Estimating the Sediment Volume of Each Sampled Holocene Layer

S1. Supplemental material : estimating the sediment volume of each sampled Holocene layer.

S1a. Simpli ed general layout of the lake. We simpli ed the shape of the lake to an 465 m oval shape of 465 x 156 m. The volume of the 156 m Water column 8 m lake was approximated with half of an ellipsoid, which represented the depth of the lling: 8 m of water and 18 m of sediment (S1a). Holocene sediment 6,2 m The volume of each layer of sediment was estimated by subtracting these half-ellipsoid volumes (S1b). Late glacial sediment Vlayer 1 = Vh-ellip1 - Vh-ellip2 11,8 m Vlayer 2 = Vh-ellip2 - Vh-ellip3

S1b. Half-ellipsoids subtraction and limitation of the uncertainty on the camber of the ellipsoid. V layer 1 h-ellip1 layer 2 Vh-ellip2

Vh-ellip3

The uncertainty of the half-ellipsoid camber could be neglected because the depth of each sampled layer was very thin (centimetric), as shown below:

Each half-ellipsoid was de ned by : (S1d.) considering the camber of the ellipsoid to be a = semi-major axis linear (noted z). This assumption was possible because b = semi-minor axis each sampled depth in the core (noted h) was su- c = semi axis (depth of the lling) ciently small (centimetric). The average angles ( a and 4 b for the major and minor axes of the ellipsoid, Volume of half-ellipsoid is : respectively) were measured (Table S1.1) and xed constant for each unit to calculate the a and b parame- 1 4 ters for any depth in each unit (S1e. and Table S1.2). Vh-ellip = . π . ai . bi . ci 2 ( 3 ) The c parameters were known because these values are Table S1.1. Graphically measured parameters for each the depth of each layer of sediment and were determined unit of sediment. when the core was sampled. Unit number Unit bo�om from top a (m) b (m) α (°) α (°) depth (m) i i a b The a and b parameters could have been graphically sediment core determined (S1c), but measuring their length at each 1 1,2 218 73 18 43 sampled depth would take too long, so we only measured 2 4,5 204,6 68 14 36 the a and b parameters for the 6 units of sediment that 3 5 202,2 67,5 11 31 were described in Bajard et al., (2016) ( rst 6,2 m of 4 5,3 200,3 67 11 30 sediment) for the Holocene period (Table S1.1). Then, we 5 5,9 197 65,8 10 28 simpli ed the geometry of the layers within each unit by 6 6,2 195 65 10 23 S1c. Graphical determination of a and b ellipsoidal parameters of each unit of sediment.

Geometry of ai parameters : major axis of the ellipsoid

465 m

a8 = 222 m 8 m 218 m 204.6 m 202.2 m 200.3 m 197 m 195 m c = 18 m

20 m

Geometry of bi parameters : minor axis of the ellipsoid 156 m 8 m b8 = 74 m 73 m 68 m 67.5 m 67 m 65.8 m 65 m c = 18 m 10 m

y ai-1 or bi-1 i-1 h². cos²( a ) a or b ai = ai-1 - h h = sampled depth z 1 - cos²( ) i a ai or bi

h². cos²( ) bi = bi-1 - b 1 - cos²( b ) y

S1d. Simpli ed geometry of the layers of each unit of S1e. Equations that was deduced from S1d. layout to sediment. determine a and b parameters of each sampled depth.

Table S1.2. Calculation of the volume of sediment of each layer by subtracting half-ellipsoids of successive depth. Then, the mass of sediment at each Mastercore Volume of half- Volume of the ai (m) bi (m) ci (m) depth was deduced according to the 3 3 depth (m) ellipsoïd (m ) layer (m ) density of each depth (from the core 0 222,00 74,00 18,00 619007,0 1038,6 sampling). This value was corrected 0,02 221,94 73,98 17,98 617968,4 1037,7 from the carbonated and organic fractions (from the loss on ignition) 0,04 221,88 73,96 17,96 616930,7 1036,9 and extended to the surface of the 0,06 221,82 73,94 17,94 615893,8 1036,0 catchment and to the time of depo- 0,08 221,75 73,91 17,92 614857,9 1035,1 sition (from the age depth model) to obtain the erosion in t.km-².yr-1. … … … … … …

Supplemental material S1. Estimating the sediment volume of Lake La Thuile considering an oval shape lake (465 x 156 m), ellipsoidal lling model and a at lake bottom. The lake is 8 m depth. The coring of the lake revealed a total sediment lling of 18 m, whose srt 6,2 m cover the Holocene period (last 12,000 years cal. BP).