The runestones were raised by the water of their time – 64 per cent within 500 metres of the water of AD 1050, against 46 per cent of the land around them

Gunnar Ekholm (1950) asked where the runestones were raised. Torun Zachrisson (1998) has linked them to farm, boundary and grave field, Lydia Klos (2009) has argued that most were raised at grave fields, and Shane McLeod (2014) has objected that this overstates the grave fields at the expense of other places, such as the roads. The bridge stones, which tell of bridges and causeways over water and wet ground, show that water belonged to the stones' setting.

Measuring the nearness to water is hard. The land has risen since the Viking Age, in the Mälaren valley by about five metres, so a stone that stood by a bay may now stand on dry land, and a stone by today's shore may have stood far from the water. SGU's shore displacement model (SGU 2022, after Påsse & Daniels 2015) gives an open model for the whole country of where the water lay at different times.

We know of no study that has measured the distance to the water of the stones' own time for the whole Swedish material, with a model of shore displacement, only stones that still stand in place, and a comparison with the land in the same district. The aim is to do so and test whether the association remains when nearness to the grave fields is held equal.

The hypotheses, measures and rules of interpretation were written down in an analysis plan on 23 October 2026, before any distance had been computed. The plan has the SHA-256 checksum 4ddb7d6cecb470a1… and was committed to version control the same evening. Before the lock, only the data sources had been fetched, the layers listed, the number of stones per province counted and the mid-years of the style groups compiled to choose the year (median 1035). A test run of the script the same evening, after the plan had its checksum but before the commit, printed only its progress and no figures. The statistics were first run on 31 October.

The direction was expected. The project's earlier measure, with a coarse land uplift and with moved stones, had already shown that the stones stand closer to water and grave fields than the land around them. The study therefore tests whether a known pattern holds with better data, not a new one.

The plan's hypotheses, verbatim (translated): H1 (primary): the stones stand closer to the water of AD 1050 than the random points. H2: the nearness is clearer relative to the water of the time than to today's. H3: the grave fields, on the same selection (the stones stand closer to registered grave fields than the random points). H1 holds if five conditions are met: p below 0.05 and AUC above 0.5 overall, the same in both halves, the same when stones within 200 metres of each other are merged into one place, the same for four other years within 200 years (750, 950, 1150 and 1250), and the same with only the sea, without the model's lakes. H3 holds if it holds overall, in both halves and for the places. H2 is called clearer only if the whole 95 % interval for the difference lies above zero.

Two addenda were locked later, both before they were computed. On 31 October (SHA-256 dc63b72b3147f9c2…), before the statistics were run: H1 and H3 are also tested on the stones that stand in place according to three sources, because the rule for moved stones had been corrected after the plan was locked. On 1 November (SHA-256 2993273eff5006c6…), when the plan's results were known: nine fixed sensitivity analyses for distances, corresponding to those the project makes for other findings. They are confirmatory in the sense that the steps and limits were locked before they were computed, but they were chosen after the main result was known.

• The stones. Swedish Viking-Age runestones in the Scandinavian Runic-text Database (Rundata), in the version locked in the plan. Of them we took those that have coordinates and have not been moved: Rundata's placement or site is not church, churchyard, wall, museum or collection. Only provinces with at least 20 such stones are tested: Uppland (623), Södermanland (234), Östergötland (89), Småland (69), Västergötland (64) and Öland (20), 1,099 in all. The 45 stones in the other provinces are not tested. Runestones only.
• The water. SGU's shore displacement model, open data version 1.0 (2022, CC BY 4.0). It is built on shore displacement curves across the country, a model of land uplift and an elevation model with a resolution of 50 metres. The water of AD 1050 is the model's sea and lakes in the slice 900 years before 1950; today's water is the slice for 1949. The distances are computed on a grid of 25-metre cells.
• The grave fields. The Swedish National Heritage Board's register of ancient monuments via K-samsök, retrieved on 12 October 2026: 18,633 registered grave fields with coordinates.
• The independent elevation model. Copernicus DEM GLO-30 (about 30 metres), with the province's land uplift since the Viking Age as the height limit.
• The dating by the stone's ornamental style (Gräslund 2006): 587 of the stones.
• The halves are the project's fixed division parish by parish: 543 and 556 stones.

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4.1 The measure and the comparison. For every stone and every point the distance is computed to the nearest water of AD 1050, to today's water and to the nearest grave field; a point in the water gets the distance zero. The comparison is a null model: for each province ten random points per stone are drawn, uniformly within five kilometres of the province's stones, and only points that are land both in AD 1050 and in 1949 are kept. The points thus lie in the districts where stones were raised, so the test concerns where in the district the stone stands, not where the districts lie. Stones and points are compared with Wilcoxon's rank-sum test (Wilcoxon 1945; Mann & Whitney 1947), one-sided. The effect is given as the medians, the share within 500 metres and the AUC, the probability that a randomly chosen stone stands closer than a randomly chosen point (0.5 means no difference). Figure 2 shows the measurement in one district.

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4.2 Correction and replication. Each hypothesis is tested in each half on its own. The provinces are tested separately with Benjamini–Hochberg correction (Benjamini & Hochberg 1995) over the six, and the secondary tests likewise over the five. The difference between the water of the time and today's (H2) gets a 95 % interval from 2,000 bootstrap samples in which the stones are drawn place by place.

4.3 The sensitivity analyses. The plan's seven tests, the addendum's test and the nine steps of the addendum plan are given with their results in 5.2 and 5.3. The parish as a random effect is a logistic regression (Bates et al. 2015) in which the points get the parish of the nearest stone. When stones are removed, the points that belong to them are removed too. A step holds when the AUC is above 0.5 and p below 0.05; without one province at a time, significance is required only where at least two thirds of the stones remain.

4.4 Deviations from the plan. The statistics were run eight days after the distances were computed, after the addendum for moved stones; the plan's tests are unchanged. The 15,590 distances that reached the edge of the computation tiles, nearly all for the sea alone far inland, were recomputed on a grid of 100-metre cells over the whole province; the plan gives no cell size. The sensitivity analyses of the addendum plan were added after the plan's results were known (see 2). Otherwise none.

5.1 The water of AD 1050 (H1), the grave fields (H3) and today's water (H2). Of the 1,099 stones, 64 per cent stand within 500 metres of the water of AD 1050, against 46 per cent of the 10,990 points; within one kilometre 83 against 76 per cent. The median distance is 305 against 555 metres, AUC 0.629, p < 0.0001. Half A (543 stones in 190 parishes): AUC 0.639, p < 0.0001. Half B (556 stones in 187 parishes): AUC 0.619, p < 0.0001. With 935 places instead of stones: AUC 0.626, p < 0.0001. H1 holds on all five conditions. In plain words: compare a runestone with a randomly chosen place in the same district, and the runestone stands closer to the water of its time in 63 cases out of 100; almost two in three stones stand within 500 metres of the water, against fewer than half of the land.

By province, with q corrected over the six: AUC 0.63 in Uppland, 0.67 in Södermanland, 0.59 in Östergötland, 0.59 in Småland and 0.61 in Västergötland, all q < 0.01, and 0.61 on Öland, q = 0.057, where the 20 stones go the same way but are not enough.

The grave fields (H3): 67 per cent of the stones stand within 500 metres of a registered grave field, against 24 per cent of the points; the median is 304 against 1,045 metres, AUC 0.784, p < 0.0001. Half A: AUC 0.790, half B: AUC 0.778, the places: AUC 0.771, all p < 0.0001. H3 holds, in the same five provinces as H1. In plain words: two in three stones stand within 500 metres of a grave field, against one in four places in the district.

Today's water (H2): relative to the water of 1949, 60 per cent of the stones stand within 500 metres, against 42 per cent of the points, AUC 0.620. The difference in AUC from the water of the time is 0.009, 95 % interval 0.002 to 0.017. The whole interval lies above zero, so by the plan the nearness is clearer relative to the water of the time, but the difference is small. It shows in Uppland, Södermanland, Östergötland and on Öland; in Småland and Västergötland today's water fits as well or slightly better.

5.2 The plan's sensitivity tests. All hold for H1:

• Other years: AUC between 0.62 and 0.64 for all six, from AD 550 to AD 1550, all p < 0.0001.
• Only the sea, without the model's lakes: 1,099 stones, median 2,975 against 4,496 m, AUC 0.57, p < 0.0001. In Östergötland, Småland and Västergötland the test says nothing, since most stones and points there lie tens of kilometres from the sea.
• Without the 108 stones that stand in the model's water of AD 1050: 991 stones, median 372 against 555 m, AUC 0.59, p < 0.0001.
• The independent elevation model: land uplift one metre lower: median 335 against 570 m, AUC 0.63, p < 0.0001; equal: median 301 against 560 m, AUC 0.64, p < 0.0001; one metre higher: median 285 against 552 m, AUC 0.65, p < 0.0001. The pattern thus does not depend on SGU's elevation model.
• Points within two kilometres: AUC 0.628; within ten kilometres: AUC 0.627.

Water and grave fields together, in a logistic regression with the province held constant: when the distance to the water halves, the odds that the place has a runestone become 1.72 times as large (95 % interval 1.58–1.87), with the distance to the grave fields equal; for the grave fields the odds ratio is 3.46 (3.18–3.76). Both hold in both halves and after correction. In plain words: the water is not merely a sign that the grave fields lie there, nor the other way round; the grave fields weigh about twice as much.

The other secondary tests do not hold: the share within one kilometre of the water is the same for early and late stones (84 per cent), Christian markers make no difference (p = 0.47), and the bridge and road stones stand somewhat closer (207 against 309 metres in median) but not reliably (p = 0.059, q = 0.10).

5.3 The sensitivity analyses. Only stones that stand in place according to three sources (657 stones): water AUC 0.629, grave fields AUC 0.786, both p < 0.0001. In the nine analyses of the addendum plan every step holds for both the water and the grave fields. For the water:

• The halves: see 5.1.
• Without one province at a time: AUC between 0.62 and 0.63, also without Uppland, when only 43 per cent of the stones remain (p < 0.0001).
• The parish as the unit (373 parishes): 373 stones, median 336 against 555 m, AUC 0.61, p < 0.0001.
• The parish as a random effect: odds ratio 1.82 when the distance halves (95 % interval 1.68–1.98), p < 0.0001; the model converged without warnings. In plain words: also compared with the land in the same parish, the stones stand closer to the water.
• One stone per certain carver (67 carvers): 767 stones, median 313 against 570 m, AUC 0.62, p < 0.0001.
• Without re-erected stones (101 removed): 998 stones, median 307 against 559 m, AUC 0.63, p < 0.0001.
• Without stones that Runor gives as moved (177 removed): 922 stones, median 305 against 553 m, AUC 0.63, p < 0.0001.
• With today's rule for moved stones (295 removed): 804 stones, median 305 against 562 m, AUC 0.63, p < 0.0001.
• The early stones, before AD 1050: 335 stones, median 276 against 575 m, AUC 0.65, p < 0.0001. The late: 252 stones, median 336 against 534 m, AUC 0.62, p < 0.0001.

For the grave fields the AUC lies between 0.72 and 0.80 in every step, without one province at a time between 0.74 and 0.79, and the parish as a random effect gives the odds ratio 3.57 (3.27–3.89), p < 0.0001.

The study stands alone, so the correction applies within the plan and not among the associations the project adjusts jointly. No hand-coded traits are involved; the source of error is the model of the water (see 6).

What the finding says. Those who raised the runestones chose places near the water. The comparison concerns the land in the same district, and the association remains parish by parish, so it is not only because the districts lay by water. Nor does it depend on a few provinces, places or carvers, on stones that have been moved or re-erected, or on SGU's elevation model. It held both before and after AD 1050. The grave fields draw more strongly, and the two associations are independent of each other: the most likely places for a runestone lay near both water and grave fields.

The water of the time and today's. That the water of the time fits only slightly better than today's does not mean that the land uplift is unimportant. In most districts the shore has moved less than the hundreds of metres that separate the stones from the points (figure 2), and the lakes remain. The model for AD 1050 is needed above all so that the finding is not an artefact of today's coast.

Against earlier research. That two in three stones stand within 500 metres of a grave field supports Klos (2009) and agrees with stones in family groups standing at grave fields more often than single stones (Kvant 2026b). That the water remains when the grave fields are held equal at the same time bears out McLeod (2014) that the grave fields are not the whole answer. The finding fits with the stones' text being turned mostly towards the road (Kvant 2026c): the stones were raised where people travelled, by land and by water. That is an interpretation; the study shows where the stones stand, not why.

Weaknesses, in order of importance.

1. The model of the water is coarse. SGU's model is regional, with an elevation model of 50 metres, time errors of up to 500 years and approximate lakes; rivers, streams and wetlands are missing. 108 stones fall in the model's water of AD 1050, where no stone can have been raised. Without them the association is weaker but clear (AUC 0.59), and the independent elevation model gives the same result.
2. The direction was expected. The project's earlier measure showed the same with coarser data. The study confirms a known pattern; it discovers nothing new.
3. The sensitivity analyses were chosen after the main result. The steps and limits were locked before they were computed, but after the plan's results were known.
4. The stones' places. A stone may have been moved a little without it being recorded. Three sources for moved stones and a word list for re-erected stones give the same result, but do not catch everything.
5. The grave fields are unevenly surveyed in different districts and mostly undated; some may be younger or older than the stones.
6. Öland has too few stones to show anything for certain.

What would overturn the finding. If the association with water disappeared with a finer elevation model, for example Lantmäteriet's at one metre's resolution, and shore displacement curves for each district, or if the stones' original places on the oldest maps lay systematically farther from the water.

The runestones were raised by the water of their time: almost two in three stand within 500 metres of the water as it lay in AD 1050, against fewer than half of the land around them, and the association withstands every sensitivity analysis. The grave fields drew even more strongly, and the two associations are independent. The next step is a finer elevation model and the oldest maps.