Net with the dynamical downscaling model (i.e., Local Data Assimilation and Prediction System) and support vector regression‐based statistical downscaling model at the seen and unseen stations for forecasting the next‐day Tmax and Tmin, DeU‐Net showed the highest spatial correlation and the lowest root‐mean‐square error in all cases. In a qualitative evaluation, DeU‐Net successfully produced a detailed spatial distribution most similar to the observations. A further comparison extending the forecast lead time to 7 days indicated that the proposed DeU‐Net is a better downscaling approach than support vector regression, regardless of the forecast lead time. These results demonstrate that bias‐corrected high spatial resolution air temperature forecasts with relatively long forecast lead times in summer can be effectively produced using the proposed model for operational forecasting.

A novel wind model downscaling with statistical regression and forecast for the cleaner energy

This study proposes an innovative statistical downscaling approach for meteorological wind models, enhancing cleaner energy applications. This method efficiently and swiftly incorporates topographic and meteorological data into wind park modelling by integrating regression-diagnosis-feature selection with statistical regression and forecasting techniques. The novel method in a de facto reduce predictive errors by approximately an average of 23.21%, lower errors downscaling and forecasting by 5.94% compared to existing methods, and optimise the balance between computational efficiency and forecasting accuracy. In an Arctic wind park case study, we systematically analysed meteorologically simulated wind speeds at various geographic points surrounding the park. The method enabled effective meteorological statistical mapping and accurate wind speed forecasting. Additionally, using the Granger causality test, the model identified a significant influence of surrounding terrain features on wind patterns. This research contributes by 1. introducing a novel downscaling technique surpassing current methods, 2. providing a resource-efficient approach for managing wind energy in complex terrains, and 3. offering a versatile model applicable to broader scenarios. The work marks an advancement in the field, promoting more reliable and efficient wind power and highlighting the importance of statistical downscaling in meteorological contexts for cleaner applications.

Leveraging Deep Learning for Climate Change Prediction Models : A Dive into Cutting-Edge Techniques

In recent years, deep learning has emerged as a promising approach for statistical downscaling, which involves generating high-resolution climate data from coarse low-resolution variables. However, concerns persist regarding the ability of these models to generalize to future climate change scenarios, primarily due to the assumption of stationarity. In this study, we propose the use of deep ensembles as a straightforward method to enhance the uncertainty quantification of statistical downscaling models. By improving the representation of uncertainty, these models offer superior planning capabilities against extreme weather events, which can have significant negative social and economic impacts. Given the absence of observational future data, we rely on pseudo-reality experiments to evaluate the effectiveness of deep ensembles in quantifying the uncertainty of climate change projections. The adoption of deep ensembles facilitates more robust risk assessment, addressing critical needs in various sectors for adapting to climate change challenges.

A Multi-stage Stochastic Approach for Statistical Downscaling of Rainfall

A Multi-stage Stochastic Approach for Statistical Downscaling of Rainfall

Review on spatial downscaling of satellite derived precipitation estimates

The present work aims at reviewing and identifying gaps in knowledge and future perspectives of satellite-derived precipitation downscaling algorithms. Here, various aspects related to statistical and dynamical downscaling approaches of the precipitation data sets from the Tropical Rainfall Measuring Mission (TRMM) and its successor Intergraded Multi-Satellite Retrievals for Global Precipitation Measurement (IMERG–GPM) mission are reviewed and the existing downscaling methods are categorized and analysed, to highlight the usefulness and applicability of the produced downscaled precipitation data sets. In addition, a critical comparison of the various statistical and dynamical methods for spatial or spatiotemporal downscaling of GPM and TRMM precipitation estimates was conducted, in terms of their advantages and disadvantages, simplicity of application and their suitability at different regional and temporal scales and hydroclimatic conditions. Finally, the adequacy of downscaling remotely sensed precipitation estimates as an effective way to obtain precipitation with sufficient spatial and temporal resolution is discussed and future challenges are highlighted.

Downscaling ERA5 wind speed data: a machine learning approach considering topographic influences

Energy system modeling and analysis can provide comprehensive guidelines to integrate renewable energy sources into the energy system. Modeling renewable energy potential, such as wind energy, typically involves the use of wind speed time series in the modeling process. One of the most widely utilized datasets in this regard is ERA5, which provides global meteorological information. Despite its broad coverage, the coarse spatial resolution of ERA5 data presents challenges in examining local-scale effects on energy systems, such as battery storage for small-scale wind farms or community energy systems. In this study, we introduce a robust statistical downscaling approach that utilizes a machine learning approach to improve the resolution of ERA5 wind speed data from around 31 km × 31 km to 1 km × 1 km. To ensure optimal results, a comprehensive preprocessing step is performed to classify regions into three classes based on the quality of ERA5 wind speed estimates. Subsequently, a regression method is applied to each class to downscale the ERA5 wind speed time series by considering the relationship between ERA5 data, observations from weather stations, and topographic metrics. Our results indicate that this approach significantly improves the performance of ERA5 wind speed data in complex terrain. To ensure the effectiveness and robustness of our approach, we also perform thorough evaluations by comparing our results with the reference dataset COSMO-REA6 and validating with independent datasets.

Development of statistical downscaling methods for the assessment of rainfall characteristics under climate change scenarios

Abstract The objective of this research was to develop a statistical downscaling approach in the Phetchaburi River Basin, Thailand, consisting of two main processes: predictor selection and relationship construction between predictors and local rainfall. Super predictor (SP) and stepwise regression (SR) were employed with principal component analysis (PCA) and the statistical downscaling model (SDSM) was applied later. Four statistical models were finally used: M1 (SP–PCA), M2 (SP–SDSM), M3 (SR–PCA), and M4 (SR–SDSM) with 26 large circulation indices generated by CanESM2 (CMIP5). Finally, the characteristics of extreme rainfall events were observed under three climate change scenarios during three different periods (2020–2040, 2041–2070, and 2071–2100). The results revealed that rainfall and geostrophic airflow velocity were the best predictors for rainfall downscaling, followed by divergence, meridional velocity, and relative humidity. Three objective functions (R2, NSE, and RMSE) were applied to evaluate model performance. The M4 model presented the highest performance while M1 showed the lowest skill. The average annual rainfall specifically increased compared with the historical rainfall for RCP2.6, RCP4.5, and RCP8.5 scenarios in future periods. Very to extremely wet years determined by the standardized anomaly index (SAI) occurred more often in the far-future while severe to extremely dry years frequently occurred in the mid- and far-future.

The understory microclimate in agroforestry now and in the future – a case study of Arabica coffee in its native range

Climate change is having a major impact on crop production and food security worldwide, and particularly so for smallholder farmers. As agroforestry is common with smallholder farmers, it is important to not only model the macroclimate, but also the microclimate that crops experience below the canopies. However, there are few high-resolution spatiotemporal climate projections for forest understories, because of constraints related to the lack of i) development of models for downscaling global climate projections, ii) high-resolution gridded datasets of environmental factors influencing microclimate, and iii) spatially replicated in-situ microclimate measurements. We focused on a landscape in southwestern Ethiopia where Arabica coffee originated, and, in the present day, is commonly grown as a shade crop. We first examined the relative contribution of in-situ field measurements vs. GIS-derived estimates of vegetation and topographic features in explaining in-situ microclimate. Second, we used a statistical downscaling approach to obtain past and future microclimate maps at 30-meter spatial resolution for the part of the landscape that is covered by trees. Predictive models using in-situ variables performed equal to models with GIS variables, indicating that remote sensing data might substitute for in-situ field measurements. Vegetation and topographic features were both important in explaining microclimatic variation. Our spatiotemporal projections of the microclimate indicate that coffee farming might have to relocate to higher altitudes due to increasing temperatures, that vegetation might buffer the macroclimate at middle altitudes to some extent, and that decreasing trends in relative humidity at the beginning of the wet season might become problematic for coffee production. Taken together, our findings demonstrate that we can rely on remote sensing data to create microclimate maps in landscapes where in-situ field measurements are challenging, and we suggest how these microclimate projections can be used as a tool to promote climate-resilient agriculture at the local and landscape levels.

Added value of an atmospheric circulation pattern‐based statistical downscaling approach for daily precipitation distributions in complex terrain

AbstractReliable prediction of heavy precipitation events causing floods in a world of changing climate is crucial for the development of appropriate adaption strategies. Many attempts to provide such predictions have already been conducted but there is still much potential for improvement left. This is particularly true for statistical downscaling of heavy precipitation due to changes present in the corresponding atmospheric drivers. In this study, a circulation pattern (CP) conditional downscaling to the station level is proposed which considers occurring frequency changes of CPs. Following a strict circulation‐to‐environment approach we use atmospheric predictors to derive CPs. Subsequently, precipitation observations are used to derive CP conditional cumulative distribution functions (CDFs) of daily precipitation. Raw precipitation time series are sampled from these CDFs. Bias correction is applied to the sampled time series with quantile mapping (QM) and parametric transfer functions (PTFs) as methods being tested. The added value of this CP conditional downscaling approach is evaluated against the corresponding common non‐CP conditional approach. The performance evaluation is conducted by using Kling–Gupta Efficiency (KGE), root mean squared error (RMSE), and mean absolute error (MAE) metrics. In both cases the applied bias correction is identical. Potential added value can therefore only be attributed to the CP conditioning. It can be shown that the proposed CP conditional downscaling approach is capable of yielding more reliable and accurate downscaled daily precipitation time series in comparison to a non‐CP conditional approach. This can be seen in particular for the extreme parts of the distribution. Above the 95th percentile, an average performance gain of +0.24 and a maximum gain of +0.6 in terms of KGE is observed. These findings support the assumption of conserving and utilizing atmospheric information through CPs can be beneficial for more reliable statistical precipitation downscaling. Due to the availability of these atmospheric predictors in climate model output, the presented method is potentially suitable for downscaling precipitation projections.

Constructing GRACE-Based 1 km Resolution Groundwater Storage Anomalies in Arid Regions Using an Improved Machine Learning Downscaling Method: A Case Study in Alxa League, China

Using the Gravity Recovery and Climate Experiment (GRACE) satellite to monitor groundwater storage (GWS) anomalies (GWSAs) at the local scale is difficult due to the low spatial resolution of GRACE. Many attempts have been made to downscale GRACE-based GWSAs to a finer resolution using statistical downscaling approaches. However, the time-lag effect of GWSAs relative to environmental variables and optimal model parameters is always ignored, making it challenging to achieve good spatial downscaling, especially for arid regions with longer groundwater infiltration paths. In this paper, we present a novel spatial downscaling method for constructing GRACE-based 1 km-resolution GWSAs by using the back propagation neural network (BPNN) and considering the time-lag effect and the number of hidden neurons in the model. The method was validated in Alxa League, China. The results show that a good simulation performance was achieved by adopting varying lag times (from 0 to 4 months) for the environmental variables and 14 hidden neurons for all the networks, with a mean correlation coefficient (CC) of 0.81 and a mean root-mean-square error (RMSE) of 0.70 cm for each month from April 2002 to December 2020. The downscaled GWSAs were highly consistent with the original data in terms of long-term temporal variations (the decline rate of the GWSAs was about −0.40 ± 0.01 cm/year) and spatial distribution. This study provides a feasible approach for downscaling GRACE data to 1 km resolution in arid regions, thereby assisting with the sustainable management and conservation of groundwater resources at different scales.

The Vatic Weather File Generator (VWFG v1.0.0)

To combat climate change caused by cities, it is necessary to perform urban physics modeling to assess mitigation solutions suitable for future climates. A challenge is availability of reliable future weather files for investigation of future climate scenarios. This research aims to develop the Vatic Weather File Generator (VWFG) using the statistical downscaling approach to fill this gap. VWFG is novel by using a history of weather files over multiple years to downscale future climate model data utilizing (1) quantile–quantile bias corrections, (2) record matching by the Finkelstein–Schafer method, and (3) shifting–stretching corrections. This implementation of VWFG uses the CanRCM4 climate model (1980–2100) and the ERA5 reanalysis data product as weather files (1980–2020) for two Representative Concentration Pathway (RCP) scenarios. VWFG outputs are consistent with findings in other studies. Further, to investigate the performance of VWFG for a case in Toronto, Canada, the Vertical City Weather Generator (VCWG) is forced by VWFG to predict the building sensible heating and cooling energy demands for a two-story single-family residential house. It is found that the building sensible heating demand is reduced over time (∼ 15%–30% for RCP 4.5–8.5 W m−2) and the building sensible cooling demand is increased over time (∼ 20%–50% for RCP 4.5–8.5 W m−2). The amount of change is greater for RCP 8.5 than RCP 4.5 W m−2. VWFG is a simple, practical, and widely applicable tool for urban physics simulations of future climates, particularly in cases where reliable forcing data is lacking otherwise.

A Stepwise Regression and Statistical Downscaling Approach for Projecting Temperature Variations under Multiple RCP Scenarios

With the rapid development of Central China, the temperature in this region is continuously increasing. Extreme weather events (e.g., high-temperature weather for many consecutive days) are becoming frequent. In order to provide future theoretical guidance on the direction of local development and the prevention of extreme natural disasters, the daily datasets of 12 meteorological stations in three provinces were collected. The corresponding predictors from 25 large-scale climatic factors were then screened using stepwise regression. A stepwise regression and statistical downscaling (SRSD) approach was developed to establish the statistical relationship. The future temperature results were projected by the weather generator, and the probability of extreme weather occurrence was analyzed by extreme values. The results indicate that future temperature in Central China shows an increasing trend from 2036 to 2065 and 2066 to 2095, with the representative concentration pathway 4.5 (RCP4.5) scenario showing a greater increase in temperature than the representative concentration pathway 8.5 (RCP8.5) scenario. Hunan Province has the largest temperature increase, followed by Hubei Province and Henan Province. The average annual duration of heat waves in Central China is 74.7 days.

A novel statistical downscaling approach for analyzing daily precipitation and extremes under the impact of climate change: Application to an arid region

Statistical downscaling of daily precipitation is important for assessing the impact of climate change. However, some issues (e.g., the underestimation of extremes) limit statistical downscaling methods to project daily precipitation accurately. This study develops an integrated hybrid statistical downscaling model combined with bias correction and Bayesian model averaging (named as SDBC-BMA) for improving the ability of downscaling methods. The performance of SDBC is compared with regression based statistical downscaling methods (RSDM) and hybrid statistical downscaling methods (HSDM) based on different performance metrics and extreme indices. SDBC-BMA is applied to the Amu Darya River Basin (ADRB) to demonstrate its feasibility and capability, where three GCMs and two CMIP6 scenarios are considered. Several findings can be summarized: (1) compared to RSDM and HSDM, SDBC is more robust with a higher computation ability of daily precipitation and a lower mean bias (i.e., +0.10 mm/day); (2) compared with the simple model average, the biases of extreme indices and mean daily precipitation obtained from SDBC-BMA could decrease by 49 % and 88 %, respectively; (3) the mean precipitation values would increase by 56 ± 41 mm/year (under SSP245) and 126 ± 55 mm/year (under SSP585) during 2076–2100, higher than mean precipitation in the historical period (1979–2005); (4) extreme indices under multi-GCMs (except for consecutive wet days) would increase by 20 % (under SSP245) and 22 % (under SSP585), implying that extreme events in ADRB are more frequent in the 21st century. The increasing risk of precipitation extremes would exacerbate threats to agricultural and ecological security.

Deep Learning-Based Downscaling of Temperatures for Monitoring Local Climate Change Using Global Climate Simulation Data

The impact of climate change on the environment has become increasingly visible today, and foreseeing future climate events, which involves long-term prediction of climate variables (e.g., temperature, wind speed, precipitation, etc.) at a local small scale in a local region, is crucial for disaster risk management. General Circulation Models (GCMs) allow for the simulation of multiple climate variables, decades into the future (often till the year 2100). GCM simulations, however, are at a global large scale (from 100 km to 600 km) and are too coarse to monitor climate change at the local small scale. Statistical downscaling approaches are often applied to the GCM simulations to allow the evaluation of the GCM outputs at the local scale. Machine learning-based techniques are popular approaches for statistical downscaling. In this paper, we provide an overview of GCM downscaling with machine learning and present a case study that leverages deep learning to downscale weekly averages of the daily minimum and maximum temperatures in the Hackensack–Passaic watershed in New Jersey.

Estimating multisite precipitation by a stepwise NHMM-VAR model considering the spatiotemporal correlations of precipitation amounts

• This study developed a stepwise NHMM-VAR multisite downscaling model. • The NHMM-VAR model performs well not only in modeling the statistical properties but also in reproducing the spatiotemporal correlations of multisite precipitation amounts. • The temporal intermittency of precipitation amounts is considered in the NHMM-VAR model. The nonhomogeneous hidden Markov model (NHMM) is a popular statistical downscaling (SD) approach that can be used for multisite precipitation estimation. However, the NHMM assumes that once the days with the occurrence of precipitation have been determined, the precipitation amounts at each station on wet days can be conditionally independently generated from a parametric family of probability distribution functions without considering impacts from the past precipitation amounts and the neighboring stations. Such assumptions may lead to underestimation of the spatiotemporal correlations (i.e., temporal autocorrelation and spatial cross-correlations) of actual multisite precipitation amounts. Thus, this study developed a stepwise downscaling method called the nonhomogeneous hidden Markov model - vector autoregressive (NHMM-VAR), in which the NHMM is used to determine the probabilities for dry days and marginal distributions of multisite daily mean precipitation amounts on wet days based on the large-scale atmospheric covariates, then a VAR(1) model is used to reproduce the spatiotemporal correlations of actual multisite daily mean precipitation amounts. A comparison experiment between the NHMM and the NHMM-VAR models is conducted against outputs from eight global climate models (GCMs) of Phase 6 of the Coupled Model Intercomparison Project (CMIP6) over the Pearl River basin (PRB) of South China. The results show that the NHMM-VAR model performs well not only in modeling the statistical properties but also in reproducing the spatiotemporal correlations of actual multisite daily mean precipitation amounts, which indicates that the VAR model does add value to the NHMM downscaling method.

Predictive Modelling of Statistical Downscaling Based on Hybrid Machine Learning Model for Daily Rainfall in East-Coast Peninsular Malaysia

In recent years, climate change has demonstrated the volatility of unexpected events such as typhoons, flooding, and tsunamis that affect people, ecosystems and economies. As a result, the importance of predicting future climate has become even direr. The statistical downscaling approach was introduced as a solution to provide high-resolution climate projections. An effective statistical downscaling scheme aimed to be developed in this study is a two-phase machine learning technique for daily rainfall projection in the east coast of Peninsular Malaysia. The proposed approaches will counter the emerging issues. First, Principal Component Analysis (PCA) based on a symmetric correlation matrix is applied in order to rectify the issue of selecting predictors for a two-phase supervised model and help reduce the dimension of the supervised model. Secondly, two-phase machine learning techniques are introduced with a predictor selection mechanism. The first phase is a classification using Support Vector Classification (SVC) that determines dry and wet days. Subsequently, regression estimates the amount of rainfall based on the frequency of wet days using Support Vector Regression (SVR), Artificial Neural Networks (ANNs) and Relevant Vector Machines (RVMs). The comparison between hybridization models’ outcomes reveals that the hybrid of SVC and RVM reproduces the most reasonable daily rainfall prediction and considers high-precipitation extremes. The hybridization model indicates an improvement in predicting climate change predictions by establishing a relationship between the predictand and predictors.

Intercomparison of Dynamically and Statistically Downscaled Climate Change Projections over the Midwest and Great Lakes Region

AbstractDownscaling of global climate model (GCMs) simulations is a key element of regional-to-local-scale climate change projections that can inform impact assessments, long-term planning, and resource management in different sectors. We conduct an intercomparison between statistically and dynamically downscaled GCMs simulations using the hybrid delta (HD) and the Weather Research and Forecast (WRF) Model, respectively, over the Midwest and Great Lakes region to 1) validate their performance in reproducing extreme daily precipitation (P) and daily maximum temperature (Tmax) for summer and winter and 2) evaluate projections of extremes in the future. Our results show the HD statistical downscaling approach, which includes large-scale bias correction of GCM inputs, can reproduce observed extremePandTmaxreasonably well for both summer and winter. However, raw historical WRF simulations show significant bias in both extremePandTmaxfor both seasons. Interestingly, the convection-permitting WRF simulation at 4-km grid spacing does not produce better results for seasonal extremes than the WRF simulation at 12 km using a parameterized convection scheme. Despite a broad similarity for winter extremePprojections, the projected changes in the future summer storms are quite different between downscaling methods; WRF simulations show substantial increases in summer extreme precipitation, while the changes projected by the HD approach exhibit moderate decreases overall. The WRF simulations at 4 km also show a pronounced decoupling effect between seasonal totals and extreme dailyPfor summer, which suggests that there could be more intense summer extremes at two different time scales, with more severe individual convective storms combined with longer summer droughts at the end of the twenty-first century.