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This study compares the frequency spectra of seasonal precipitation during the last millenium from climate model simulations, tree-ring-based reconstructions, and gauge-based gridded observations of the twentieth century. Climate model simulations are from phase 6 of the Coupled Model Intercomparison Project (CMIP6) past1000 experiment, while tree-ring reconstructions are derived from the North American Seasonal Precipitation Atlas (NASPA). NASPA and CMIP6 model output are analyzed to understand their unique frequency biases in high-, mid-, and low-frequency ranges for both paleo-climatic millennium and recent centennial time series across North America. This was accomplished by first extracting signals from periodic ranges of 2–6, 4–15, 10–30, 20–50, and 30–110 years and then analyzing the result in Fourier space. This study reveals that the NASPA shows better alignment with observations in the frequency domain than global climate models (GCMs) even for low-frequency components. Moreover, the spatial distributions of the spectral biases indicate that there are significant disagreements between NASPA and GCMs in the east during cool seasons and in the west of North America during warm seasons for both historical centennial and preindustrial millennial periods. This is likely caused by NASPA tree-ring sensitivity, as its distribution roughly mirrors NASPA skill metrics. Notably, the spatial patterns of spectral biases differ between the modern and preindustrial eras, suggesting a changing bias through time. This study provides a new frequency-based metric to evaluate climate models and reconstructions and provides a first comparison of the two for North America. 

Numerous drought indices originate from the Standardized Precipitation Index (SPI) and use a moving-average structure to quantify drought severity by measuring normalized anomalies in hydroclimate variables. This study examines the theoretical probability of annual minima based on such a process. To accomplish this, we derive a stochastic model and use it to simulate 10 ×106 years of daily or monthly SPI values in order to determine the distribution of annual exceedance probabilities. We believe this is the first explicit quantification of annual extreme exceedances from a moving-average process where the moving-average window is proportionally large (5 %–200 %) relative to the year, as is the case for many moving-window drought indices. The resulting distribution of annual minima follows a generalized normal distribution rather than the generalized extreme-value (GEV) distribution, as would be expected from extreme-value theory. From a more applied perspective, this study provides the expected annual return periods for the SPI or related drought indices with common accumulation periods (moving-window length), ranging from 1 to 24 months. We show that the annual return period differs depending on both the accumulation period and the temporal resolution (daily or monthly). The likelihood of exceeding an SPI threshold in a given year decreases as the accumulation period increases. This study provides clarification and a caution for the use of annual return period terminology (e.g. the 100-year drought) with the SPI and a further caution for comparing annual exceedances across indices with different accumulation periods or resolutions. The study also distinguishes between theoretical values, as calculated here, and real-world exceedance probabilities, where there may be climatological autocorrelation beyond that created by the moving average.