Tree rings track the broad drought pattern in a gridded record and almost none of its local detail

Tree-ring records are often judged one site at a time, by comparing a site's yearly growth with the climate at that spot. We split that comparison by spatial scale as well as by time scale. The data are June to August drought from a gridded index, 399 one-degree cells in western North America from 1903 to 2025, and 664 tree-ring chronologies in the same box, complete from 1903 to 1980. In each of 30 cells of a grid of six spatial bands and five time bands we measured the share of the drought signal that the trees pass. The preregistered contrast between broad, slow scales and fine, fast scales was 0.64 (95% interval 0.62 to 0.68), against 95th percentiles of 0.22 for phase-randomised trees and 0.30 for trees moved to shuffled sites, with an adjusted p of 0.035.c1 Below 55 km the trees pass at most 0.004 of the drought signal in every time band, and above 460 km they pass 0.27 to 0.91.c2 Three tests on the drought field came back null. The drought field's covariance did not depart from a space part times a time partc3, no travel speed could be claimedc4, and broad and fine patterns lose their memory at the same rate, with a dispersion exponent of 0.02 (-0.12 to 0.17)c5. Measured against this gridded record, the trees act as a spatial filter; the grid is itself smooth, so the filter may belong partly to the grid.

Tree rings are the main yearly record of drought before instruments. Cook, Meko, Stahle and Cleaveland [1] built a grid of summer drought reconstructions for the United States from a dense network of chronologies. Their method fits each grid point from the chronologies within a search radius, and they reported that a radius of 450 km worked best. The method is local by design. The reconstructed maps still kept the spatial features of drought, which the authors noted although the method was not built to be spatial.

The drought index in this study is the self-calibrating Palmer Drought Severity IndexDefinition: self-calibrating Palmer Drought Severity Index (scPDSI) climatology and hydrology A monthly drought index computed from temperature and precipitation through a simple soil water balance, with its constants calibrated to each location so values compare across places. Excludes a direct measurement of soil moisture or of streamflow. full definition, the scPDSI. Wells, Goddard and Hayes [2] built it so that the index's constants are fitted to each location and values compare across places. We use the gridded scPDSI distributed with CRU TS 4.10 [3] [4].

Averaging trees over larger areas raises the shared climate signal. Duchesne and colleagues [6] showed with forest inventory cores in Quebec that averaging at the stand level removes tree-level noise, that covariance among tree-ring signals falls with distance, and that reliable climate signals can be drawn at scales from 37 to 583,000 square km.

These studies treat space and time one at a time. A field that varies in both can hold structure neither view shows: a front that moves, patterns that fade faster at fine scales than at broad ones, or a filter that passes some combinations of scale and period and not others. This paper asks four questions at once. Does the drought field's covariance split into a space part times a time part? Do its anomalies travel? Do broad and fine patterns fade at different rates? And which combinations of spatial scale and period do the tree rings pass?

The design was written and hash-locked on 30 October 2026 at 04:20 UTC as a preregistrationDefinition: preregistration research method (metascience) A dated written plan for a study, fixed and hash-locked before any result is seen: the question, the data, the statistic, the controls, the forecasts and the kill conditions. Excludes a plan revised after looking at the outcome, and a protocol written only for the record after the run. full definition, before any outcome on the real data was computed. The real data had been built and only their counts printed: 399 cells by 123 years and 664 sites by 78 years. Development before the lock used synthetic fields only, and the preregistration lists every change made during that development.

The locked file names the data, the band edges, the four statistics, the synthetic fields, each controlDefinition: control statistics and experimental design A version of the data with the structure under test removed (for example shuffled or pooled), scored the same way as the real data, so the real score has something to beat. Excludes a comparison with a different dataset or with a published number. full definition, the forecasts and the pass rules. No unit, constant or rule changed after the lock. Each test passes only when the real statistic beats the 95th percentile of every scoring control, its adjusted p value is 0.05 or less, and, where named, its bootstrap interval sits above 0. A test that misses any rule fails, which makes each rule a kill conditionDefinition: kill condition research method (our usage, after Popper's falsification) A rule written in the preregistration that, when met, decides against the claim with no room for judgement after the run. Excludes reasons found after the run for setting a result aside. full definition.

Forecasts, written before the run: the covariance would be non-separable in both fields; travel would most likely be null or slow, under 100 km per year; the dispersion exponent would lie between 0 and 2, closer to the diffusion reading than the wave reading; and the tree filter test would pass.

Drought: the June to August mean of the scPDSI from CRU TS 4.10 [4], 0.5 degree, 1903 to 2025, released under the Open Database License 1.0. The 0.5 degree cells between 125 and 100 W and 30 and 50 N were averaged to one-degree cells, keeping only one-degree cells with all four parts present in every year. This gave 399 cells and 123 years.

Trees: standard ring-width chronologiesDefinition: tree-ring chronology dendrochronology One yearly series for a site, built by dating many cores, removing each core's age and size trend and averaging the results into a dimensionless index. Excludes a single core's raw ring widths, and a climate reconstruction made from the series. full definition from the International Tree-Ring Data Bank [5], held by NOAA's National Centers for Environmental Information as public data, with a request to cite the contributors. Every site in the same box with a gap-free index from 1903 to 1980 entered, which gave 664 sites and 78 years. Each site was paired with the June to August scPDSI of its nearest 0.5 degree cell with data in every year.

Every series had its linear trend removed and was scaled to unit variance. Positions were projected about 112.5 W and 40 N, in km. The input files and code were hashed into the run record before the run.

Let $K_b$ be the matrix that smooths values on the sites with a normalised Gaussian of width ${\sigma }_b$, where ${\sigma }_1$ to ${\sigma }_5$ are 55, 95, 160, 270 and 460 km. Set $K_0$ to the identity and $K_6$ to zero. The space bandDefinition: space band signal processing on irregular sites, our usage The part of a field that varies between two spatial scales, taken as the difference of two Gaussian smoothings of the site values with the two scales as widths. The six bands add back to the field. Excludes an exact wavenumber, because the bands overlap and the sites are irregular. full definition $b$ of a field $x$ is

for $b$ from 0 to 5, so band 0 holds what varies below about 55 km and band 5 what varies above about 460 km, and the six bands add back to $x$. Each band is then split into five time bandsDefinition: time band spectral analysis The part of a yearly series whose period lies between two limits, kept by masking its Fourier transform. Excludes a moving average or a fitted trend. full definition by masks on the Fourier transform in time with edges at 1/32, 1/16, 1/8 and 1/4 cycles per year: periods over 32 years, 16 to 32, 8 to 16, 4 to 8 and 2 to 4 years. Each masked series is kept as a complex analytic signal.

For space band $b$ and time band $w$, let $a_x$ and $a_y$ be the analytic signals of drought at the tree sites and of the trees. Summing over sites and years, with $S_{\mathrm{xy}}$ the sum of $a_x^{*}{a}_y$ and $S_{\mathrm{xx}}$, $S_{\mathrm{yy}}$ the sums of squared magnitudes, the passed share is the squared coherenceDefinition: squared coherence spectral analysis The share of one series' variance in a frequency band that a linear filter of the other series can account for, between 0 and 1. Here pooled over sites and years in each space and time band and called the passed share. Excludes a nonlinear dependence, and the size of the response, which the gain measures. full definition

The preregistered contrast $C$ is the mean of ${\gamma }^2$ over the four broad, slow cells (space bands 4 and 5, periods over 16 years) minus its mean over the four fine, fast cells (space bands 0 and 1, periods under 8 years):

where $L$ and $F$ name those two sets of cells.

Let $\rho (h,u)$ be the correlation between sites at distance $h$ and time lag $u$, pooled over pairs in 100 km bins to 800 km and lags of 1, 2 and 3 years. A separable covarianceDefinition: separable covariance space-time statistics A covariance that factors into a function of distance times a function of time lag, so the spatial pattern of correlation has the same shape at every lag. Excludes fields whose anomalies travel or whose broad and fine patterns fade at different rates. full definition has $\rho (h,u)=g_u\rho (h,0)$. With $g_u$ fitted by least squares, the statistic is the sum of squared departures

For site pairs within 1,200 km, the lag-one cross-correlation $s_{\mathrm{ij}}$ between site $i$ in one year and site $j$ in the next is pooled in 40 km offset cells and fitted as

where $d_{\mathrm{ij}}$ is the offset between the sites, ${\rho }_0$ the lag-zero correlation profile by distance, $c$ a scale and $v$ a velocity in km per year.

The decay rate of a pattern with wavenumber $k$ is modelled as $a{k}^{\beta }$, with power per log wavenumber proportional to $k^{-q}$ and white site noise. Each interior space band's autocovariance at lags 0 to 8 is predicted from the measured transfer of plane waves through that band on these exact sites and from the bias that linear detrending adds, and the four parameters are fitted to bands 1 to 4. The dispersion exponentDefinition: dispersion exponent physics of waves and diffusion The power beta in a decay rate that grows as the wavenumber to the power beta. Diffusion gives about 2, and fine and broad patterns that fade at the same rate give 0. Excludes a speed of travel, which the propagation test measures. full definition $\beta$ is the result. Beside it, each band's e-folding time is reported.

Each control was drawn 200 times, or 300 times for the separable surrogate. A test's p value is the larger of its scoring controls' p values from drawsDefinition: p value (from draws) statistics The share of control draws scoring at or above the real score, with one added to the count and to the number of draws, so it is never zero. Excludes a p value from a textbook formula that assumes a distribution. full definition,

with $k$ the draws at or above the real value and $n$ the draws. The seven real-data tests were adjusted together with the Benjamini-Hochberg correctionDefinition: Benjamini-Hochberg correction statistics, multiple testing A rule that adjusts a set of p values so that, among the tests called significant, the expected share of false calls stays at or below a chosen rate q, here 0.05. Excludes the stricter Bonferroni rule, which bounds the chance of any false call. full definition at q 0.05. Every 95% intervalDefinition: 95% interval statistics The range that holds the middle 95 percent of the bootstrap or control draws for a number. We report it beside the number. Excludes a statement that the true value lies inside with 95 percent probability. full definition comes from a moving-block bootstrapDefinition: moving-block bootstrap statistics An interval made by recomputing a statistic many times on years resampled in runs of consecutive years, here 10, so that year-to-year correlation is kept inside each run. Excludes resampling single years as if they were independent. full definition in time with blocks of 10 years.

A planted checkDefinition: planted check research method; called a positive control in experimental science Data built so that the right answer is known, run through the same analysis before the real data, to show that the analysis can see the effect when it is there. Excludes any check made on the real data, which cannot show what the analysis misses. full definition ran in the same job before the real data were scored. Synthetic fields were built on the drought cell positions with 123 years, from 800 random plane waves plus white noise: a separable field, a wave field carried at 120 km per year toward 30 degrees, a diffusing field and pure noise. Five realisations of each were scored, and the instrument passed every rule. The separable field read as separable (median p 0.37 against a rule of at least 0.05), the planted wave's speed came back within 2.2 percent and its direction within 0.66 degrees, and the diffusion exponent read 1.88 against a rule of 1.7 to 2.3.c6 On the separable field the exponent read -0.01 and on the wave 0.81. Those are the readings a real exponent is compared against. The same checks on the tree positions with 78 years were reported and not scored; there the diffusion exponent read 1.64, below the scoring range.

The real statistics faced these controls, each drawn many times to give a null distributionDefinition: null distribution statistics The spread of scores a control gives over many random draws. Its 95th percentile is the bar a real score must clear. Excludes a theoretical distribution assumed rather than drawn from the data. full definition. Separability: a fitted separable field with the same lag-zero spatial covariance and pooled time autocovariance, and shuffled site coordinates. Travel: the lag-one matrix with its symmetric part kept and its antisymmetric part permuted across pairs in 100 km distance bins, and shuffled coordinates. Dispersion: shuffled coordinates and phase randomisationDefinition: phase randomisation time-series statistics, surrogate data A copy of a series with the same power spectrum and random Fourier phases, so it keeps the series' autocorrelation and loses its timing against any other series. Excludes shuffling years, which also destroys the autocorrelation. full definition of each site. The tree filter: phase-randomised trees, and tree series reassigned to shuffled sites. The last control keeps every tree's own spectrum and breaks only the link to its place.

The preregistered contrast between broad, slow scales and fine, fast scales was 0.64 (95% interval 0.62 to 0.68), against 95th percentiles of 0.22 for phase-randomised trees and 0.30 for trees moved to shuffled sites, with an adjusted p of 0.035.c1 Figure 1 sets the contrast against both controls.

Figure 2 gives the passed share in all 30 cells. The share rises with spatial scale in every time band. Below 55 km the trees pass at most 0.004 of the drought signal in every time band, and above 460 km they pass 0.27 to 0.91.c2 Between 95 and 160 km they pass 0.07 to 0.11, and between 270 and 460 km 0.13 to 0.53. Time scale matters most at the broad end: above 460 km the share is 0.91 at periods over 32 years and 0.27 at 2 to 4 years. The phase-randomised control's mean share is at most 0.18, in that broadest and slowest cell.

The trees' own variance sits where their filter is closed. Figure 3 shows that 0.61 of the tree variance lies in the finest space band, against 0.12 for the drought index at the same sites.

Table 1 gives all seven real-data tests. The drought field's covariance did not depart from a space part times a time partc3: its departure $D$ was 0.0011 against a separable-surrogate 95th percentile of 0.0085 (p 0.72). The trees came back the same way (p 0.35). Both fields beat shuffled coordinates, which only shows that nearby sites are alike.

The drought fit for travel pointed 34.5 degrees north of east at 64.5 km per year and beat the antisymmetric shuffle (95th percentile 32 km per year). It did not beat shuffled coordinates, whose 95th percentile was 467 km per year, and no travel speed could be claimedc4: the bootstrap interval of the speed along the fitted direction ran from -41 to 242 km per year and the adjusted p was 0.66.

Broad and fine patterns lose their memory at the same rate, with a dispersion exponent of 0.02 (-0.12 to 0.17)c5. The e-folding times of the four interior bands were 0.76 to 0.83 years. Neither 1 nor 2 lies inside the interval, and the reading sits at the instrument's separable value, not at its wave or diffusion value. The trees' dispersion fit stopped at its lower bound of -1 and says nothing.

Table 2 sets the forecasts against the outcomes.

The tree-ring network behaves as a spatial filter. It passes most of the drought signal at scales of several hundred km and above and almost none below 55 km. A site-by-site comparison of rings and local drought averages over that whole range and reports one number, so it reads as a weak recorder. In our findings ledgerDefinition: findings ledger our records The append-only, dated list of every result we have recorded, numbered F1, F2 and on. Corrections are added as new dated lines and old entries are never rewritten. Excludes a curated list of successes. full definition, a site-by-site comparison in time only, run after the preregistered test as exploration, gave pooled coherence of 0.24, 0.18, 0.19, 0.20 and 0.07 across the five time bands: weak and nearly flat in period. The space-time view places the weakness in space.

The result agrees with two lines of earlier work. Cook and colleagues [1] found that chronologies within a radius of about 450 km predict a drought grid point best, which is the scale at which our filter opens. Duchesne and colleagues [6] found that averaging over stands removes tree-level noise and that the shared signal falls with distance. Our grid puts both on one map of scale and period, with a number and an interval in each cell.

The three nulls on the drought field matter in their own right. If the field had a travelling front or patterns that faded at scale-dependent rates, separate views in space and in time would miss them. Here they did not: at one-degree cells and yearly steps, the separate views lose nothing a coupled view could find. The coupling that matters lies in the tree filter. Each null is an honest nullDefinition: honest null research method (our usage) A working test that found no effect beyond a stated bound, a forecast that failed, or an error of our own found and fixed, reported with the same weight as a positive result. Excludes a void run, and a result left unpublished because it was negative. full definition from a working instrument, since the planted wave and the planted diffusion were each recovered.

Part of the filter may belong to the reference grid. Harris and colleagues [3] report that CRU interpolates precipitation with a correlation decay distance of about 450 km, close to where the filter opens, so a smooth reference would itself pass little fine-scale signal to agree with. The next study needs station data or a denser grid as the reference.