Determination of the Lithosphere-Asthenosphere Boundary using program LABWA2015

Lithosphere-Asthenosphere Boundary (LAB) is a lower boundary of the lithospheric plate, so, it is an important tectonic boundary. We present the package of numerical program LABWA2015 developed for simple calculations of position of LAB. It assumes isostatic state and uses gravity as well as topographic data. However, program provides better results if additional geophysical data are used, e.g. seismic data about position of Moho. If position of LAB is determined by other methods, the package can be used for determining density or thermal properties of lithosphere and asthenosphere. Contrary to earlier methods, LABWA2015 uses the full equation of thermal conduction. Problem of the lack of isostasy in some regions is also discussed.


INTRODUCTION
Lithosphere, asthenosphere and LAB (lithosphere-asthenosphere boundary) are popular terms in the physics of the solid Earth, however their meaning are differently understand by different specialists. In the plate tectonics theory the lithosphere is a layer divided on tectonic plates, i.e. the units which could move one in respect to another. The asthenosphere is a layer of low viscosity detaching the lithospheric plates from the mantle below. Unfortunately, the motion of some plates is very slow and we do not know the true motion of the mantle material below the plates. The models of mantle convection have limited resolution. Therefore sometimes this definition is difficult to be used. A few other definitions of the lithosphere, asthenosphere and LAB are: -seismic definition (asthenosphere is a layer of low velocity and high attenuation of seismic waves, lithosphere is a layer consisting of the crust and mantle above the asthenosphere), -thermal definition (lithosphere is a thermal boundary layer of the mantle convection cell, asthenosphere is a layer where temperature is close to the temperature of solidus), -magnetotelluric definition (asthenosphere is a high electrical conductivity layer), -rheologic definition (the lithosphere is elastic or brittle (especially the crust), while the asthenosphere deforms viscously and could accommodate strain through plastic deformation.
Of course, provided that we consider the rate of deformation corresponding to geodynamic processes).
Note also that even in the seismology a few sub-definitions exist. Jones et al. (2010) indicate a few types of LAB obtained by different seismological methods: i.e. by the receiver functions method (sLABrf) and by the determination of seismic anisotropy change (sLABa), in addition to (eLAB) obtained by magnetotelluricse.g. Eaton et al. (2009).
Tectonics also suggests a few possibilities. Below moving plate one has an 'active asthenosphere' with a large vertical gradient of the horizontal velocity (this definition corresponds to the definition used in the plate tectonics). If the gradient is low but the viscosity is also low, then one can use the term: a 'potential asthenosphere' (i.e. it would be an active asthenosphere if the plate were in motion). Moreover, the possibility of asthenospheric layers of thermal and mechanical origin were also indicated -Czechowski and Grad (2015 a, b).
Fortunately, most of above definitions concern different properties of the same layer.
Therefore, a few methods could be used independently for determination position of LAB and conditions in the asthenosphere. It is the main idea of program LABWA2015. It uses the 'classical' method of determination of LAB based on isostasy -e.g. Krysiński et al (2013Krysiński et al ( , 2015, Grinc et al.. (2014). Moreover, it includes also full equation of heat transport (according to our best knowledge it is the first such program). It can use also seismic data in the equation of state and enables to introduce some corrections resulting from the absence of isostatic equilibrium in some regions (Czechowski 2017). Compare also with the model presented by Jones et al. (2014).??

LOCAL ISOSTASY
The idea of local isostasy is based on the assumption that upper layer of the Earth could be treated as series of rigid columns (blocks) that float on a liquid layer. In hydrostatic state the pressure in a given liquid depends only on the depth. It means that the total mass above some level (known as a compensation level) is the same for each column. This idea was proposed in XIX century and was used in many investigations. For some time the compensation level was placed close to the Moho (e.g. Turcotte and Schubert, 2002, p. 74). However, according to the plate tectonics theory, the asthenosphere (instead of the lithosphere just below of Moho) is a layer which could be treated as a liquid for slow tectonic motions. Hence, the compensation level should be placed inside the asthenosphere. It means that the rigid columns correspond to the lithosphere, while the 'liquid' layer corresponds to the asthenosphere. It is presented on Fig.   1 for the continental lithosphere (when the solid surface is above sea level) and for the oceanic lithosphere (when rocks are covered by a water layer).
where g=9.81 m s -2 , G= 6.67 10 -11 N m 2 kg -2 , and Δρ(z) is the density contrast in respect to a given reference column. Integration is from the surface down to the compensation level. For the presented above simplified model of the lithosphere, the geoid anomaly N is given by (Fullea et al., 2006): where N0 is an integration constant.
Topography E is known and could be taken from different databases. The geoid data for

LIMITATION OF THE METHOD
The set of equations (1)  The assumption of isostasy is justified if the crust contains many preexisting faults.
These preexisting zones of weakness could be reactivated under the tectonic stresses (e.g.  give equations with additional unknown parameters introduced by additional, hypothetical block with different properties.

ISOSTASY AND DYNAMICAL PROCESSES IN THE MANTLE
It is said above that the isostasy (isostatic equilibrium) is a state when solid blocks of the 2. Oceanic spreading centers, where the lithosphere is thin, but the upward flow of the mantle material below the lithosphere causes large vertical forces that may not allow for isostasy.
3. Hot spots. They are areas located above the rising mantle plumes that causes large vertical forces that may not allow for isostasy.
4. Some zones of horizontal flow in asthenosphere. The flow could be forced by motion the moving oceanic plates. In such a case the state of isostasy could be approximately attained.
Horizontal flow in the asthenosphere could be also a result of gradient of the pressure in the asthenosphere. In such a case some modifications of the procedure in LABWA2015 are necessary (see Czechowski 2017). The simplest one is introduction of pressure gradient in the asthenosphere.

EXTENSION OF THE METHOD IN LABWA2015
The method based on the principle of isostasy is sensitive for the distribution of density.
This distribution depends on the distributions of pressure, temperature and composition, so they could be also determined (at least to some degree).  The density is coupled with the temperature and pressure by the following equation of state: where α is the coefficient of thermal expansion and K is the bulk modulus.
It means that some seismic data from the lithosphere could be also taken into account.
The number of parameters of thermal model for a given column depends on the complexity of the assumed model. It could contain several different layers if the properties of these layers are given. Unknown properties introduce unknown parameters in the numerical package. Presently, thermal model is 1D but model 3D could be incorporated into the numerical code.
In the region where position of LAB is determined by other methods (e.g. by the seismology or by the magnetotellurics), the package could be used to determine some other properties of the lithosphere and asthenosphere. The program LABWA2015 allows also to choose the unknowns in the system of equations, e.g. k, Q, hf, α, E, B, Φ0 could be treated as unknowns. The user of the program must divide the studied region into blocks to provide a sufficient number of independent equations.

EXAMPLES OF APPLICATION OF THE MODEL
A few examples of the results of the model are presented below.
The Fig. 3   Geoid is from the model EGM2008. A flat topography (100 m) is used. Moreover, the radiogenic heat production in the upper crust is 2 10 -6 W m -3 , the density of the upper crust is 2760 kg m -3 , the coefficient of thermal conductivity in the upper crust is 2.6 W m -1 K -1 .    6. Reaction of LAB to changing parameters. The lower surface presents the LAB calculated using the same parameters as in Fig. 3. The upper surface shows the LAB calculated using a higher density of the upper crust (100 kg m -3 higher). The values of other parameters are the same as in Fig. 3.

CONCLUSIONS
We Realistic values of the thermal and mechanical properties should be used in the model.
Note, that some of these conditions could be not satisfied in some regions (e.g. the isostasy of the blocks or the position of the sources of geoid anomalies). Therefore the best way to check the reliability is to compare the results of LABWA2015 with the results of an independent method. For example, determination of LAB using seismic method in a single point of the considered region could give some insight into reliability of LABWA2015 for the whole region.