### Regional magnetic feature and geology

A high-resolution aeromagnetic survey over the Ikogosi Warm Spring and its surroundings was conducted between 2004 and 2008 and published by the Geological Survey of Nigeria (GSN) on a map scale of 1:100,000 series. The various map sheets obtained were processed and merged together to a common dataset. They were transformed to a total field anomaly dataset and gridded at 0.5 km. The analysis of magnetic field used reduced to the pole data (RTP). The RTP correction applied assumed a declination of −3° and an inclination of −10° for this region utilizing the method of Silva ([1986]). Figure 3 shows the geomagnetic anomaly field map of the study area. The map had the regional geomagnetic field and the effects of diurnal magnetic variations removed.

The magnetic map and geologic map shows a good correlation between exposed geologic units and magnetic signatures. The strong variations in magnetic intensity suggest a wide variety of different magnetic properties. Notable (positive) anomalies are observed at the Oshogbo, Okemesi and Ilesha locations (Figure 3) reaching values of 60 to 112 nT. This tends to correspond with the exposed undifferentiated metasediments noted in these locations (Figure 2). Positive anomalies of 50 to 70 nT are also observed at the Ijero-Ekiti location whose probable source could be traced to the undifferentiated metasediments in this region. Within the towns of Ado-Ekiti and Ikere, positive magnetic anomalies of 80 to 110 nT were obtained. We traced this observation to the Older Granites domiciled in this region. In other regions of the map encompassing the towns of Erinmo, Ifewara, Oriade, Ikogosi and Okeigbo, negative magnetic anomalies ranging from −30 to −156 nT have been noted. The belt of quartzite, quartz-mica schist and granulitic migmatite encapsulated by the quartzite belt in the region is identified with this observation. This quartzite is part of the Okemesi quartzite member of the Effon Psammite formation (east of Ilesha) belonging to the Ife-Ilesha schist belt of the Precambrian basement complex of Nigeria (Loehnert [1985]; Adegbuyi et al. [1996]). The Ikogosi town (study location) has magnetic lows of −0.46 to −35 nT, extending westwards. This location is also part of the fractured quartzite unit in the formation.

One of the methods of examining thermal structure of the crust is the estimation of the CPD, using aeromagnetic data (Dolmaz et al. [2005]). Various studies have shown correlations between Curie temperature depths and average crustal temperatures, leading to viable conclusions regarding lithospheric thermal conditions in a number of regions around the world (Ross et al. [2006]). The mathematical model on which our analysis is based is a collection of random samples from a uniform distribution of rectangular prisms, each prism having a constant magnetization. Two fundamental methods serve as a basis of all subsequent analysis, first provided by Spector and Grant ([1970]), estimating the average depths to the top of the magnetized bodies from the slope of the log power spectrum and second by Bhattacharyya and Leu ([1975]) for obtaining the depth to the centroid of the causative body. The method can provide valuable information about the regional temperature distribution at depths not easily examined using other methods (Okubo et al. [1985]). The Curie temperature isotherm corresponds to the temperature at which magnetic minerals lose their ferromagnetism (approximately 580°C for magnetite at atmospheric pressure). Magnetic minerals warmer than their Curie temperature are paramagnetic and from the perspective of the earth’s surface are essentially nonmagnetic (Ross et al. [2006]). Thus the Curie temperature isotherm corresponds to the basal surface of magnetic crust and can be calculated from the lowest wavenumbers of magnetic anomalies, after removing the approximate regional field from the aeromagnetic data (e.g. Bhattacharyya and Morley [1965]; Spector and Grant [1970]; Mishra and Naidu [1974]; Byerly and Stolt [1977]; Connard et al. [1983]; Hamdy et al. [1984]; Blakely [1988]; Tanaka et al. [1999]; Salem et al. [2000]; Ross et al. [2006]).

We applied the methods of Spector and Grant ([1970]), Okubo et al. ([1985]) and Trifonova et al. ([2006]), which examined the spectral knowledge included in subregions of magnetic data for our analysis.

### Spectral analysis

The earliest papers on Curie point depth determination based on spectral analysis of geomagnetic data are those of Byerly and Stolt ([1977]) where analyses for different areas of USA have been published. More recently, investigations have been made for parts of the territory of Japan (by Okubo [1985, 1989, 1994]), USA (by Mayhew [1985]; Blakely [1988]), Greece (Tsokas et al. [1998]; Stampolidis and Tsokas [2002]), Portugal (Okubo et al. [2003]), Bulgaria (Trifonova et al. [2006, 2009]) and Turkey (Dolmaz et al. [2005]; Maden [2009]). Authors consider the power spectrum of the total geomagnetic field intensity anomaly over a single rectangular block using the expression, which was first given by Bhattacharyya ([1965]). The equation was transformed into polar wavenumber coordinates (S,ψ), and the average depths to the top of magnetized bodies from the slope of the log power spectrum were calculated. The model has proven successful in estimating average depths to the tops of magnetized bodies (Trifonova et al. [2006]).

One principal result of Spector and Grant’s analysis is that the expectation value of the spectrum for the model is the same as that of a single body with the average parameters for the collection (Okubo et al. [1985]).

In polar coordinates (S,ψ) in frequency space, this spectrum has the form

\mathit{F}\left(\mathrm{S},\mathrm{\psi}\right)=2\mathrm{\pi}\mathit{JA}\left[\mathit{N}+\mathit{i}\left(\mathit{L}cos\mathrm{\psi}+\mathit{M}sin\mathrm{\psi}\right)\right]\times \left[\mathit{n}+\mathit{i}\left(\mathit{l}cos\mathrm{\psi}+\mathit{m}sin\mathrm{\psi}\right)\right]\phantom{\rule{0.25em}{0ex}}

(1)

\times sin\mathit{c}\left(\mathrm{\pi}\mathit{sa}cos\mathrm{\psi}\right)sin\mathit{c}\left(\mathrm{\pi}\mathit{sb}sin\mathrm{\psi}\right)

\times exp\left(-2\mathrm{\pi}\mathit{is}\left({\mathit{x}}_{\mathrm{o}}cos\mathrm{\psi}+{\mathit{y}}_{\mathrm{o}}sin\mathrm{\psi}\right)\right)\phantom{\rule{0.25em}{0ex}}

\times \left[exp\left(-2\mathrm{\pi}\mathit{s}{\mathit{z}}_{\mathrm{t}}\right)-exp\left(-2\mathrm{\pi}\mathit{s}{\mathit{z}}_{\mathrm{b}}\right)\right]\phantom{\rule{0.25em}{0ex}}

where *J* = magnetization per unit volume; *A* = average cross-sectional areas of the bodies; *L*, *M*, *N* = direction cosines of the geomagnetic field; *l*, *m*, *n* = direction cosines of the average magnetization vector; *a* and *b* = average body *x*- and *y*-directions, *x*_{o} and *y*_{o} = average body *x*- and *y*-centre locations and *z*_{t} and *z*_{b} = average depths to the top and bottom of the bodies, and where

\mathrm{sinc}\left(\mathit{x}\right)=\frac{sin\mathit{x}}{\mathit{x}}

Following Bhattacharyya and Leu ([1975, 1977]), estimation of the bottom depths could be approached in two steps: first, find the centroid depth *z*_{o} and second, determine the depth to the top *z*_{t}. The depth to the bottom (inferred CPD) is calculated from these values:

{\mathit{z}}_{\mathrm{b}}=2{\mathit{z}}_{\mathrm{o}}-{\mathit{z}}_{\mathrm{t}}

(2)

The terms involving *z*_{t} and *z*_{b} can be recast into a hyperbolic sine function of *z*_{t} and *z*_{b} plus a centroid term. At very long wavelengths, the hyperbolic sine tends to unity, leaving a single term containing *z*_{o}, the centroid. At somewhat shorter wavelengths, the signal from the top dominates the spectrum and an estimate of the depth to the top can be obtained (Okubo et al. [1985]). If we begin with the centroid, at very long wavelengths (compared to the body dimensions), the terms involving the body parameters (*a*, *b*, and *z*_{b-}*z*_{t}) may be approximated by their leading terms, to yield

\mathit{F}\left(\mathrm{S},\mathrm{\psi}\right)=4\mathrm{\pi}\mathit{VJs}\left[\mathit{N}+\mathit{i}\left(\mathit{L}cos\mathrm{\psi}+\mathit{M}sin\mathrm{\psi}\right)\right]\times \left[\mathit{n}+\mathit{i}\left(\mathit{l}cos\mathrm{\psi}+\mathit{m}sin\mathrm{\psi}\right)\right]\phantom{\rule{0.25em}{0ex}}

(3)

\times exp\left(-2\mathrm{\pi}\mathit{is}\left({\mathit{x}}_{\mathrm{o}}cos\mathrm{\psi}+{\mathit{y}}_{\mathrm{o}}sin\mathrm{\psi}\right)\right)\phantom{\rule{0.25em}{0ex}}

\times \phantom{\rule{0.5em}{0ex}}exp\left(-2\mathrm{\pi}\mathit{s}{\mathit{z}}_{\mathrm{o}}\right),

where *V* is the average body volume.

Equation 3 can be recognized as the spectrum of a dipole.

In effect, the ensemble average at these very low frequencies is that of a random distribution of point dipoles. What follows, therefore, is independent of the details of the body parameters (prisms, cylinders or whatever), provided that the dimensions in all directions are comparable. The method of Okubo et al. ([1985]) was used in estimating *z*_{o} from Equation 3:

If \mathit{G}\left(\mathrm{S},\mathrm{\psi}\right)=\frac{1}{\mathit{s}}\mathit{F}\left(\mathrm{S},\mathrm{\psi}\right)

First, average the square amplitude of *G* over an angle in the frequency plane

{\mathit{H}}^{2}\left(\mathit{s}\right)=\frac{1}{2\mathrm{\pi}}{\displaystyle \underset{-\mathrm{\pi}}{\overset{\mathrm{\pi}}{\int}}}{\left|\mathit{G}\left(\mathrm{S},\mathrm{\psi}\right)\right|}^{2}\mathit{d}\mathrm{\psi}

(4)

Then *H*(*s*) has the form

\mathit{H}\left(\mathit{s}\right)=\mathit{A}exp\left(-2\mathrm{\pi}\mathit{s}{\mathit{z}}_{\mathrm{o}}\right)

if *F* satisfies Equation 3, where *A* is a constant. Hence,

ln\mathit{H}\left(\mathit{s}\right)=ln\mathit{A}-2\mathrm{\pi}\mathit{s}{\mathit{z}}_{\mathrm{o}}

(5)

holds. The centroid depth *z*_{o} can now be estimated by least-squares fitting ln *H*(s) with a constant and a term linear in *s*.

The second step is the process of estimating the depth to the top. For this purpose, we return to Equation (1) and assume that a range of wavelengths can be found for which the following approximations hold:

\mathrm{sinc}\left(\mathrm{\pi}\mathrm{\pi a}cos\mathrm{\psi}\right)\approx 1,

\mathrm{sinc}\left(\mathrm{\pi}\mathrm{\pi a}sin\mathrm{\psi}\right)\approx 1,

and

exp\left(-2\phantom{\rule{0.25em}{0ex}}\mathrm{\pi}\mathit{sa}{\mathrm{z}}_{\mathrm{b}}\right)\approx 0.

For these approximations to make sense, the bodies must in general be large in depth compared to their horizontal dimensions. However, if the distribution of horizontal body dimension is very broad, a similar effect will be produced by variability in terms corresponding to the horizontal body dimensions.

If the above approximation holds, the spectrum reduces to the form

\mathit{F}\left(\mathrm{S},\mathrm{\psi}\right)=2\mathrm{\pi}\mathit{JA}\left[\mathit{N}+\mathit{i}\left(\mathit{L}cos\mathrm{\psi}+\mathit{M}sin\mathrm{\psi}\right)\right]\times \left[\mathit{n}+\mathit{i}\left(\mathit{l}cos\mathrm{\psi}+\mathit{m}sin\mathrm{\psi}\right)\right]

(6)

\times exp\left(-2\mathrm{\pi}\mathit{is}\left({\mathit{x}}_{\mathrm{o}}cos\mathrm{\psi}+{\mathit{y}}_{\mathrm{o}}sin\mathrm{\psi}\right)\right)

\times exp\left(-2\mathrm{\pi}\mathit{s}{\mathit{z}}_{\mathrm{t}}\right)

which is very similar to Equation 3, except for a factor of *s*. Equation 6 is in fact the spectrum of a monopole.

Estimation of *z*_{t} is therefore done using

{\mathit{K}}^{2}\left(\mathit{s}\right)=\frac{1}{2\mathrm{\pi}}{\displaystyle \underset{-\mathrm{\pi}}{\overset{\mathrm{\pi}}{\int}}}{\left|\mathit{F}\left(\mathrm{S},\mathrm{\psi}\right)\right|}^{2}\mathit{d}\mathrm{\psi}

(7)

from which

\mathit{k}\left(\mathit{s}\right)=\mathit{B}exp\left(-2\mathrm{\pi}\mathit{s}{\mathit{z}}_{\mathrm{t}}\right)

follows, where *B* is a sum of constants independent of *s*,

then from

ln\mathit{K}\left(\mathit{s}\right)=ln\mathit{B}-2\mathrm{\pi}\mathit{s}{\mathit{z}}_{\mathrm{t}}

(8)

and fit ln *K* (*s*) with a constant and a term linear in *s*.

The reliability of this method has been proven in many cases (e.g. Okubo [1994]; Tsokas et al. [1998]; Trifonova et al. [2006]). Fast Fourier transform (FFT) estimates Fourier components between zero frequency and the Nyquist limit imposed by the grid cell size. The Nyquist frequency is the highest frequency (short wavelength) that is possible to measure given a fixed sample interval (Yawsangratt [2002]). It is defined by the expression (Clement [1972]; Billing and Rechards [2000]),

\mathit{N}=\frac{1}{\left(2\mathit{\Delta x}\right)}

(9)

where ∆*x* is the sampling interval.

The sampling interval used during the analysis of data in this work, 500 m (0.50 km), directly imposes a Nyquist frequency of 1 km on the data.

### Conductive heat flow

The basic relation for conductive heat transport is Fourier’s law (Tanaka et al. [1999]). In one-dimensional case under the assumptions that the direction of the temperature variation is vertical and the temperature gradient \raisebox{1ex}{$\mathrm{dT}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{dz}$}\right. is constant, Fourier’s law takes the form

\mathit{q}=\mathit{k}\frac{\mathrm{dT}}{\mathrm{dz}}\phantom{\rule{0.5em}{0ex}}

(10)

where *q* is the heat flux and *k* is the coefficient of thermal conductivity.

According to Tanaka et al. ([1999]), the Curie temperature (θ) can be obtained from the Curie point depth *z*_{b} and the thermal gradient \raisebox{1ex}{$\mathrm{dT}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{dz}$}\right.\phantom{\rule{0.25em}{0ex}} using the following equation:

\mathrm{\theta}=\left(\frac{\mathrm{dT}}{\mathrm{dz}}\right)\phantom{\rule{0.25em}{0ex}}{\mathit{z}}_{\mathrm{b}}

(11)

In this equation, it is assumed that the \raisebox{1ex}{$\mathrm{dT}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{dz}$}\right.\phantom{\rule{0.5em}{0ex}} is constant.

From Equations 10 and 11,

{\mathit{z}}_{\mathrm{b}}=\mathit{k}\frac{\mathrm{\theta}}{\mathit{q}}

(12)

Tanaka et al. ([1999]) showed that any given depth to a thermal isotherm is inversely proportional to heat flow, where *q* is the heat flow. This equation implies that regions of high heat flow are associated with shallower isotherms, whereas regions of lower heat flow are associated with deeper isotherms (Ross et al. [2006]). An average surface heat flow value was computed using Equations 10 and 11 and was based on possible Curie point temperature of 580°C using a thermal conductivity of 2.5 Wm^{−1}°C^{−1}, given by Stacey ([1977]) as the average for igneous rocks.

### Radiogenic heat generation

The presence of thorium-rich accessory minerals, such as monazite, zircon and rutile is usually linked to micaceous zones in radio-active porphyritic and coarse granite (Ragland and Roger [1961]; Gerard and Kappelymeyer [1987]). Such granitic rock bodies are found to be suitable nuclear resources including the uraniferous sandstone and marginal marine sediments. Using the scanning electron microscope to determine the spot chemical composition and empirical formulae of nearly all rock-forming minerals in the rocks of the basement complex of southwestern Nigeria, Oyinloye ([2011]) discovered the mineral, monazite. He concluded that this mineral was present as a notable accessory mineral in all the crystalline rocks of the basement complex in Ilesha area even in the amphibolites which is supposed to be igneous. Monazite is a phosphate of the rare earth elements, especially the light ones. The petrogenetic implication of the presence of monazite in the crystalline rocks of southwestern Nigeria is that the initial magma from which the precursor rocks were formed contains some input from the crustal or sedimentary source. There is therefore the possibility of a uranium/thorium bearing unit at a vertical depth in the micaceous-quartzite rock unit of the Ikogosi Warm Spring area (Adegbuyi and Abimbola [1997]). The majority of the continental heat flow originates from the decay of radioactive isotopes in the crust, thus finding areas with high isotope concentration can be equal to finding areas with high heat flow (Holmberg et al. [2012]).

Radiogenic heat production (RHP), *H* (μW/m^{3}) is related to the decay of primarily, the radioactive isotopes ^{232}Th, ^{238}U and ^{40}K and can be estimated based on the concentration (*C*) of the respective elements (Rybach [1988]; Holmberg et al. [2012]) through Equation 13:

\mathit{H}=\mathit{\rho}\left(9.52{\mathit{C}}_{\mathrm{U}}+2.56{\mathit{C}}_{\mathrm{Th}}+3.48{\mathit{C}}_{\mathrm{K}}\right){10}^{-5}

(13)

where *ρ* is the density of the rock and its concentrations in uranium (*C*_{u}), thorium (*C*_{Th}) are given in weight/parts per million and weight percent for potassium (*C*_{K}).

Heat flow depends critically on radioactive heat production in the crust. The two primary effects are thus that continental heat flow is proportional to the surface crustal radioactivity in a given region and decreases with time since last major tectonic event (Stein [1995]). Heat flow must be a continuous function inside the earth; in particular, it will be the same on both sides of the boundary separating the crust from the mantle (Masters and Constable [2013]). A plot of heat flow and heat production rate from radioactivity of rocks revealed a linear distribution which can be fitted with a linear equation of the form:

{\mathit{q}}_{\mathrm{s}}={\mathit{q}}_{\mathrm{m}}+{\mathit{q}}_{\mathrm{r}}

(14)

where *q*_{s} is the surface heat flow and *q*_{m} is the mantle heat flow (heat flow into the base of the crust). The total contribution of heat production in the crust to the surface heat flow *q*_{r} is therefore,

{\mathit{q}}_{\mathrm{r}}=\mathit{\rho}{\mathit{H}}_{\mathrm{s}}{\mathit{z}}_{\mathrm{r}}

(15)

where *ρ* is density of the crust, *H*_{s} is the heat production measured in rocks collected at the surface, *z*_{r} could be interpreted as the ‘equivalent depth extent of heat production’, that is, the extent to which the heat production measured at the surface (*H*_{s}) extends to depth if we consider distribution models for radioactivity in the crust and extent of heat production (Stuwe [2008]). Masters and Constable ([2013]) had assumed that *z*_{r} is much less than the thickness of the continental crust. Clearly, in nature, radioactivity is not constant in the crust down to *z*_{r} and zero below that, but this model gives us a fair indication of the proportion of surface heat flow that is due to radioactivity (Stuwe [2008]).