Rescaled Range and Detrended Fluctuation Analysis of Global Mean Sea Level: Anti-Persistence in Seasonally Adjusted Monthly Increments

Author: stepstep_labs

Abstract

We apply rescaled range (R/S) analysis and detrended fluctuation analysis (DFA) to the Church and White (2011) global mean sea level (GMSL) reconstruction spanning 1880–2013, after explicit seasonal adjustment and first differencing. Working with 1607 seasonally adjusted monthly increments, R/S analysis yields a Hurst exponent of H = 0.35 (95% bootstrap CI: [0.337, 0.364]), while DFA produces a scaling exponent of α = 0.21 (95% CI: [0.179, 0.237]). Both estimates fall well below their respective thresholds for uncorrelated noise (0.5), indicating significant anti-persistent behavior. Seasonal removal has negligible impact on the scaling exponents (ΔH < 0.01), confirming that the anti-persistence is not an artifact of the annual cycle. The observed H is far below the AR(1) null expectation of H ≈ 0.60 (z = −8.6, p < 0.001), ruling out short-memory autocorrelation as an explanation. Structural break analysis dividing the record at 1990 yields consistent anti-persistence in both the pre-1990 segment (H = 0.40, α = 0.25) and the post-1990 segment (H = 0.39, α = 0.18), indicating the scaling property is stable across the known acceleration in sea-level rise. These findings suggest that projection frameworks assuming independent or positively correlated residuals may overestimate the spread of natural variability at monthly to interannual scales, with implications for uncertainty quantification in sea-level projections.

1. Introduction

Global mean sea level (GMSL) is among the most consequential indicators of climate change. Over the 20th century, GMSL rose by approximately 15–20 cm, with the rate accelerating from roughly 1.5 mm/yr early in the century to over 3 mm/yr in recent decades as measured by satellite altimetry (Church and White, 2011; Nerem et al., 2018). Projecting future sea-level rise is critical for coastal planning, infrastructure design, and climate adaptation, yet projection uncertainty remains substantial—IPCC AR6 scenarios for 2100 span a range exceeding one meter depending on emission pathway and ice-sheet response (Fox-Kemper et al., 2021).

A key but often overlooked factor in projection uncertainty is the temporal correlation structure of sea-level variability. Most projection frameworks implicitly assume that the residuals around a forced trend are either independent or follow simple autoregressive processes. However, many geophysical time series exhibit long-range dependence (LRD), a property in which correlations decay as a power law rather than exponentially. This can profoundly affect the statistics of extremes, trends, and uncertainty bounds.

The concept of LRD in geophysical records originates with Harold Edwin Hurst (1951), who studied annual discharge records of the Nile River. Hurst found that the rescaled range R/S of cumulative departures from the mean grew with the observation window size n as a power law R/S ∝ n^H, where H ≈ 0.73 for the Nile—substantially above the value H = 0.5 expected for independent increments. Mandelbrot and Wallis (1968, 1969) formalized this by introducing fractional Gaussian noise and fractional Brownian motion. When H > 0.5, the process exhibits persistence (positive long-range correlations); when H < 0.5, anti-persistence (negative long-range correlations, or mean-reversion). Franzke et al. (2015) demonstrated that atmospheric circulation indices exhibit H ≈ 0.6–0.7, with implications for extreme weather predictability.

The classical R/S method has well-known limitations: it is biased by short-range correlations (Lo, 1991) and sensitive to non-stationarities (Teverovsky et al., 1999). Detrended fluctuation analysis (DFA), introduced by Peng et al. (1994), removes local polynomial trends within each analysis window, making it more robust against non-stationarities and trends. DFA has become the standard tool for scaling analysis in climate science (Kantelhardt et al., 2001; Lennartz and Bunde, 2009).

An important concern is the "Hurst phenomenon"—the observation that apparent long-range dependence can arise from structural breaks or non-stationarity in the mean rather than genuine fractal scaling (Klemeš, 1974; Koutsoyiannis, 2003). Regime shifts, trend changes, or isolated events can produce scaling that mimics long-range dependence, and distinguishing between these mechanisms remains an open challenge.

Monthly sea-level records contain a seasonal cycle driven by thermal expansion, precipitation patterns, and atmospheric circulation changes. Applying scaling analysis to first differences of a series with a strong periodic component can produce spuriously low Hurst exponents, as the cycle's mean-reverting nature introduces artificial anti-persistence. Proper seasonal adjustment is therefore essential before interpreting scaling exponents.

In the oceanographic context, Li and Zhang (2017) characterized long-range dependence in tide gauge records using a generalized Cauchy model, and Matos and Barbosa (2012) applied DFA to satellite altimetry data, finding scale-dependent Hurst exponents. However, systematic application of both R/S and DFA to the spatially integrated GMSL reconstruction—with seasonal adjustment and structural break testing—has received limited attention. We address this gap using the Church and White (2011) reconstruction.

2. Methods

2.1 Data

We use the Church and White (2011) GMSL reconstruction updated to the end of 2013, obtained from the CSIRO Sea Level Data archive. The dataset contains 1608 monthly observations of reconstructed GMSL from January 1880 through December 2013. Values are expressed in millimeters relative to a 1990 baseline. The reconstruction combines tide gauge records from the Permanent Service for Mean Sea Level (PSMSL) database with satellite altimetry-derived empirical orthogonal functions (EOFs), corrected for glacial isostatic adjustment (GIA). The series spans approximately 267 mm, from −184.5 mm (early 1880s) to +82.4 mm (late 2013).

2.2 Seasonal Adjustment

Monthly GMSL data contain a seasonal cycle reflecting annual variations in ocean thermal expansion, freshwater exchange, and atmospheric loading. To prevent the seasonal signal from biasing scaling estimates, we compute a monthly climatology by averaging GMSL values for each calendar month across all 134 years of record (134 values per month). This climatology is subtracted from each observation, yielding a seasonally adjusted GMSL series. The seasonal amplitude is modest (peak-to-trough range of approximately 2.2 mm in the climatology), consistent with the spatial averaging inherent in a global reconstruction.

2.3 First Differences

Because the GMSL series is non-stationary, we work with first differences—monthly sea-level changes Δ_t = GMSL(t+1) − GMSL(t). This yields M = 1607 monthly increments. We compute differences of both raw and seasonally adjusted series. For the seasonally adjusted series: mean = 0.149 mm/month (~1.79 mm/yr), standard deviation = 3.06 mm, lag-1 autocorrelation φ = 0.312.

2.4 Rescaled Range (R/S) Analysis

For each window size n, we divide the series into ⌊L/n⌋ non-overlapping blocks. Within each block of observations {x_1, ..., x_n}: compute the block mean m, form the mean-adjusted cumulative sums Y_k = Σ_{i=1}^{k}(x_i − m), and calculate R/S = [max(Y_k) − min(Y_k)] / S, where S is the block standard deviation. We average R/S across blocks and estimate the Hurst exponent H as the OLS slope of log(R/S) vs. log(n).

We use window sizes n ∈ {10, 15, 20, 30, 50, 75, 100, 150, 200, 300, 500}. We exclude n = 750 from the analysis because it yields only two non-overlapping blocks, insufficient for reliable estimation. Even at n = 500 (three blocks) and n = 300 (five blocks), the estimates carry substantial sampling uncertainty.

2.5 Detrended Fluctuation Analysis (DFA)

Following Peng et al. (1994) and Kantelhardt et al. (2001): (1) compute the cumulative profile of the demeaned series Y_k = Σ_{i=1}^{k}(x_i − x̄); (2) divide into non-overlapping segments of size n; (3) fit a local linear trend by OLS within each segment; (4) compute the RMS of detrended residuals within each segment; (5) average across segments to obtain the fluctuation function F(n). The DFA exponent α is the OLS slope of log(F(n)) vs. log(n). The interpretation parallels R/S: α < 0.5 indicates anti-persistence, α = 0.5 uncorrelated noise, α > 0.5 persistence. DFA is more robust against non-stationarities than R/S because it explicitly removes local trends within each window.

2.6 Bootstrap Confidence Intervals

We construct 95% CIs for both H and α using a non-parametric bootstrap with 10,000 iterations. For each iteration, we resample (with replacement) the set of block-level or segment-level values within each window size, compute the resampled average, and refit the log-log regression. The 2.5th and 97.5th percentiles form the confidence interval.

2.7 AR(1) Null Model

To assess whether the observed scaling reflects genuine long-range anti-persistence versus an artifact of short-memory correlation, we compare against an AR(1) null model calibrated to the seasonally adjusted differenced series (φ = 0.312, innovation standard deviation σ_ε = 2.91 mm). We simulate 1,000 AR(1) realizations of the same length and compute H via identical R/S procedures.

2.8 Structural Break Analysis

The GMSL record contains a well-documented acceleration around 1990. To test whether the observed anti-persistence is an artifact of this structural break, we split the seasonally adjusted differenced series at 1990 and compute both H and α separately for the pre-1990 segment (1319 observations, 1880–1989) and the post-1990 segment (288 observations, 1990–2013). If exponents are consistent across sub-periods, this supports the interpretation of anti-persistence as a stable property rather than a statistical artifact.

3. Results

3.1 Effect of Seasonal Adjustment

Table 1 compares the scaling exponents estimated from raw versus seasonally adjusted first differences.

Table 1. Comparison of scaling exponents with and without seasonal adjustment.

Seasonal adjustment has a negligible effect on both scaling exponents (ΔH = −0.004, Δα = −0.001). This confirms that the anti-persistence is not driven by the seasonal cycle and reflects intrinsic properties of the month-to-month variability. The small magnitude of the seasonal signal relative to the monthly noise (seasonal amplitude ~2.2 mm vs. monthly standard deviation ~3 mm) explains this insensitivity. All subsequent analyses use the seasonally adjusted series.

3.2 R/S and DFA Scaling of Seasonally Adjusted First Differences

Table 2 presents the combined R/S and DFA results for the seasonally adjusted differenced GMSL series.

Table 2. R/S and DFA analysis of seasonally adjusted monthly sea-level changes.

The R/S log-log relationship is linear (R² = 0.951), yielding:

H = 0.351 (95% bootstrap CI: [0.337, 0.364])

The DFA scaling yields:

α = 0.208 (95% bootstrap CI: [0.179, 0.237], R² = 0.900)

Both exponents are substantially below 0.5, indicating strong anti-persistence: months with above-average sea-level rise tend to be followed by months with below-average rise, and vice versa. The slight flattening at n = 500 (where mean R/S decreases slightly) likely reflects the limited number of blocks (three) at this window size, a known limitation of scaling estimators at the largest scales. The DFA exponent is lower than the R/S estimate, consistent with the known upward bias of R/S in the presence of positive short-range autocorrelation (Lo, 1991).

3.3 AR(1) Null Comparison

The AR(1) null model (φ = 0.312) produces a distribution of R/S Hurst exponents with mean H_null = 0.604 and standard deviation 0.029. The 95% range of null exponents is [0.546, 0.659]. The observed H = 0.351 falls far below this distribution:

• z-score: −8.64
• p-value: < 0.001 (no null simulation out of 1,000 produced H ≤ 0.351)

This demonstrates that the observed anti-persistence is highly significant and cannot be explained by the short-memory autocorrelation structure of the data. The AR(1) process, despite having positive autocorrelation (φ = 0.312), produces R/S scaling consistent with H ≈ 0.5–0.6 due to the known upward bias of R/S for positively autocorrelated series. The observed H = 0.351 represents a fundamentally different correlation structure extending beyond the lag-1 time scale.

3.4 Structural Break Analysis

Table 3 presents the scaling exponents computed separately for the pre-1990 and post-1990 segments.

Table 3. Scaling exponents by sub-period.

Both sub-periods exhibit anti-persistence (H < 0.5 and α < 0.5), with overlapping R/S confidence intervals. The post-1990 segment shows a higher mean rate (0.205 vs. 0.137 mm/month, consistent with the known acceleration) and lower variability (std = 2.58 vs. 3.15 mm), but the scaling structure is consistent. The full-record H (0.351) is somewhat lower than the sub-period values (0.403, 0.386). This difference likely reflects the additional variance introduced by the trend change: the shift in mean rate between the two periods creates extra dispersion in the combined differenced series, slightly depressing the R/S estimate. The DFA exponents show a similar pattern, with α < 0.5 in both segments.

The key finding is that anti-persistence is present in both sub-periods, indicating it is not solely an artifact of the 1990 structural break. While we cannot exclude contributions from other undetected regime shifts within each sub-period, the consistency across two segments with markedly different properties (data quality, rate of rise, variability amplitude) supports anti-persistence as a genuine feature of the GMSL fluctuation process.

3.5 Raw (Undifferenced) GMSL

For methodological context, R/S analysis of the raw (undifferenced) GMSL series yields H = 0.99 (R² = 0.993). This near-unity value is expected for a non-stationary trending process and primarily reflects the secular rise (~1.8 mm/yr) rather than the intrinsic correlation structure of the fluctuations. It confirms that first differencing is necessary to isolate the stochastic component and that the raw-series result carries no meaningful information about the fluctuation dynamics.

3.6 Summary of Key Results

Table 4. Summary of principal findings.

4. Discussion

4.1 Anti-Persistence in Sea-Level Changes

The central finding is that monthly sea-level changes exhibit significant anti-persistence after seasonal adjustment, confirmed by two independent methods: R/S analysis (H = 0.35) and DFA (α = 0.21). Both estimates are well below their thresholds for uncorrelated noise, and the R/S exponent is far below the AR(1) null. The negligible difference between raw and seasonally adjusted exponents (ΔH < 0.01) demonstrates that the anti-persistence is not an artifact of the seasonal cycle.

Anti-persistence (H < 0.5) implies mean-reversion in the detrended fluctuations. Physically, this is consistent with several well-known oceanographic processes. Ocean thermal equilibration creates negative correlations in the increment series: periods of anomalous heat uptake are followed by enhanced heat loss (or reduced uptake), with thermal relaxation time scales spanning weeks to decades depending on ocean depth. Conservative water mass redistribution between ocean basins, land storage, and the atmosphere produces compensating fluctuations at seasonal to interannual scales. Partial steric compensation—where halosteric and thermosteric sea-level changes offset each other—further reduces the net variability of GMSL increments.

4.2 Convergence of R/S and DFA

The DFA exponent (α = 0.21) is lower than the R/S Hurst exponent (H = 0.35). This discrepancy is expected: R/S is upward-biased by positive short-range autocorrelation (Lo, 1991), while DFA removes local linear trends more effectively. The positive lag-1 autocorrelation (φ = 0.31) is consistent with the direction and magnitude of this bias. Despite the quantitative difference, both methods converge on the same qualitative finding of strong anti-persistence significantly below the uncorrelated threshold.

4.3 Structural Stability and the Hurst Phenomenon

Koutsoyiannis (2003) and Klemeš (1974) emphasized that apparent long-range dependence can arise from structural breaks rather than genuine scaling processes. Our structural break analysis directly addresses this concern. If the anti-persistence were solely an artifact of the 1990 acceleration, the sub-period exponents would be substantially closer to 0.5 than the full-record value. Instead, both sub-periods show H well below 0.5 (0.40 and 0.39), comparable to or slightly above the full-record value (0.35). The slightly lower full-record H likely reflects the inter-segment mean shift rather than genuine deepening of anti-persistence. While we cannot exclude contributions from other undetected regime shifts, the evidence supports anti-persistence as a genuine property of the fluctuation process.

4.4 Implications for Uncertainty Quantification

The anti-persistence of sea-level increments has several implications for uncertainty quantification in projections, though these should be interpreted with appropriate caution:

1. Variability bounds. Models assuming independent or positively correlated residuals may overestimate the variance of natural fluctuations at monthly to interannual time scales. Anti-persistent increments imply that the process is more tightly constrained than a random walk.

2. Trend detection. Anti-persistent fluctuations around a trend make trend detection somewhat easier relative to positively correlated or independent residuals, because fluctuations are more tightly bounded. This has implications for the statistical significance of observed acceleration.

3. Scale limitations. The scaling exponents characterize fluctuations at monthly to interannual scales (windows of 10–500 months, roughly 1–40 years). Extrapolation to multi-decadal or centennial scales is not warranted, as different physical processes may dominate at longer time scales.

4. Complementary to physical models. These statistical findings complement rather than replace physically based projections. The anti-persistence structure could inform the stochastic component of semi-empirical sea-level models, but the dominant long-term uncertainties remain ice-sheet and glacier dynamics, which involve threshold behavior outside the scope of linear scaling analysis.

4.5 Limitations

Several caveats apply. The Church and White reconstruction relies on a spatially sparse tide gauge network in the early record, with uncertainties exceeding ±20 mm before 1900; reconstruction errors could introduce artifacts into the correlation structure. The estimates at the largest window sizes (n = 300 with 5 blocks, n = 500 with 3 blocks) carry substantial sampling uncertainty; these points contribute to the log-log regression but are less reliable than estimates from smaller window sizes with many blocks. The analysis is limited to monthly resolution; the correlation structure may differ at other temporal scales. We assume a single scaling exponent across all window sizes, though crossover behavior is common in geophysical systems (the R² values of 0.90–0.95 suggest a single exponent is a reasonable first-order description). More sophisticated methods such as wavelet-based estimators or multifractal analysis could provide additional confirmation and reveal richer scaling structure.

5. Conclusion

We have applied rescaled range analysis and detrended fluctuation analysis to the Church and White (2011) global mean sea level reconstruction spanning 1880 to 2013, after removing the seasonal cycle and computing first differences. Both methods reveal significant anti-persistence in seasonally adjusted monthly sea-level increments: H = 0.35 (95% CI: [0.337, 0.364]) from R/S and α = 0.21 (95% CI: [0.179, 0.237]) from DFA. The anti-persistence is not a seasonal artifact, is highly significant against an AR(1) null (z = −8.6, p < 0.001), and is stable across the 1990 structural break in the rate of sea-level rise. Physically, this is consistent with the restoring dynamics of ocean thermal equilibration and conservative water mass redistribution. For uncertainty quantification, the findings suggest that residual variability around forced trends may be more tightly bounded than models assuming independent or persistent increments would predict, at least at the monthly-to-interannual scales covered by our analysis.

Future work should apply wavelet-based methods for additional cross-validation, examine scale-dependent behavior through multifractal analysis, investigate whether the anti-persistence structure is consistent across individual tide gauge records and the satellite altimetry era, and extend the analysis to higher-frequency data to identify the temporal scales at which anti-persistence transitions to other correlation regimes.

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