F O Marques
email: fomarques@fc.ul.pt

A Hildenbrand

H Zeyen

C Cunha

S S Victória

The complex vertical motion of intraplate oceanic islands assessed in Santiago Island, Cape Verde

Keywords: uplift and subsidence rates of oceanic islands, vertical displacement in Santiago Island, Cape Verde, sea level changes, paleoshorelines, passage zones, topography response to isostasy

Dated paleo-sea level markers and eustatic sea level changes are necessary but not sufficient information to calculate vertical motion rates on oceanic islands. Therefore, we use a procedure in which we work progressively back in time to incorporate the more recent vertical motion rates implied by the youngest paleoshorelines into the vertical motion history of all older shorelines. Specifically, we calculate the time-averaged vertical motion rates required to explain the presentday elevations of the dated sequence of paleoshorelines on Santiago volcanic island (Cape Verde).

We thus obtain a vertical motion history consisting of time-averaged vertical motion rates spanning the five intervening periods between paleoshoreline formation and the present day: (1) 5.06 to 3.29 Ma -seamount growth or island subsidence because all the rocks in this period are submarine; (2) fast uplift (ca. 0.96 mm/a) from 3.29 to 2.87 Ma, mostly responsible for putting submarine lavas currently close to 410 m altitude; (3) relatively fast subsidence (ca. -0.11 mm/a) between 2.87 and 2.18 Ma; (4) stagnation from 2.18 to 0.811 Ma; (5) relatively fast uplift (ca. 0.14 mm/a) between 0.811 and 0 Ma. We numerically tested top-down (volcanic loading) and bottom-up (lithosphere thinning, underplating and mantle plume) mechanisms to explain the inferred vertical movements, and we conclude that volcanic loading and crustal underplating are capable of producing the observed subsidence and uplift, respectively.

Plain Language Summary

Oceanic volcanic islands experience vertical movements during their lifetime, which can be the consequence of several mechanisms like bending of the lithosphere due to the island's weight (subsidence), or horizontal intrusion of magma below the volcanic edifice (uplift). In order to discriminate among the possible mechanisms, we must correctly estimate the successive vertical displacements and motion rates. We account for eustatic sea level change and use dated paleoshorelines, working progressively back in time to incorporate the more recent vertical motion rates implied by the youngest paleoshorelines into the vertical motion history of all older shorelines, to obtain a step-by-step vertical motion history:: (1) 5.06 to 3.29 Ma -seamount growth or island subsidence because all the rocks in this period of time are submarine; (2) fast uplift (ca. 0.96 mm/a) from 3.29 to 2.87 Ma mostly responsible for putting submarine lavas currently close to 410 m altitude; (3) relatively fast subsidence (ca. -0.11 mm/a) between 2.87 and 2.18 Ma; (4) stagnation from 2.18 to 0.811 Ma; (5) relatively fast uplift (ca. 0.14 mm/a) between 0.811 and 0 Ma. From analytical and numerical modeling we conclude that volcanic loading and crustal underplating are likely mechanisms to explain the observed vertical displacements.

Introduction

The vertical motion of oceanic islands records the interactions between volcanic load, mantle flow and lithospheric reaction to both. An important step toward understanding the mechanisms responsible for the vertical motions in volcanic ocean islands is therefore the estimation of rates of vertical displacement. When one finds a submarine lava flow currently at 400 m altitude on an island, it means that the island was uplifted, because no specific eustatic sea level change can, by itself, explain such position. The problem is that we do not know when and how the uplift took place. As an end member, one can assume that the uplift was linear from the time of formation of the rock to present day (e.g. Ramalho et al., 2010a, c). If the rock is 4 Ma old, then 400 000 mm divided by 4,000,000 a equals 0.1 mm/a. But this assumption can be grossly wrong if geological evidence indicates that the uplift is much younger or in a much shorter period of time. If the rock is 4 Ma but uplift occurred only in the last 1Ma, then the uplift rate is 0.4 mm/a, i.e. 4 times greater than the linear estimate. If uplift occurred at 4 Ma but over a short period of time of 0.4 Ma, then the uplift rate would be 1 mm/a, i.e. one order of magnitude greater than the linear estimate. Finally, we also do not know how the submarine rock currently observed at 400 m altitude got there: was it by jerks? Or was it intercalated with periods of subsidence and/or quiescence? The unravelling of these problems is the main objective of this work. In the end, we use analytical solutions and numerical modeling to find likely mechanisms capable of explaining such vertical displacements, based on the case of Santiago Island in Cape Verde.

The calculation of vertical motion rates in an oceanic island depends critically on two parameters: position of a paleosealevel marker at a given time. Therefore, determining the current elevation as well as the age and depth of formation of a paleoshoreline indicator are required to calculate vertical motion rates. To accomplish this fundamental objective, we looked for field evidence that allows us to determine the position of a given rock in space and time: lavas in a passage zone (transition from subaerial to submarine flow defining the paleoshoreline), preferably with in situ shallow water fossils, and which are most suitable for K-Ar dating.

Paleoshorelines record relative sea level, therefore their elevation above or below current sea level results from changes in both land and/or glacio-eustatic sea level, which has changed over time (e.g. [START_REF] Bintanja | Modelled atmospheric temperatures and global sea levels over the past million years[END_REF][START_REF] De Boer | Cenozoic global ice-volume and temperature simulations with 1-D ice-sheet models forced by benthic δ18O records[END_REF][START_REF] Hansen | Climate sensitivity, sea level and atmospheric carbon dioxide[END_REF][START_REF] Miller | The Phanerozoic record of global sealevel change[END_REF][START_REF] Miller | A 180-million-year record of sea level and ice volume variations from continental margin and deep-sea isotopic records[END_REF].

Therefore, when analyzing vertical motions in a volcanic island, the eustatic variations of sea level must be considered.

The use of paleoshorelines and glacio-eustatic sea level curves has been used in previous works, as the methodology to assess an island's vertical motion rate (e.g. [START_REF] Cas | Volcanic successions. modern and ancient: a geological approach to processes, products and successions[END_REF][START_REF] Jones | Pillow lavas as depth indicators[END_REF][START_REF] Jones | The flow of basalt lava from air into water, its structural expression and stratigraphic significance[END_REF][START_REF] Porebski | Lava-fed Gilbert-type delta in the Polonez Cove Formation (Lower Oligocene), King George Island, West Antarctica[END_REF][START_REF] Ramalho | Coastal evolution on volcanic oceanic islands: a complex interplay between volcanism, erosion, sedimentation, sealevel change and biogenic production[END_REF]. The Cape Verde Archipelago comprises ten islands located about 700 km west of Dakar in Senegal (Fig. 1). They lie on Jurassic-Cretaceous seafloor, which is the basement above which the Cape Verde Rise and islands rise 2 to 8 km, respectively. Santiago Island, the target of the study reported here, is one of the islands where evidence for large vertical motion can be found. Previous work (Ramalho et al. 2010a, c) deduced a linear relationship between the vertical displacement and age of paleoshorelines on the island, implying a constant uplift rate over the past ~4 Ma. Such a relation seems incompatible with the isostatic response of the lithosphere to the complex evolution of an ocean volcanic island, which encompasses major episodes of construction and erosion/destruction (top-down forcing) as well as mantle/crustal processes like crustal underplating (bottom-up forcing).

The main objective was to introduce a new methodology to estimate rates of vertical motion at time intervals between paleoshoreline formation, and discuss possible mechanisms responsible for vertical displacement. To accomplish these objectives, we used the following methodology: analysis of aerial imagery for geomorphological and geological interpretation; fieldwork to recognize the major unconformities and volcanic units, to unravel the geometry and position of main volcanoes, and to find the critical paleodepth markers (paleoshorelines); sampling for highprecision K-Ar dating, used to calibrate the volcanic stratigraphy and to obtain the ages of the critical paleodepth markers.

Geological setting

Santiago is the largest of the ten inhabited islands in Cape Verde. It is elongated NNW-SSE, it shows rugged topography with large (km wide) and deep (up to 1 km) canyons (Fig. 2A), and peaks at 1394 m at Pico da Antónia, in the center-south of the island. The northern two thirds of the island are cut by deep valleys, but, in places (e.g. west of Assomada village, center of the island), rugged paleotopographies can still be recognized comprising major unconformities filled with younger lavas (Fig. 3). [START_REF] Serralheiro | A Geologia da ilha de Santiago (Cabo Verde)[END_REF] with simplified volcanic stratigraphy, sample locations (green stars), sample references, and ages obtained in the present work.

The volcanic stratigraphy of the island, based on unconformity bounded stratigraphic units, is well established [START_REF] Serralheiro | A Geologia da ilha de Santiago (Cabo Verde)[END_REF]. (Fig. 2). However, the geological map shows important inconsistencies, as suggested previously by [START_REF] Holm | An 40 Ar-39 Ar study of the Cape Verde hot spot: Temporal evolution in a semistationary plate environment[END_REF] and indicated here for, at least, the Porto Rincão and Águas Belas creeks. The volcanic stratigraphy of the island comprises five main volcanic complexes [START_REF] Serralheiro | A Geologia da ilha de Santiago (Cabo Verde)[END_REF], the local names of which have no meaning to the common reader. Therefore, we introduce here more intuitive names for the different volcanic complexes, which are from bottom to top (Fig. 2B):

(1) the Basement Complex (former Complexo Antigo of [START_REF] Serralheiro | A Geologia da ilha de Santiago (Cabo Verde)[END_REF] -deeply eroded unit mostly composed of intensely altered igneous rocks pervasively intruded by dykes and small pockets of plutonic rocks. The Basement Complex may represent an early submarine edifice [START_REF] Serralheiro | A Geologia da ilha de Santiago (Cabo Verde)[END_REF], but its original setting (submarine or sub-aerial) is still disputed. Biotite from a foidal gabbro, and biotite/phlogopite from carbonatites in the Basement Complex were dated at ca. 10 Ma [START_REF] Bernard-Griffiths | Geochronologie: Donnés radiometriques potassium/argon sur quelques formations magmatiques des îlles de l'archipel du Cap Vert[END_REF], which can be regarded as the minimum age for the Basement Complex because the dated rocks occur high in this unit.

(2) Lower Volcanic Complex (former Flamengos Formation) -this unit lies on the Basement Complex through a major unconformity, and is, according to current knowledge, exclusively composed of submarine lavas that outcrop from current sea level to an altitude of c. 410 m.

According to [START_REF] Holm | An 40 Ar-39 Ar study of the Cape Verde hot spot: Temporal evolution in a semistationary plate environment[END_REF], this major submarine complex formed in a few hundred thousand years around 4.6 Ma.

(3) Intermediate Volcanic Complex (former Pico da Antónia Formation) -this unit overlies the Basement Complex and the Lower Volcanic Complex through a major unconformity, in places with a thick conglomerate, and is composed of subaerial flows that pass into submarine flows close to the current coast. This unit outcrops from current sea level to an altitude of c. 1394 m at Pico da Antónia, and was dated in the range 3.3 to 2.2 Ma [START_REF] Holm | An 40 Ar-39 Ar study of the Cape Verde hot spot: Temporal evolution in a semistationary plate environment[END_REF].

(4) Upper Volcanic Complex (former Assomada Formation) -this unit is mostly composed of thick subaerial lava flows unconformably overlying all previous units. The volcanic products of this unit fill valleys deeply carved in the Lower and Intermediate Volcanic Complexes, in places putting the Upper Volcanic Complex directly in contact with the Basement Complex (e.g. Águas Belas Creek). According to all previous work (e.g. [START_REF] Serralheiro | A Geologia da ilha de Santiago (Cabo Verde)[END_REF][START_REF] Holm | An 40 Ar-39 Ar study of the Cape Verde hot spot: Temporal evolution in a semistationary plate environment[END_REF]Ramalho et al., 2010a, c), the lava flows of this later volcanic period apparently did not reach the sea (no pillow lavas recognized in this unit), as can be confirmed on the geological map [START_REF] Serralheiro | A Geologia da ilha de Santiago (Cabo Verde)[END_REF].

(5) Young Volcanic Complex (former Monte das Vacas Formation of [START_REF] Serralheiro | A Geologia da ilha de Santiago (Cabo Verde)[END_REF]) -this unit is made of Strombolian cones lying unconformably on all previous units, and it represents the latest episode of volcanism. [START_REF] Holm | An 40 Ar-39 Ar study of the Cape Verde hot spot: Temporal evolution in a semistationary plate environment[END_REF] considered these two younger units (UVC and YVC) together as "late volcanism", for which they attributed an age between 1.1 and 0.7 Ma.

Field data

Field work carried out in the critical areas of the Flamengos (east Santiago), Porto Rincão and Águas Belas (west Santiago) creeks revealed the existence of major unconformities, which helped us better constrain the volcanic stratigraphy. We also found the critical outcrops comprising passage zones, locally with in situ shallow water fossils (mouth of the Porto Rincão Creek), with very fresh rock appropriate for reliable and precise dating by K-Ar. The main geology in the deep Águas Belas valley (Fig. 3) shows a major unconformity, filled with a very thick conglomerate, between the Intermediate Volcanic Complex and the underlying Lower Volcanic Complex. This is not consistent with the geological map where the underlying lavas are mapped as the submarine equivalent of the overlying Intermediate Volcanic Complex.

The critical outcrops at the mouth of the Porto Rincão Creek show clear examples of passage zones currently at ca. 43 m altitude (Supp. Fig. 1). Further upstream in the creek, we could distinguish a different unit, still made of subaerial lavas passing into submarine, but with many dykes cutting through the unit. The new ages confirmed that it is actually a different unit (cf. Fig. 4).

In the Flamengos valley, east Santiago, opposite to the Águas Belas valley, we can observe two different sequences of submarine lava flows: (1) at the mouth of the Flamengos Creek, a well preserved passage zone, ca. 20 m above current sea level, dated by [START_REF] Holm | An 40 Ar-39 Ar study of the Cape Verde hot spot: Temporal evolution in a semistationary plate environment[END_REF] at ca. 2.85 Ma (their samples 121367 and 121371), so confirming the geological map of [START_REF] Serralheiro | A Geologia da ilha de Santiago (Cabo Verde)[END_REF];

(2) a very thick sequence (Lower Volcanic Complex, former Flamengos Formation) of gently east dipping submarine lava flows and pyroclasts up to ca. 420 m altitude, where we can observe the unconformable contact with the overlying Upper Volcanic Complex (former Assomada Formation).

We could not find a passage zone in this unit, neither here nor in the Águas Belas valley, so we just collected the uppermost submarine rocks just below the conglomerates making up the major unconformity.

K-Ar dating

Sampling strategy

We looked for passage zones and in situ shallow water fossils on pillow lavas (the critical sea level markers) and collected samples in both submarine and subaerial lavas in the Porto Rincão creek. In the absence of these ideal markers, we looked for submarine lavas and collected samples from the base (oldest) and top (youngest) of the lava pile, which was the case in the Águas Belas (western Santiago, immediately north of Porto Rincão creek) and Flamengos (eastern Santiago) creeks.

Sampling was systematically carried out on well-identified lava flows. The central massive part of the lava was extracted by means of hammers and chisels, and broken in-situ with a sledgehammer to reach the freshest, bubble-free core of the flow and reject any vesicle-rich part. We carefully examined the rock in the field and discarded samples showing traces of weathering.

Analytical techniques -K-Ar dating on fresh separated groundmass

Thin sections of all samples were scrutinized to characterize the texture, and ensure the freshness of samples. Most samples are mafic in composition, so the volcanic groundmass was chosen for subsequent geochronological analyses. After crushing and sieving at an adequate grain size (typically 63-125 µm or 125 -250 µm), samples were washed in dilute nitric acid and rinsed with deionized water. Magnetic separator and heavy liquids were systematically used to eliminate phenocrysts (olivine, pyroxene, plagioclase), which may carry unsuitable excess-argon, and to extract the unaltered and homogeneous fraction of the groundmass to obtain a meaningful eruption age. The samples were dated in Laboratory GEOPS (Univ. Paris-Sud, Orsay, France) with the unspiked K-Ar Cassignol-Gillot technique [START_REF] Gillot | The Cassignol technique for potassium-argon dating, precision and accuracy -examples from the Late Pleistocene to recent volcanics from southern Italy[END_REF][START_REF] Gillot | The K-Ar dating method: principle, analytical techniques and application to Holocene volcanic eruptions in Southern Italy[END_REF], which allows the precise determination of small amounts of radiogenic argon ( 40 Ar*) and is particularly appropriate to date young volcanic rocks, including late quaternary low-K basalts and andesites (e.g., [START_REF] Hildenbrand | Evidence for a differentiated ignimbritic activity ending the building-stage of Tahiti-Nui (French Polynesia)[END_REF][START_REF] Hildenbrand | Evidence for a persistent uplifting of La Palma (Canary Islands) inferred from morphological and radiometric data[END_REF][START_REF] Hildenbrand | Volcano-tectonic and geochemical evolution of an oceanic intra-plate volcano: Tahiti-Nui (French Polynesia)[END_REF]2008;2012;2014;2018;[START_REF] Salvany | Geomorphic evolution of the Piton des Neiges volcano (Réunion Island, Indian Ocean): competition between volcanic construction and erosion since 1.4 Ma[END_REF][START_REF] Germa | Volcanic evolution of the back-arc Pleistocene Payun Matru volcanic field (Argentina)[END_REF][START_REF] Germa | The K-Ar Cassignol-Gillot technique applied to western Martinique lavas: a record of Lesser Antilles arc activity from 2 Ma to Mount Pelée volcanism[END_REF][START_REF] Quidelleur | Causal link between Quaternary paleoclimatic changes and volcanic islands evolution[END_REF][START_REF] Boulesteix | Eruptive response of oceanic islands to giant landslides: New insights from the geomorphologic evolution of the Teide-Pico Viejo volcanic complex (Tenerife, Canary)[END_REF]2013, Costa et al., 2014;;2015;[START_REF] Sibrant | Construction and destruction of a volcanic island developed inside an oceanic rift: Graciosa Island, Terceira Rift, Azores[END_REF]Sibrant et al., , 2015a,b;,b;[START_REF] Ricci | Volcanic evolution of central Basse-Terre Island revisited on the basis of new geochronology and geomorphology data[END_REF]Ricci et al., , 2018;;[START_REF] Silva | Palaeomagnetic study of a subaerial volcanic ridge (São Jorge Island, Azores) for the past 1.3 Myr: evidence for the Cobb Mountain Subchron, volcano flank instability and tectonomagmatic implications[END_REF][START_REF] Silva | Volcano-tectonic evolution of a linear volcanic ridge (Pico-Faial Ridge, Azores Triple Junction) assessed by paleomagnetic studies[END_REF][START_REF] Marques | Ground motion and tectonics in the Terceira Island: Tectonomagmatic interactions in an oceanic rift (Terceira Rift, Azores Triple Junction)[END_REF][START_REF] Marques | Evolution of a volcanic island on the shoulder of an oceanic rift and geodynamic implications: S. Jorge Island on the Terceira Rift, Azores Triple Junction[END_REF]Bablon et al., 2018Bablon et al., , 2019)). The technique has even been extended to the last millennium with an uncertainty of only a few centuries in the case of high-K lavas (Quidelleur et al., 2001). In oceanic environments, it has also been successfully applied to date precisely submarine flows of various ages [START_REF] Bonneville | Temporal evolution and geochemical variability of the South Pacific superplume activity[END_REF]Sibrant et al., 2015;Marques et al., 2019).

The analytical data and the new K-Ar ages are presented in Table 1, Supp. Table 1, and Figs. 2 to 5. Decay constants of Steiger and Jäger (1977) were used. Full details on the analytical procedure and performances are given elsewhere [START_REF] Gillot | The K-Ar dating method: principle, analytical techniques and application to Holocene volcanic eruptions in Southern Italy[END_REF]. The relative uncertainty (σ age )

on each individual age determination is obtained as follows: , from the quadratic sum of all three independent sources of uncertainty involved in the calculation: (1) the relative uncertainty on the K-content determination ( ), ( 2 (1) The relative uncertainty on K-content measurement is about 1%, from repeated measurements on standards MDO-G and ISH-G [START_REF] Gillot | Two reference materials, trachytes MDO-G and ISH-G, for argon dating (K-Ar and Ar-40/Ar-39) of Pleistocene and Holocene rocks[END_REF], and BCR-2 (Rackzec et al., 2001).

For a given sample, repeated K measurements (usually twice) on distinct aliquots are achieved

σage=(σ K) 2 +(σ cal) 2 +(σ Ar*) 2 σ K σ cal σ Ar*
until reaching an average value with a relative standard deviation (RSD) better than 1% (see supplementary Table 1).

(2) The calibration of the 40 Ar signal on our mass-spectrometer is obtained by systematic measurements of an air pipette, which is routinely compared to the HD-B1 biotite international standard with its recommended age of 24.21 Ma [START_REF] Hess | Compilation of K-Ar measurements on HD-B1 standard biotite. Phanerozoic Time Scale[END_REF], recently confirmed at 24.18 ± 0.09 Ma [START_REF] Schwarz | Intercalibration of 40Ar-39Ar age standards NL-25, HB3gr hornblende, GA1550, SB-3, HD-B1 biotite and BMus/2 muscovite[END_REF]. The relative uncertainty on the calibration, including the standard uncertainty, is 1%.

(3) The radiogenic argon content (% 40 Ar*) is measured by comparison of the 40 Ar/ 36 Ar ratio of the sample with an air-pipette, measured under strictly similar Ar pressure conditions. This can be achieved because of the very stable analytic conditions of our mass spectrometer. The limit of detection of the radiogenic Ar content is presently of 0.1%, which allows ages as young as 2 ka to be obtained with only a few centuries uncertainty in favorable cases [START_REF] Gillot | The K-Ar dating method: principle, analytical techniques and application to Holocene volcanic eruptions in Southern Italy[END_REF].

For samples with low radiogenic yield (<10%), the uncertainty on the age is dominated by the uncertainty on the atmospheric contamination, and can reach several tens of %. For 40 Ar* higher than 10%, the uncertainty on the correction of atmospheric contamination becomes negligible, and the total age uncertainty rapidly converges towards 1.4% (i.e.

).

When the individual ages overlap within their range of uncertainties, the mean age is calculated by weighting each individual age with the amount of radiogenic argon. This method is conservative, as we apply the typical relative uncertainty on K (1%) and 40 Ar (1%), rather than calculating the age uncertainty from duplicates of each sample, as two or three measurements are not statistically representative. Note that such approach prevents the systematic underestimation of uncertainties when duplicates yield strictly identical values.

Nine independent measurements on HD-B1 standard [START_REF] Hess | Compilation of K-Ar measurements on HD-B1 standard biotite. Phanerozoic Time Scale[END_REF] as an

(1) 2 + (1) 2

unknown in the course of the present study yielded a mean age of 24.18 ± 0.11 Ma (RSD = 0.46%, 1σ), in full agreement with the recommended value of 24.21 Ma [START_REF] Hess | Compilation of K-Ar measurements on HD-B1 standard biotite. Phanerozoic Time Scale[END_REF], ruling

out a significant potential bias linked with calibration of our mass spectrometer. [START_REF] Miller | A 180-million-year record of sea level and ice volume variations from continental margin and deep-sea isotopic records[END_REF].

Supplementary

Rates of vertical motion

Assuming the uncertainties in both sea level and age, it is possible to estimate a mean rate of vertical movement (and its uncertainty) for each period between the age of the sample and presentday, as follows (see Supp. Table 2): (1) From the mean K-Ar age of a sample emplaced at sea level (or close), and the available sea level dataset [START_REF] Miller | A 180-million-year record of sea level and ice volume variations from continental margin and deep-sea isotopic records[END_REF], based on δ 18 O for the period analyzed here), we can estimate maximal, minimal and median sea levels over the period of interest. For instance, our sample ST17X yields a mean age of 811 ± 12 ka, i.e. an age in the interval 799 -823 ka. From the dataset of [START_REF] Miller | A 180-million-year record of sea level and ice volume variations from continental margin and deep-sea isotopic records[END_REF], the median sea level during this period was between -100 m and -45 m, i.e. a sea level value of -72 ± 28 m.

(2) This sea level value is subtracted from the present altitude of the sample to calculate a mean amount of vertical displacement, i.e. uplift (if positive) or subsidence (if negative). In the case of (3) The rate of vertical motion is then simply calculated by dividing the amount of vertical displacement by the mean age of the sample. The relative uncertainty on the rate of vertical motion (in %) is obtained as the sum of the relative uncertainty on the vertical displacement and the relative uncertainty on the age (very small compared to the former). For our sample ST17X, a total relative uncertainty on the rate of uplift of 25% is thus obtained. A (conservative) mean rate of uplift 0.14 ± 0.04 mm/a is thus estimated for the period between 811 ± 12 ka and presentday.

The same approach for each sample gives rates of vertical displacement over different periods and for different sectors of the island (Suppl. Table 2). The rates thus obtained are all positive suggesting at first examination that there has been only uplift during the last 5 Ma.

However, a closer examination shows that the vertical motion history can be more complex. For instance, the rates obtained over the last 5 Ma (samples ST17 I and J) are one order of magnitude lower than rates obtained on the period 3.3 -0 Ma (sample ST17L) in the same sector. This suggests a more complex history, where fast uplift may have been followed/preceded by periods with either no uplift at all, or even subsidence. It is thus necessary to use a step-by-step reconstruction to elucidate the more complex history. The new method introduced here works as follows (Figs. 5 and6): we calculate the original position of the youngest rock unit (UVC, former Assomada), and use it to estimate the vertical displacement and the rate of vertical motion in the most recent period of time. Given that the rock is the youngest, all older rocks must follow the same motion path, which we call the anchor path (Figs. 5 and6). Following this stage, the number of rock units is reduced by one. And then we repeat the process using the second youngest rock unit, until we reach the oldest unit, which must follow all vertical motions of the younger units. More critically, we used paleo-shorelines (inferred from passage zones) at different age and altitude to set anchor points (Fig. 5). Given that all older rocks should show the same rate (assuming only pure vertical motion, i.e no differential movement/tilting), then the calculated path becomes an anchor path, which should be identical to all rocks (cf. blue, black and red dash-dot lines drawn parallel to the anchor green dashed line in Fig. 5). From this step onwards, all rates for older time intervals must be consistent with the anchor points and paths. Given that the effect of eustatic sea level uncertainty (in the considered time periods) on rate calculations is much greater than the effect of age error, we only consider the former in the calculations. For each period, we can use average or median sea level calculated from the dataset of [START_REF] Miller | A 180-million-year record of sea level and ice volume variations from continental margin and deep-sea isotopic records[END_REF], which differ at most by only a few meters. Positive and negative vertical motion rates denote uplift and subsidence, respectively.

Supplementary Figure 2. Sea level curves from [START_REF] Miller | A 180-million-year record of sea level and ice volume variations from continental margin and deep-sea isotopic records[END_REF] for the periods indicated in the plots. Mean sea level for each period represented by blue dashed line with respective value.

The detailed description of the procedure we used to constrauct Figures 5 and6 illustrate how the minimum uplift rate between 3.29 and 2.87 Ma can be estimated. B -sea level curve between 2.6 and 3.2 Ma [START_REF] Miller | A 180-million-year record of sea level and ice volume variations from continental margin and deep-sea isotopic records[END_REF] to show sea level variation in the interval of the 2.87 ± 0.21 Ma age [START_REF] Holm | An 40 Ar-39 Ar study of the Cape Verde hot spot: Temporal evolution in a semistationary plate environment[END_REF]. The average sea level in the period is ca.

-20 m, which means that the current altitude of ca. 20 m of the passage zone corresponds to sea level. LVC -Lower Volcanic Complex (former Flamengos Formation of Serralheiro, 1976);

IVC -Intermediate Volcanic Complex (former Pico da Antónia Formation, which can be subaerial -pink lines, and pass into submarine -blue lines); UVC -subaerial Upper Volcanic

Complex (former Assomada Formation). Red dashed line -major unconformity between LVC and UVC; green dashed line -major unconformity between LVC and IVC.

Discussion

New ages

From field work in the target area, geomorphological analysis and the new ages reported here (Fig. 4), we have updated and improved previous mapping of the volcanic units: ( 1

Rates of vertical motion

We present a new methodology to deduce vertical motion histories, which can be applied to any island or coastline hosting sequences of paleoshorelines of determinate age. Here we use this methodology and new K-Ar chronostratigraphy to infer the vertical motion history of Santiago Island. We complemented the new methodology with an exercise based on state-of-the-art geochronology and a paleo sea level curve [START_REF] Miller | The Phanerozoic record of global sealevel change[END_REF][START_REF] Miller | A 180-million-year record of sea level and ice volume variations from continental margin and deep-sea isotopic records[END_REF]. Sensitivity tests indicate that the vertical displacement rates we deduce in a given interval are relatively insensitive to our choice of eustatic sea level curve [START_REF] Bintanja | Modelled atmospheric temperatures and global sea levels over the past million years[END_REF][START_REF] De Boer | Cenozoic global ice-volume and temperature simulations with 1-D ice-sheet models forced by benthic δ18O records[END_REF][START_REF] Hansen | Climate sensitivity, sea level and atmospheric carbon dioxide[END_REF], and came to the conclusion that none can change our results significantly, i.e. none can change subsidence into uplift or vice-versa. In fact, we already show this when we estimate vertical displacement rates using the end-member values in a given interval.

Using the geological map of [START_REF] Serralheiro | A Geologia da ilha de Santiago (Cabo Verde)[END_REF], Ramalho et al. (2010a, c) estimated vertical motion in Santiago, but were unable to find and date passage zones, the critical criterion for an ideal marker of paleo sea level (cf. 3.87 Ma -cf. Fig. 4), which means that they belong to the Lower Volcanic Complex (former Flamengos). Moreover, the new stratigraphy calibrated by isotopic ages indicates that the passage zone within the Intermediate Volcanic Complex (former Pico da Antónia) lies at ca. 80 m altitude in the Porto Rincão valley (immediately to the south of the Águas Belas valley), which is significantly lower than the 270 m used by Ramalho et al. (2010a, c). Therefore, the rates estimated by Ramalho et al. (2010a, c) are incorrect. Because they were unable to find passage zones, Ramalho et al. (2010a,b,c) dated outcrops of subaerial volcanic units and assumed that underlying submarine units formed at the same time and at a minimum depth below sea level equivalent to the thickness of the submarine unit. However, our fieldwork and dating show that the contact between the subaerial and submarine units is an erosional unconformity and that the subaerial volcanics are at least 2 Ma younger than the underlying submarine rocks. The rates calculated by Ramalho et al. (2010a,b,c) therefore underestimate both the minimum paleodepth (since the thickness of the eroded submarine volcanics is unknown) and the age of the paleoshoreline indicator. Given that the whole unit is made up of submarine flows, the Lower Volcanic Complex (former Flamengos) can be interpreted in two contrasting ways: (1) the edifice was in a seamount stage, i.e. volcanic growth under sea level, therefore there is no argument in favor of any vertical motion -the older lavas simply formed deeper in the ocean; (2) the edifice was on a subaerial building stage, so that the subaerial lavas at the rim of the proto-island passed into submarine; this means that the submarine lavas formed all at similar depth in the ocean, but were gradually carried to depth by island subsidence between ca. 5 and 3.3 Ma. We do not have an answer to this question because we could neither find passage zones in the LVC, nor fossils on the pillows that could indicate depth. Ramalho et al. (2010a, b, c) concluded for a "general uplift trend over the last 5 Ma" for Santiago. The use of precisely dated passage zones, at different ages (0.811, 2.18 and 2.87 Ma) and in different places on the island led us to a contrasting conclusion: instead of general uplift at a constant rate of ca. 0.092 mm/a (Ramalho et al., 2010a, b, c), our new data and procedure indicate that Santiago experienced periods of uplift, at different rates (0.95 to 0.14 mm/a), intercalated with at least one period of subsidence (-0.11 mm/a), so improving the temporal resolution of Santiago's vertical motion history.

The subsidence period here recognized corresponds to major volcanic construction, i.e. the construction of the most widespread unit in Santiago -the Intermediate Volcanic Complex (former Pico da Antónia). Subsidence may also have occurred during the deposition of the Lower Volcanic Complex, so explaining why this unit is entirely made of submarine lavas.

Mechanisms responsible for the vertical displacements of Santiago Island

Having estimated displacements and rates, we now discuss the possible mechanisms responsible for the vertical motions of Santiago Island. Here we will concentrate on two main mechanisms, one top-down (island loading and unloading) and the other bottom-up (lithosphere thinning, underplating, thermal mantle plume), both directly related to isostasy.

Top-down: insights from isostatic calculation

For the top-down mechanism, we do not have enough unloading (erosion in all its forms) data to quantitatively justify the inferred uplifts. However, we have geological information enough to quantitatively explain the subsidence episode in Santiago between ca. 2.9 and 2.2 Ma, which corresponds to the time when the voluminous IVC was being deposited on top of a major unconformity affecting both BC and LVC. These complexes underwent major erosion prior to the deposition of the IVC, thus comprising one of the main unconformities whose topography peaks at ca. 500 m altitude (cf. Fig. 2B). Given that the IVC peaks currently at ca. 1400 m, we can use a thickness of 900 m to calculate the maximum subsidence due to loading. We can estimate the isostatic effect of constructional loading by assuming Airy isostasy (i.e. the lithosphere has zero strength and all loads are locally compensated) and a maximum 900 m thickness of the IVC: ε = 900 -(900*ρ l /ρ a ) = 900 -900*2800/3200 = 900 -788 = 112 m where ε is the topography, ρ l is the density of the lavas, and ρ a is the density of the asthenosphere.

112 m is an upper bound calculation of the isostatic response to constructional loading, therefore indicating that loading by the IVC could potentially account for the inferred ca. 80 m of subsidence in this period.

Bottom-up: insights from numerical modeling

Different bottom-up mechanisms can be responsible for the inferred vertical movements (up to 400 m): thinning of the lithosphere by basal erosion, and/or crustal underplating, and/or arrival of a hot, low-density material at the base of the lithosphere (thermal mantle plume). We performed a set of 1-D numerical modeling tests to simulate these bottom-up mechanisms by solving the 1-D heat transport equation and calculating the uplift resulting from density variations following crustal thickening or temperature changes. We designed these modeling experiments to best reproduce the observed uplift of Santiago, assuming reasonable input parameters, to assess the viability of each mechanism.

The heat equation to be solved is the following: for the mantle (peridotite) [START_REF] Robertson | Thermal properties of rocks[END_REF]. Heat production was 0.2 µW/m 3 for the crust and 0.02 µW/m 3 for the mantle.

𝜌𝑐 !" !" = 𝐴 + 𝑘 ! ! ! !! !
Density is related to temperature through the following equation:

𝜌 𝑇 = 𝜌 𝑇 ! • 1 -𝛼 𝑇 -𝑇 ! Eq. 2
where ρ is density [kg/m 3 ], α the thermal expansion coefficient [1/K], and T 0 a reference temperature at which density is ρ0. This reference temperature is taken as 0 ºC for crustal materials and 1300 ºC for the mantle. The thermal expansion coefficient is taken as 3.5 * 10 -5 K -1 for all materials [START_REF] Afonso | Thermal expansivity and elastic properties of the lithospheric mantle: results from mineral physics of composites[END_REF]. As reference density we used ρ 0 =2800 kg/m 3 for the crust and 3200 kg/m 3 for the mantle.

Several works have investigated the nature and properties of the lithosphere in the Cape Verde Swell (Ali et al., 2013;[START_REF] Carvalho | Evidence for high temperature in the upper mantle beneath Cape Verde archipelago from Rayleigh-wave phase-velocity measurements[END_REF][START_REF] Dash | Geophysical investigation of the Cape Verde archipelago[END_REF][START_REF] Hellfrich | Transition zone structure under a stationary hot spot: Cape Verde[END_REF][START_REF] Lodge | Depleted swell root beneath the Cape Verde Islands[END_REF]McNuttt, 1988;[START_REF] Pim | Crustal structure and origin of the Cape Verde Rise[END_REF][START_REF] Vinnik | Cape Verde hotspot from the upper crust to the top of the lower mantle[END_REF][START_REF] Wilson | Uplift at lithospheric swells I: seismic and gravity constraints on the crust and uppermost mantle structure of the cape verde mid-plate swell[END_REF][START_REF] Wilson | Uplift at lithospheric swells II: is the Cape Verde mid-plate swell supported by a lithosphere of varying mechanical strength?[END_REF]. Fundamental for the modeling are the thicknesses suggested in those works for the lithosphere and the oceanic crust, which vary from 60 km [START_REF] Carvalho | Evidence for high temperature in the upper mantle beneath Cape Verde archipelago from Rayleigh-wave phase-velocity measurements[END_REF] to 100 km [START_REF] Lodge | Depleted swell root beneath the Cape Verde Islands[END_REF] for the former, and from 7 km (e.g. [START_REF] Pim | Crustal structure and origin of the Cape Verde Rise[END_REF][START_REF] Wilson | Uplift at lithospheric swells I: seismic and gravity constraints on the crust and uppermost mantle structure of the cape verde mid-plate swell[END_REF] to 15 km for the latter [START_REF] Dash | Geophysical investigation of the Cape Verde archipelago[END_REF]. Therefore, we tested initial lithosphere configurations with crustal thicknesses of 7 km and 15 km, and lithosphere thickness of 80 and 100 km. Topography was calculated assuming local isostatic equilibrium using the following equation [START_REF] Lachenbruch | Continental extension, magmatism and elevation: Formal relations and rules of thumb[END_REF]:

𝜀 = ! ! ! ! !! ! ! ! !! ! ! ! 𝐻 + 𝜀 ! Eq. 3
where ε is the topography [m], ρ a the asthenospheric density (3200 kg/m 3 ), ρ L the average lithospheric density, ρ W the density of sea water (1030 kg/m 3 ), H the thickness of the lithosphere

[m], and ε 0 the reference topography of an unloaded asthenosphere (-2400 m).

In the results presented below, the beginning of the calculations corresponds to 3.3 Ma ago, because this is the age of the first recognized major uplift. We present for the different models 2 to 3 calculations in order to show that different initial configurations combined with different structural modifications can explain the uplift.

Model 1: Lithospheric thinning

The temperature at the initial depth (80 or100 km) is maintained constant at 1300 ºC during the whole calculation. Between 0 and 0.4 Ma in the simulation (3.3 to 2.9 Ma in nature), the base of the lithosphere rises from 100 km to 60 km in steps of 5 km/0.05 Ma (blue line in Fig. 8A) or from 80 to 44 km in steps of 4 km/0.05 Ma (orange line in Fig. 8A), resulting in a total topographic uplift of ca. 470 m. From 0.4 to 2.5 Ma (2.9 to 0.8 Ma in nature), the temperature relaxes, and the lithosphere thickens slowly, resulting in a subsidence of 50 m. At 2.5 Ma (0.8 Ma in nature), a new pulse of lithosphere heating is simulated by thinning the lithosphere again to 60 km / 44 km, and maintaining the base of the lithosphere (i.e. a temperature of 1300 ºC) at this depth until 3.3 Ma (0 Ma in nature), which results in another uplift of ca. 200 m. The topography rises higher than at 0.4

Ma because the crust has meanwhile been heated as well. Both models give very similar results. The crust thickens by a total of 800 m (from 7 to 7.8 km, blue line in Fig. 8B, and from 15 to 15.8 km, orange line in Fig. 8B) between 0 and 0.4 Ma (3.3 to 2.9 Ma in nature). Material arrives at 1100 °C, the typical temperature of a basaltic magma, its temperature is maintained constant until 0.4 Ma (2.9 Ma in nature) to simulate a constant supply and heating of the surrounding layers during the whole process. This results in a topographic uplift of 460 m. After 0.4 Ma (2.9 Ma in nature), the temperature relaxes and topography decreases slightly. At 2.5 Ma (0.8 Ma in nature), another pulse of thickening of 100 m (7.8 to 7.9 km) is simulated, again with a temperature of 1100 °C that is maintained during 0.1 Ma. After 0.8 Ma (at 3.3 Ma = 0 Ma in nature), it results in an uplift of 100 m, including the partial temperature relaxation between 2.6 and 3.3 Ma (0.7 to 0 Ma in nature). The effect of relaxation is stronger for the initially thinner crust, which, when taking into account the observation of roughly stable topography during this period, would give a preference for the model with thicker initial crust.

Model 3: Rise of a thermal plume

In this model, hot material (thermal mantle plume) arrives at the base of the lithosphere and replaces it in a variable thickness depending on plume temperature. We modeled temperature differences between 100 and 300 ºC, i.e. plume material at 1400 to 1600 °C. The temperature is maintained identical until 0.4 Ma (2.9 Ma in nature), then it relaxes. In order to achieve the desired total topographic uplift of about 400 m, the thickness of the plume material layer must vary between 25 km for a 1400 °C plume and 16 km for a 1600 °C plume. The effect is thus nearly linear with the temperature difference. However, subsidence is much more important for hotter material. A simple model as presented here would therefore not be compatible with the observations. Nevertheless, a continuous supply of hot material between 2.9 and 0.8 Ma ago would maintain the topography at the observed height. We did not model the second pulse at 2.5 Ma (0.8 Ma in nature), since the other models show that the necessary topography variation can be easily achieved.

For the quantitative interpretation, a critical condition is the one of local isostasy.

Significant elastic strength would reduce the topographic response. However, on the one hand, the volcanism in Cape Verde indicates a locally hot and therefore weaker lithosphere. On the other hand, especially for models 2 and 3, the structural variations used for the models are quite modest:

doubling crustal thickening or the thickness of the hot plume layer would certainly not be exaggerated. It would be more difficult to consider much stronger thinning of the lithosphere in Model 1 without the additional effect of hot rising plume material. Anyhow, also a combination of all models would be able to explain the inferred uplift. The question now is application of the model results to Santiago in the context of the archipelago and the Cape Verde Rise. Most largescale geophysical studies carried out in the Cape Verde Swell suggest that upwelling within the asthenosphere supports the anomalous topography of the Cape Verde Swell, with minor contributions from local thickened oceanic crust and, perhaps, partial thermal rejuvenation of the lithospheric mantle. If large-scale swelling (dynamic uplift at the scale of the whole archipelago; e.g. [START_REF] Huppert | Hotspot swells and the lifespan of volcanic ocean islands[END_REF] were the mechanism responsible for island uplift , then one would expect to find similar uplift in islands with similar age, which is not the case (e.g. Ramalho et al., 2010a, b, c). Therefore, the solution to explain geological evidence should come from local mechanisms.

Model 1 is intended to simulate extension, which does not seem a sensible option in the geodynamic setting of Cape Verde where rifting has never been recognized. In contrast, Model 3 looks like a good option, because this model simulates the scenario whereby the mantle plume dynamically uplifts the lithosphere underlying the Cape Verde Islands during Santiago's evolution, but is inconsistent with what is known about the differential vertical motion among the Cape Verde islands (e.g. Ramalho et al., 2010a, b, c), because some islands show large uplifts (e.g. Santiago and S. Nicolau), but other islands do not record uplift (e.g. Santo Antão and Fogo). Therefore, the largescale plume (regional mechanism) cannot explain the differential uplift. The solution is therefore to look for a local mechanism, which is the case of magma underplating (Model 2), as already suggested to exist beneath the islands (e.g. [START_REF] Lodge | Depleted swell root beneath the Cape Verde Islands[END_REF]Ramalho et al., 2010b;[START_REF] Vinnik | Cape Verde hotspot from the upper crust to the top of the lower mantle[END_REF][START_REF] Wilson | Uplift at lithospheric swells I: seismic and gravity constraints on the crust and uppermost mantle structure of the cape verde mid-plate swell[END_REF].

Conclusions

We use paleoshorelines at different altitude and age to constrain the vertical motion history of Santiago Island. Specifically, we calculate the time-averaged vertical motion rates required to explain the present-day elevations of the dated sequence of paleoshorelines. We account for eustatic sea level changes and work progressively back in time to incorporate the more recent vertical motion rates implied by the youngest paleoshorelines into the vertical motion history of all older shorelines. We thus obtain a vertical motion history consisting of time-averaged vertical motion rates spanning the five intervening time periods between paleoshoreline formation and the present day: (1) > 6 Ma -this stage is certainly mostly submarine (seamount growth), but there is no consensus regarding the submarine or subaerial nature of the rocks of the Basement Complex currently cropping out onshore: all submarine, or subaerial in part? Did the seamount grow into an island with subaerial volcanism? (2) 5.06 to 3.29 Ma -given that all known rocks in this period are submarine, they can represent seamount growth or island subsidence. (3) 3.29 to 2.87 Ma -fast uplift (0.96 mm/a) mostly responsible for putting submarine lavas currently close to 410 m altitude.

(4) 2.87 to 2.18 Ma -relatively fast subsidence (-0.11 mm/a), most likely due to the fast volcanic growth of the most voluminous volcanic unit on the island, the Intermediate Volcanic Complex (former Pico da Antónia Formation). ( 5) 2.18 to 0.811 Ma -stationary island (0.00 mm/a). ( 6) 0.811 to 0 Ma -relatively fast uplift (0.14 mm/a).

The advantage of using dated passage zones, at different ages (0.811, 2.18 and 2.87 Ma) and in different places on the island, and a different procedure is that we find a more detailed history of vertical motion consistent with geological observation. Instead of a simple general uplift at a constant rate of ca. 0.092 mm/a (Ramalho et al., 2010a, b, c), our new data and methodology indicate that Santiago experienced several periods of uplift, at different rates (0.95 to 0.14 mm/a), intercalated with at least one period of subsidence (-0.11 mm/a).

We numerically tested top-down (volcanic loading) and bottom-up (lithosphere thinning, underplating and mantle plume) mechanisms to explain the inferred vertical movements, and we conclude that volcanic loading and crustal underplating are capable of producing the observed subsidence and uplift, respectively.

Figure 1 .

 1 Figure 1. Annotated Google Earth image showing geographic location and tectonic setting of the Cape Verde archipelago.

Figure 2 .

 2 Figure 2. A -digital elevation model showing the main morphological features of Santiago, and relevant ages (in Ma) from previous studies. B -geological map of study region,(Serralheiro,

Figure 3 .

 3 Figure 3. Annotated photographs taken to north (A) and south (B) of the Águas Belas creek with volcanic stratigraphy based on main unconformities and calibrated by K-Ar dating. In both images, especially in B, it is clear that the lava flows of the UVC fill deep valleys carved in both IVC and LVC, which stand higher than the UVC despite being older. From bottom to top: BC -Basement Complex unconformably overlain by LVC (yellow dashed line) and by UVC (out of the image); LVC -Lower Volcanic Complex (submarine flows) unconformably overlain by IVC (green dashed line) and by UVC (red dashed line); IVC -Intermediate Volcanic Complex (subaerial in the image) unconformably overlying the LVC through a thick conglomerate, and unconformably overlain by UVC; UVC -Upper Volcanic Complex made of subaerial flows filling a rugged topography carved in BC, LVC and IVC, and passing into submarine flows close to the coast (west, out of the image).

Supplementary Figure 1 .

 1 Images from the Porto Rincão creek: A -passage zone with subaerial flows passing into submarine, with in situ shallow water oyster fossils, as shown in B, and sampled for geochronology; C -passage zone observed and sampled on the coast immediately south of Porto Rincão. Location of samples given in Fig. 4.

Figure 4 .

 4 Figure 4. A -geological map of study region[START_REF] Serralheiro | A Geologia da ilha de Santiago (Cabo Verde)[END_REF] annotated with ages reported here. B -Perspective view of the topography, distribution of volcanic units, and their geomorphological expression. From the new ages and geomorphology, we infer that the mapped volcanic units are wrongly placed: (1) the Lower Volcanic Complex (LVC, former Flamengos Formation, orange in the geological map) in the Porto Rincão creek is actually the submarine part of the Intermediate Volcanic Complex (IVC, former Pico da Antónia Formation); (2) the submarine IVC in the Porto Rincão creek (blue horizontal stripes in the geological map) is actually the submarine part of the Upper Volcanic Complex (UVC, former Assomada Formation), which is recognized here for the first time; (3) part of the subaerial IVC (PA and peach colour in the geological map) is actually UVC, which form quite well preserved lava deltas; (4) the submarine IVC in the Águas Belas creek (blue horizontal stripes in the geological map) is actually the entirely submarine LVC.

  ) the relative uncertainty on the calibration of the 40 Ar signal ( ), and (3) the relative uncertainty on the correction of the atmospheric contamination ( = 0.1 / 40 Ar* × 100), 40 Ar* being the radiogenic content.

Figure 5 .

 5 Figure 5. Sketch to show the use of anchor points (passage zones) to construct anchor paths (marked with blue arrows), which must ultimately converge to a check point. The first anchor path calculated to constrain the rates of vertical motion is the youngest (green line), because the most recent vertical motion must be identical to all units in the case of vertical motion alone. By going back in time, we obtain the paleo-positions and vertical motion rates. Note that older rocks follow apath identical to the younger rocks to guarantee that the whole island follows anchor paths. All ages were obtained in the present work, except the 2.87 Ma age reported in[START_REF] Holm | An 40 Ar-39 Ar study of the Cape Verde hot spot: Temporal evolution in a semistationary plate environment[END_REF].

Figure 6 .

 6 Figure 6. Sketch to illustrate the methodology introduced here to estimate vertical motion rates. The blue dashed lines represent current sea level. The sums (black bold numbers) represent vertical displacement (positive for uplift and negative for subsidence). The blue numbers represent the corresponding vertical motion rates.

Figure 7 .

 7 Figure 7. Sketch of the geological setting in the Flamengos valley (A) and graph (C) to

  the temperature [ºC], t the time [s], z the depth [m], ρ the density [kg/m 3 ], c the mass heat capacity [J/(kg*K)], and k the thermal conductivity [W/(m*K)]. The boundary conditions are fixed temperatures at the top and bottom of the model domain, respectively 0 ºC and 1300 ºC (considered to be the normal asthenospheric temperature). We used standard values of k = 2.2 W/(m*K) and c = 1000 J/(kg*K) for the crust (gabbro), and k = 3 W/(m*K) and c = 1000 J/(kg*K)

Figure 8 .

 8 Figure 8. Numerical modeling of the topography variation over time resulting from the Airy isostatic response to (A) lithospheric thinning as specified in Model 1, (B) underplating as specified in Model 2, and (C) excess buoyancy associated with a mantle plume as specified in Model 3.

Table 1 .

 1 Summary of the isotopic ages reported here, with sample locations and references, geographical coordinates, and altitudes. For each sample, the mean age is obtained by weighing by the amount of radiogenic argon. Uncertainties quoted at the 1σ level.

	Locality and sample reference	Latitude (N)	Longitude (W)	Altitude (m)	K (%)	40Ar* (%)	40Ar* (10 12 at/g)	Age (Ma)	Uncertainty (Ma)
	Flamengos valley
	(eastern Santiago)
	ST17I	15.16244 -23.60938	48	1.699 15.6 9.079	5.11	0.08
						15.5 8.889	5.00	0.08
							mean	5.06	0.08
	ST17J	15.15405 -23.63730 115	1.101 27.5 5.742	4.99	0.07
						28.0 5.779	5.02	0.07
							mean	5.00	0.07
	ST17L	15.13284 -23.66905 409	0.701 13.4 2.385	3.26	0.05
						10.1 2.448	3.34	0.06
							mean	3.29	0.05
	Porto Rincão
	and Águas Belas
	valleys
	(western Santiago)
	ST17U	15.05471 -23.76369	20	1.598 19.3 1.401	0.839	0.013
						16.0 1.404	0.841	0.013
							mean 0.840	0.013
	ST17X	15.06421 -23.76259	43	1.608 18.0 1.359	0.809	0.012
						29.5 1.363	0.812	0.012
							mean 0.811	0.012
	ST17W	15.07013 -23.75175	80	2.042 19.1 4.611	2.16	0.03
						18.0 4.687	2.20	0.03
							mean	2.18	0.03
	ST17Y	15.10868 -23.73271 324	0.542 4.3	2.163	3.82	0.10
						4.6	2.223	3.92	0.10
							mean	3.87	0.10

Table 1 .

 1 K-content of our samples measured by flame absorption photometry in GEOPS Laboratory (Univ.Paris-Sud, Orsay, France). For each sample, the mean is used as the final K-content for age calculation when the relative standard deviation (numbers into brackets) is better than 1%, in order to account for the total uncertainty including K-solution content and the uncertainty on BCR2 and MDO-G standards (see text).

	Sample	K (%)
	ST17i	1.694
		1.704
		Mean: 1.699 ± 0.007 (0.42%)
	ST17J	1.099
		1.104
		Mean: 1.101 ± 0.004 (0.32%)
	ST17L	0.702
		0.700
		Mean: 0.701 ± 0.001 (0.16%)
	ST17U	1.598
		1.600
		1.595
		Mean: 1.598 ± 0.003 (0.17%)
	ST17X	1.612
		1.599
		1.612
		Mean: 1.608 ± 0.007 (0.45%)
	ST17W	2.062
		2.037
		2.026
		Mean: 2.042 ± 0.018 (0.91%)
	ST17Y	0.544
		0.541
		0.542 ± 0.02 (0.40%)

Table 2 .

 2 Data necessary to calculate the rates of vertical motion. Sea level data from

  our sample ST17X, presently at an altitude of 43 m (uncertainty on present altitude is here neglected), the estimated mean amount of uplift is thus 43-(-72) ± 28 m = 115 ± 28 m. The relative uncertainty on the amount of vertical motion is thus (28/115)*100 = 24%.

												Relative
											Uncertainty	uncertainty	Rate
				Age	Relative age	Lowest	Highest	Median	Uncertainty	Vertical	vertical	vertical	vertical
		Altitude	Age	uncertainty	uncertainty	sea level	sea level	sea level	on sea level	displacement	displacement	displacement	motion
	Sample	(m)	(Ma)	(Ma)	(%)	(m)	(m)	(m)	(m)	(m)	(m)	(%)	(mm/y)
	ST17X	43	0.811	0.012	1.5	-100	-44.67	-72	28	115	28	24	0.14
	ST17U	20	0.840	0.013	1.5	-59	-18	-38	20	58	20	35	0.07
	ST17W	80	2.18	0.03	1.5	-67	-9	-38	29	118	29	25	0.05
	121367* 20	2.87	0.11	3.7	-33	6	-13	19	33	19	57	0.01
	ST17Y	324	3.87	0.10	2.7	-35	-7	-21	14	345	14	4	0.09
	ST17L	409	3.29	0.05	1.6	-61	-12	-36	24	445	24	5	0.14
	ST17I	48	5.06	0.08	1.6	-29	5	-12	17	60	17	29	0.01
	ST17J	115	5.00	0.07	1.5	-29	2.67	-13	16	128	16	12	0.03

* age data from

[START_REF] Holm | An 40 Ar-39 Ar study of the Cape Verde hot spot: Temporal evolution in a semistationary plate environment[END_REF]

, with uncertainty quoted at 1 s

  to the same volcanic unit and so had to undergo the same vertical displacement, which is that of the youngest rock (ca. 3.29 Ma). The highest submarine rocks of ÁguasBelas (ca. 324 m) are not as high as the highest in the opposing Flamengos valley (ca. 410 m), but the age of the two

	rocks is also not the same -greater in the Águas Belas valley (ca. 3.87 Ma) than in the Flamengos
	valley (ca. 3.29 Ma), which, together with differential erosion, could justify the current difference
	in altitude (410 -324 = 86 m).
	can be
	found in the supplementary material.
	Given that we could not find passage zones in the Lower Volcanic Complex at Águas Belas
	and Flamengos creeks, the estimated uplift rate is a minimum. At ca. 3.29 Ma, the rocks dated at

3.29, 3.87 and 5.00 Ma were at -29, -115 and -354 m depth, respectively (cf. Figs.

5 and 7

), because they belong

  ) the Lower Volcanic Complex (former Flamengos Formation, orange in the geological map) in the Porto Rincão creek is actually the submarine part of the Intermediate Volcanic Complex (former Pico da Antónia Formation); (2) the submarine Intermediate Volcanic Complex in the Porto Rincão creek (blue horizontal stripes in the geological map) is actually the submarine part of the Upper Volcanic Complex (former Assomada Formation), which is recognized here for the first time; (3) part of the subaerial Intermediate Volcanic Complex (former Pico da Antónia, peach color in the geological

map) is actually Upper Volcanic Complex, which form quite well preserved lava deltas; (4) the submarine Intermediate Volcanic Complex in the Águas Belas creek (blue horizontal stripes in the geological map) is actually the entirely submarine Lower Volcanic Complex. The recognition that the geological map is incorrect in critical places suggests that previous work based on the geological map may be erroneous.

  Table4ofRamalho et al., 2010c). They used submarine lavas at ca. 270 m altitude in the Águas Belas Valley to estimate paleo sea level at ca. 2.8 Ma, because the geological map attributes these rocks to the submarine Intermediate Volcanic Complex (former Pico da Antónia Formation). However, those submarine lavas are dated here about 1 Ma older (ca.

Acknowledgments

This is a contribution to Project MEGAHazards2 (PTDC/GEO-GEO/0946/2014), funded by FCT, Portugal. The sea level data used in this study was kindly supplied by Kenneth Miller, and can be found here: https://www.ncdc.noaa.gov/paleo-search/study/13947. Thorough and constructive reviews by five anonymous Reviewers helped to clarify and improve the quality of this manuscript.