The model on stress changes in the SISZ

$\\$ Introduction $\\$Movements on faults occur in earthquakes, aseismic slip or inelastic creep. There are sequences of seismic quiescence followed by seismicity and so on. Stresses are accumulated by plate motion and released in earthquakes and/or the mentioned processes. In the following, a model is described that accounts for plate motions and earthquakes on Iceland.

The SISZ is situated between two sections of the Mid-Atlantic Ridge, the Reykjanes Ridge (RR) and the eastern volcanic zone (EVZ). Even though the angle between the SISZ and the neighbouring ridges is far from 90$^{\circ }$, it is considered as a transform fault (for the geological structures, see Figure 2). Following the transform fault hypothesis, left-lateral shear stress is expected along the E-W striking zone. This is equivalent to right-lateral shear stress with N-S orientation. In fact, earthquakes seem to occur on N-S trending faults ([,]. They are located side by side between the Hengill triple junction, where the RR meets the low activity western volcanic zone (WVZ), and Hekla volcano, a part of the EVZ [].

The questions to be solved are:

• $\bullet$ Do these events, placed on parallel faults, release all the energy stored in the 3-D volume of the SISZ?
• $\bullet$ Do the earthquakes always take place in areas of high stress?
• $\bullet$ What is the critical stress level? How large is its variability?
• $\bullet$ Where are the highest stresses nowadays?

  
  The method $\\$The elasticity theory of dislocations is used to calculate the stress changes induced by earthquakes with a double-couple strike-slip mechanism on an extended rupture plane [61].

  The area investigated extends from 18 to 24$^{\circ }$W and from 63 to 65$^{\circ }$N. The origin of the coordinate system is set to 24$^{\circ }$W, 64$^{\circ }$N (Figure 36) it includes the SISZ, north and south of 64$^{\circ }$N,
  

    

  Figure: Map of Iceland and surrounding area. Thick red lines indicate Mid-Atlantic Ridge segments, as used in the modelling. The smaller box shows the region of the model on the SISZ. The SISZ extends approximately between (125, 0) to (200, 0). The large box gives the region for the Iceland rift model.

  
  
  the SW edge of the EVZ (extending NE from about 18.7$^{\circ }$E, 64$^{\circ }$N), and the north eastern most part of RR (approximated by a straight line trending from about 23$^{\circ }$W, 64$^{\circ }$N at the SISZ in SW direction). This latter assumption is a strong simplification as the RR changes strike direction all along Reykjanes Peninsula up to Hengill triple junction where it finally mainly strikes E-W as the transform fault itself.

  The initial stress field is determined as follows: a tensional stress acting N103$^{\circ }$E (nearly parallel to the SISZ [] is assumed, due to ridge push or basal drag of the adjacent plates. The stress magnitude, which is unknown, is set to a value that produces left-lateral shear stresses in E-W direction as large as the stress drop determined for the largest event (M=7.1) in the studied earthquake sequence. Insofar, this stress is the minimum stress to be expected. The amount of this minimum shear stress is 2.7 MPa (consistent with Gudmundsson [32], who determined maximum values of 12 MPa from maximum displacement of superficial fault traces found in the field). This shear stress corresponds to an uniaxial tensional stress of -13.1 MPa. Tensional stresses at both ridges are modelled as constantly being released to have zero values at the rifts. These are the major disturbance of the unknown background stresses.

  On this initial field, the stress changes due to earthquakes are iteratively superposed as well as the stress changes due to further spreading at the ridge segments based on an opening of 2 cm/year. This value is taken from []. The stress field before every event is thus the sum of the initial field, the stress drop of all preceding events, and the plate tectonic stress build-up since the starting time of the model, which is set to 1706, as discussed below.

  Results were calculated for 56 x 44 test points covering 280 km in E-W direction and 220 km in N-S direction; i.e. the spacing is 5 km in both coordinates. Stresses were computed for a homogeneous half-space (as a starting model) with both Lamé's constants being 39 GPa. Although surface stress changes are calculated, these should be representative for crustal stresses using these values for the moduli, that are typical for oceanic crust [] and not for sedimentary layers at the surface.

  Figures 37 and 38 give an example to illustrate the modelled effects: the
  

    

  Figure: Demonstration of the effect on the shear stress field by a sample earthquake at position (164 km, 0 km) with magnitude 7.1, a rupture plane of 35 km length and 14 km vertical extend, as well as an average co-seismic displacement of 1.9 m. The initial stress field is shown in Figure 40. Values at the isolines of stress are in MPa.

  
  
  background stress level is about 2.7 MPa as explained above. The shear stress is reduced to very low up to negative values in the ridge segments (lower left and upper right) where the tensile stresses, oriented N103$^{\circ }$E, have been reduced to zero by rifting. At the tips of the ridge segments and on the line
  
    

  Figure: Same as Figure 37. The area around (130, 0) is enlarged.

  
  
  connecting them high shear stresses are induced. In the framework of an initial stress field as in Figure 40 an earthquake happened at (164, 0) releasing the stresses there to values around zero (see especially Figure 38 where the area around the epicenter has been enlarged). Please note that the yellow isolines in the figures correspond to different values: those near the ridge and earthquake rupture planes show stresses of -1MPa (Figure 40), those off the rupture plane tips give values of 3.2 MPa. Besides the reduction of stress at the fault plane, there is an increase off its tips and at intermediate distances along the strike of the rupture.

  
  The earthquake data $\\$All events in the SISZ with M $\ge$ 6 since 1706 were used (Table 2), following [,,]. Even though the time of first 3 events is not exactly known, their occurrence seems to be secured (in contrast to other events []) and the catalogue is supposed to be complete from 1706 for earthquakes with M $\ge$ 6 [].
  

   

  Table: 1) Data taken from []. 2) Position in the model coordinate system with origin at N, W. 3) Calculated via the magnitude moment relationship $\log M_0$ [dyne cm] $= 1.5M_S - (11.8 - \log(\sigma_a/\mu))$ with the apparent stress $\sigma _a = 150$ MPa and the shear modulus $\mu =0.39\cdot 10^{11}$ Pa after [], followed by using the values of $\mu$ above, the rupture length as given in the table as well as a vertical fault width of 14 km east of W and 7 km between W and W. Finally, the values were reduced by a factor of 2, following the discussion in []. 4) Calculated using $\log L$ [km] = 0.5 M -2 (after [60]) which results in slightly lower values compared to e.g. [70].

  Date1	Magnitude1	Epicenter1		South end of		Co-seismic	Rupture
  				rupture2		slip3	length4
  		Lat. $^{\circ }$N	Lon. $^{\circ }$W	x [km]	y [km]	U0 [m]	L [km]
  20.04.1706	6.0	64.0	21.2	131	-5	0.30	10
  07.09.1732	6.7	64.0	20.1	183	-11	0.77	22
  22.03.1734	6.8	63.9	20.8	150	-23	0.96	25
  14.08.1784	7.1	64.0	20.5	164	-18	1.9	35
  16.08.1784	6.7	63.9	20.9	145	-22	0.77	22
  26.08.1896	6.9	64.0	20.2	178	-14	1.2	28
  27.08.1896	6.7	64.0	20.1	183	-11	0.77	22
  05.09.1896	6.0	63.9	21.0	140	-16	0.30	10
  05.09.1896	6.5	64.0	20.6	159	-9	0.48	18
  06.09.1896	6.0	63.9	21.2	131	-16	0.30	10
  06.05.1912	7.0	63.9	20.0	187	-27	1.5	32

   
  
  

  The shape of their damage distribution and the strike of surface breaks (Figure 39) further confirms strike-slip events on N-S trending rupture planes []. Nevertheless, all but one event are not recorded instrumentally. The damage areas from historical records are not gathered by scientists and are as usually biased by uneven population density. So the magnitudes and locations are not very accurate.
  

    

  Figure: The SISZ showing mapped surface breaks and regions in which over half of the buildings were destroyed in historic seismic events []. The N-S dashed line near Vatnafjöll indicates the estimated location of the fault on which the May 25, 1987, earthquake occurred []. The structural features and coastline are from [].

  
  
  Due to the unknown initial stress field, the model results (especially the pre-seismic stress level) can only be checked at places where more than one event occurred in the sequence. Therefore, a long time span should be and was used.

  Determination of seismic moment, size of the fault plane, and average co-seismic slip from the magnitudes, was done as indicated in Table 2. As also discussed in [], there is the problem that fault lengths observed (e.g. 1912) are rather short compared to the magnitude of the events. Therefore a formula of Qian [60] was used here to calculate the rupture length. Additionally, the vertical fault width is limited by a brittle crustal thickness of about 15 km (cf. also the hypocenter vs. longitude plot of Stefánsson et al. [], their Figure 8) -- even though the effective short term crustal thickness reacting in brittle manner might be thicker for high frequency events as earthquakes compared to slow geological processes. For this first model, however, the co-seismic slip values were estimated to be lower (by a factor of 2) than from the global relation [] combined with the fault lengths used here. Hackman et al., had a reduction by a factor of 1.4 []. The reduction in slip values is equivalent to slightly reducing the magnitudes.

  Earthquake fault planes near the Hengill triple junction were assumed to extend vertically to 7 km depth, for events further east 14 km were used (cf. the above mentioned hypocenter depth distribution). All ruptures were set to be oriented N-S.

  
  The results $\\$The stress fields at 18 dates were calculated: the pre- and post-seismic situation for all 11 events and the stresses in 1998. The time before 5 events was too short to accumulate appreciable plate tectonic stresses since the preceding event. In these cases, the post-seismic stress field of the preceding event was set equal to the pre-seismic stress field of these earthquakes.

  Figure 40 gives the initial stress field in 1706. The rift segments, lower left and upper right, with about zero tensile stresses also show low shear
  

    

  Figure: Shear stress field in the SISZ and its surroundings as assumed in 1706 -- the starting field for the model calculations. The background stress is 2.7 MPa. It is increased between the rift tips which are at (60,-10) and (250, 0). -- Note that yellow isolines have two meanings: 3.2 and -1.0 MPa.

  
  
  stresses. The area between the tips of the rifts shows increased left-lateral E-W shear stresses (equivalent to right-lateral N-S shear stresses). The SISZ as a strike-slip fault zone extends from 23$^{\circ }$W, 64$^{\circ }$N or (125, 0) to 20.5$^{\circ }$W, 64$^{\circ }$N or (200, 0). Concerning the yellow isolines, please note the remarks on Figures 37 and 38. Figure 41 gives the stress field after the 1706, M=6, earthquake took place at (131, 0) reducing stresses there to values below 2.2 MPa.
  
    

  Figure: The stress field after the 1706, M=6, earthquake occurred at (131, 0).

  
  
  

  26 years later, at the break of the next large event, the stresses had slightly increased, see e.g. between (160, 0) and (175, 0) in comparing Figures 41 and 42. The 1732, M=6.7 event then hit at (183, 0) in an area of high stress.
  

    

  Figure: The stress field before the 1732 event.

  
  
  After this, the 1734, M=6.8 earthquake hit the high stress area at (150,-12), just aside the 1706 event (Figure 43). Then the largest shock, the August 14,
  
    

  Figure: The stress field after the 1734, M=6.8 earthquake occurred at point (150,-12).

  
  
  1784 earthquake, occurred in the high stress segment between the 1734 and 1732 deformation regions ((164, 0) (Figures 44 and 45).
  
    

  Figure: The stress field before the August 14, 1784 event.

  
  
  
  
    

  Figure: The stress field after the August 14, 1784, M=7.1 earthquake occurred at (164, 0).

  
  
  

  The two August 26 and 27, 1896, M=6.9 and M=6.7 events occurred at the same position as the 1732, although the stresses at (178-183, 0) had not yet recovered beyond the background value of 2.7 MPa, besides two very small areas at (183,${\pm}$17). Figures 46, 47 and 48 show the pre- and
  

    

  Figure: The stress field before the August/September 1896 event series.

  
  
  
  
    

  Figure: The stress field after the August 26, 1896, M=6.9 earthquake occurred at (178, 0).

  
  
  
  
    

  Figure: The stress field after the August 27, 1896, M=6.7 earthquake occurred at (183, 0).

  
  
  post-seismic situation. The September 1896 events occurred further west, but did not hit small high stress spots at about (150, 15) and (105,${\pm}$20).

  The stress field 16 years later, before the 1912 earthquake (M=7, at (187, -11)) had not changed much compared to 1896, but the stresses in the south part of the epicentral area were rather high. Figure 49 shows the stress field after the
  

    

  Figure: The stress field after the May 6, 1912, M=7.0 earthquake occurred at (187, -11).

  
  
  1912 event.

  Figure 50 summarizes the mean shear stress level before each of the earthquakes
  

    

  Figure: Cross plot of the pre-seismic shear stress level at the site of the impending earthquakes vs. occurrence time. The stress values at 2 to 14 test points near the surface trace of the rupture plane were averaged.

  
  
  at the area of the impending event. The stress level for the 1706, 1732, 1734, the first 1784, the first and last 1896, and the 1912 events are rather high ($\ge$2.0 MPa). The second 1784 earthquake (two days later, 0.4 in magnitude smaller, 19 km away) might be an aftershock and therefore situated in a lower stress area (1.2 MPa). The same might apply to the second, third and fourth of the 1896 events, whereas the last 1896 shock hit a high stress area (2.5 MPa) -- the place of the 1706 earthquake, where the stress had recovered since then.

  Finally, the stress field was extrapolated to spring 1998, with the additional stresses due to plate motion since 1912. Figures 51 and 52 show the shear
  

    

  Figure: Extrapolation of the shear stress distribution from 1912 to 1998.

  
  
  
  
    

  Figure: Same as Figure 51. The area around (150, 0) is enlarged. The SISZ extends approximately between (125, 0) to (200, 0).

  
  
  stress distributions. The shear stress inside the SISZ (Figure 52) are only high at (150, 10) and at some places at the northern and southern margins of the seismically active zone. West and east of the zone there are high stresses where the SISZ meets the ridge segments and where stresses may be release by earthquakes or by aseismic creep due to high temperature near the surface at changing places and variable in time (this is not yet included in the model). Interestingly, in 1987, there was a strong earthquake, M_S=5.8, at $63.91^\circ$N $19.78^\circ$W (198,-9) near Vatnafjöll (Figure 39). This event was not included in the modelling as its magnitude was below M=6. Nevertheless, it occurred in a region of high stress as shown in Figures 51 and 52.

  
  Discussion $\\$The pre-seismic stress for most main shocks is high, 2.0-2.8 MPa. It is fairly stable. As the initial stress field is unknown, the stress level at a site where the earthquakes recur in the sequences provides a good test for the right assumptions concerning the stress build-up due to plate movements: 1706 compared to September 6, 1896 and 1912; 1732 compared to August 1896. The tendency with time towards slightly lower values, is an indication that the stress increase due to rifting might have been assumed too low, i.e. the spreading rate between 1706 and 1912 might be higher than 2 cm/year.

  Even though the earthquake rupture planes strike N-S, the stress changes calculated here affect the whole area of the SISZ. The areas west and east of the SISZ show constant stress increase in most parts. Here, the model should be improved to account for an extension of the spreading movements into these regions. Moreover, the initial stress field of 1706 could be reduced in the eastern part and the central part, where the first events did not occur before 1732 and 1734, respectively.

  Smaller, instrumentally recorded events (as e.g. the above named Vatnafjöll event) will be included in a model with a denser test point distribution, based on the stress field determined here for 1912.

  The known layering of the crust and upper mantle below Iceland is now introduced into the model, especially to account for inelastic layers, i.e. to include post-seismic relaxation processes. This will be compared to continuous GPS crustal deformation data as soon as these are available. A new code is prepared for this much faster, more accurate, and capable of including even more layers than the existing one.

  
  Conclusions $\\$The fact that almost all events occur in high stress areas indicates that this first and simplified model can already explain the main features of the behaviour of the SISZ. This is especially astonishing, when the fact is kept in mind, that most (all but one) events used are not instrumentally recorded.

  The model goes beyond the standard earthquake moment release analysis as it includes the spatial location and extension of the events and provides an extrapolation to the present situation.