Patent Publication Number: US-10316643-B2

Title: High resolution distributed temperature sensing for downhole monitoring

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     The present application is related to U.S. patent application Ser. No. 14/062561, filed Oct. 24, 2013, now abandoned, the contents of which are hereby incorporated herein by reference in their entirety. 
     BACKGROUND OF THE DISCLOSURE 
     1. Field of the Disclosure 
     The present application relates to methods for increasing a resolution of measurements obtained downhole and, in particular, to methods for increasing resolution of temperature measurements obtained using a distributed temperature sensing system in a wellbore. 
     2. Description of the Related Art 
     Temperature measurements obtained in a wellbore can be useful in performing downhole operations such as determining a placement of an injection fluid, determining an injection profile, determining a production profile, determining an oil/liquid interface, etc. One method of obtaining temperature measurements downhole includes the use of a distributed temperature sensing (DTS) system. DTS systems measure temperatures by means of one or more optical fibers functioning as distributed sensor arrays. The one or more optical fibers are generally run along the wellbore. Temperatures are recorded along the optical fiber as a continuous profile. The DTS system generally provides a temperature measurement having a spatial resolution from about 0.5 meters to about 1 meter and a temperature resolution from about 1.5° C. to about 0.5° C. when measured at a scan rate of one to several minutes. At a deep downhole location, the geothermal environment is thermally stable. Microvariations in temperature occurring downhole may be indicative of a geological event, a wellbore operation, a well integrity issue, a flow assurance problem, or a change in the status of downhole control devices, etc. The microvariations associated with these events, issues and/or operations are generally below the level of resolution directly provided by current DTS systems. 
     SUMMARY OF THE DISCLOSURE 
     In one aspect, the present disclosure provides a method of obtaining a temperature profile of a wellbore, the method including: obtaining raw temperature data from the wellbore using a distributed temperature sensor system, the raw temperature data including noise; performing a numerical decomposition of the raw temperature data within a dynamic window in measurement space of the raw temperature data to obtain decomposition terms of first order and higher; applying an adaptive filter to the dynamic window to reduce noise from the decomposition terms of first order and higher; and using the filtered decomposition terms of first order and higher to obtain a temperature profile of the wellbore. 
     In another aspect, the present disclosure provides a system for obtaining a temperature profile at a downhole location, the system including: a distributed temperature system configured to obtain raw temperature data from the downhole location, wherein the raw temperature data includes noise; and a processor configured to: perform a numerical decomposition of the raw temperature data within a dynamic window in measurement space of the raw temperature data to obtain decomposition terms of first order and higher; apply an adaptive filter to the dynamic window to reduce noise from the decomposition terms of first order and higher; and use the filtered decomposition terms of first order and higher to obtain a temperature profile of the wellbore. 
     In yet another aspect, the present disclosure provides a computer-readable medium having instructions stored thereon that are accessible to a processor and enable the processor to perform a method for obtaining a temperature profile at a downhole location, the method including: obtaining raw temperature data from the downhole location from a distributed temperature sensor system; performing a numerical decomposition of the raw temperature data within a dynamic window in measurement space of the raw temperature data to obtain decomposition terms of first order and higher; applying an adaptive filter to the dynamic window to reduce noise from the decomposition terms of first order and higher; and using the filtered decomposition terms of first order and higher to obtain a temperature profile of the wellbore. 
     summarized rather broadly in order that the detailed description thereof that follows may be better understood. There are, of course, additional features of the apparatus and method disclosed hereinafter that will form the subject of the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present disclosure is best understood with reference to the accompanying figures in which like numerals refer to like elements and in which: 
         FIG. 1  shows a wellbore system having a distributed temperature system for determining a temperature at a downhole location in an exemplary embodiment of the present disclosure; 
         FIG. 2  shows an alternate embodiment of a wellbore system suitable for temperature measurements according to the present disclosure; 
         FIG. 3  shows an exemplary data boundary of a localized two-dimensional subspace of the measurement space; 
         FIG. 4  shows a schematic diagram of an iterative self-adaptive algorithm of the present disclosure; 
         FIG. 5  shows a flowchart illustrating an exemplary method of correcting bi-directional DTS temperature measurements for asymmetric signal loss; 
         FIG. 6  show a flowchart illustrating an exemplary method of reducing system level noises in the DTS data; and 
         FIG. 7  shows various thermal gradient data sets obtained using a distributed temperature system measurements. 
     
    
    
     DETAILED DESCRIPTION OF THE DISCLOSURE 
       FIG. 1  shows a wellbore system  100  having a distributed temperature sensing system  110  for determining a temperature at a downhole location in an exemplary embodiment of the present disclosure. The exemplary wellbore system  100  includes a tubular member  102  disposed in a wellbore  104  formed in a formation  106 . The wellbore  104  may be lined with a casing string  108  and the member  102  may be a casing string or disposed inside the casing string  108 . In the latter case, the member  102  may be a production tubing, a coiled tubing, or a downhole tool in various embodiments. 
     The wellbore system  100  further includes a distributed temperature sensing (DTS) system  110  that is used to obtain a temperature profile along the wellbore  104  over a selected time interval. The DTS system  110  includes fiber optic cable  112  that extends downhole, generally from a surface location. In the embodiment of  FIG. 1 , fiber optic cable  112  is disposed alongside member  102 . In other embodiments, the fiber optic cable  110  may be disposed along the casing string  108  or between the casing string  108  and the formation  106 . Thus, the fiber optic cable may be either permanently deployed or may be removable from the wellbore along with the removable member to which it is attached. 
     The DTS system  110  includes an optical interrogator  114  which is used to obtain raw temperature measurements from the fiber optic cable  112 . The optical interrogator  114  includes a laser light source  118  that generates a short laser pulse that is injected into the fiber optic cable  112  and a digital acquisition unit (DAU)  120  for obtaining optical signals from the fiber optic cable  112  in response to the laser pulse injected therein. The obtained optical signals are indicative of temperature. In one embodiment, Raman scattering in the fiber optic cable  112  occurs while the laser pulse travels along the fiber, resulting in a pair of Stokes and anti-Stokes peaks. The anti-Stokes peak is highly responsive to a change in temperature while the Stokes peak is not. A relative intensity of the two peaks therefore provides a measurement indicative of temperature change. The back-reflected Raman scattering (i.e., the Stokes and anti-Stokes peaks) may thus transmit the temperature information of a virtual sensor while the laser pulse is travelling through the fiber optic cable  112 . The location of the virtual sensor is determined by the travel time of the returning optical pulse from the interrogator  114  to the signal detector  120 . 
     The DAU  120  obtains raw temperature measurement data (raw data) and sends the raw data to a data processing unit (DPU)  116 . The DPU  116  performs the various methods disclosed herein for increasing a resolution of temperature measurements, among other things. The DPU  116  may include a processor  122  for performing the various calculations of the methods disclosed herein. The DPU  116  may further comprise a memory device  124  for storing various data such as the raw data from the DAU  120  and various calculated results obtained via the methods disclosed herein. The memory device  124  may further include programs  126  containing a set of instructions that when accessed by the processor  122 , cause the processor  122  to perform the methods disclosed herein. The DPU  116  may provide results of the calculations to the memory device  124 , display  127  or to one or more users  128 . In various embodiments, the DPU  116  may wrap the resulting high-resolution DTS data into a managed data format that may be delivered to the users  128 . The DPU  116  may be in proximity to the DAU  120  to reduce data communication times between the DPU  116  and DAU  120 . Alternatively, the DPU  116  may be remotely connected to the DAU  120  through a high-speed network. 
     The raw data obtained at the DAU  120  may include noises at levels that are in a range from one to several degrees Celsius. Such noises may originate due to attenuation loss, noise in the data acquisition system, environmental temperature variations of the fiber optic cable, etc. In one embodiment, the present disclosure provides an adaptive filter to reduce those noises to thereby increase a resolution of the temperature measurements. In one embodiment, the temperature resolution of the data after the filtering methods described herein may be greater than the resolution of the raw temperature measurement data. In an exemplary embodiment, a resolution of raw temperature measurement data that is from about 0.5° C. to about 1.5° C. may be processed using the methods disclosed herein to obtain a post-filtered resolution of about ten millidegrees Celsius. In general, an increase in temperature resolution may be about two orders of magnitude. 
       FIG. 2  shows an alternate embodiment of a wellbore system  120  suitable for temperature measurements according to the present disclosure. The alternate wellbore system  120  includes a member  132  having a DTS system  134  attached thereto in which a fiber optic cable  136  of the DTS system  134  is a dual-ended cable. The fiber optic cable  136  has a first leg  136   a  that extends from a surface location  140  to a bottom location  142  along one side of the member  132  and a second leg  136   b  that may extend from the bottom location  142  back to the surface location  140  along a same side of the member  132 . A third segment  136   c  of the fiber optic cable  136  may wrap around the bottom of the member  132 . Both ends of the fiber optic cable  136  are coupled to the interrogator unit  144 . Thus, source laser light generated at the interrogator unit  134  may enter the fiber optic cable at point A and propagate in one direction, referred to herein as a forward direction and indicated by arrows  144 , to return to the interrogator unit  134  at point B. Temperature measurements may thus be obtained for the laser light propagating in the forward direction. Alternatively, source laser light may enter the fiber optic cable at point B and propagate in an opposite direction, referred to herein as a backward direction and indicated by arrows  146 , to return to the interrogator  134  at point A. Temperature measurements may be obtained for the laser light propagating in the backward direction. 
     The raw temperature measurements obtained from the DTS systems of  FIGS. 1 and 2  exist in a locally-compact measurement space that is correlative and expandable. A two-dimensional measurement space in time and depth for the temperature measurements may be written as:
 
 R ( t,z| 0 &lt;t&lt;∞,−∞&lt;z &lt;∞)  Eq. (1)
 
for which there exists a subspace
 
 R   i,j ( t,z|t   i−n     t     &lt;t&lt;t   i+n     t     ,z   j−n     z     &lt;z&lt;z   j+n     z   )  Eq. (2)
 
(also referred to herein as R ij ) where 2n t  and 2n z  are respectively the dimensions for a window defining this subspace within the two-dimensional measurement space.
 
       FIG. 3  shows an exemplary data boundary of a localized two-dimensional subspace R ij  of the measurement space. The data boundary may be related to raw temperature measurement data and may be used in the exemplary filtration method described herein to filter the temperature measurements input into the filter. Signal point  302  is plotted as a function of the variables time (t) and depth (z), with the time plotted along the x-axis and the depth plotted along the y-axis. As shown in  FIG. 3 , exemplary signal point  302  is located at (i,j). In one aspect, window  304  is drawn around and centered at the exemplary signal point  302  to the selected subspace R ij . The dimension of the window  304  may define parameters of the applied filter. The window  304  has dimensions of 2n t +1 along the time axis and 2n z +1 along the depth axis and extends from i−n t  to i+n t  along the time axis and from j−n z  to j+n z  along the depth axis. The dimensions of the window  304  may affect a finite impulse response of a filter defined over the measurement subspace. 
     If n t  and n z  are of a selected size, for a raw temperature measurement T iΔi,j+ which falls into the subspace R ij , a Taylor series expansion may be used to correlate measurements for the current window with that of the center point T i,j  of the subspace using the following expression: 
                     T       i   +     Δ   ⁢           ⁢   i       ,     j   +     Δ   ⁢           ⁢   j           =       T     i   ,   j       +         (       ∂   T       ∂   t       )       i   ,   j       ⁢   Δ   ⁢           ⁢     id   t       +         (       ∂   T       ∂   z       )       i   ,   j       ⁢   Δ   ⁢           ⁢     jd   z       +         (         ∂   2     ⁢   T       ∂     t   2         )       i   ,   j       ⁢         (     Δ   ⁢           ⁢     id   t       )     2     2       +         (         ∂   2     ⁢   T       ∂     z   2         )       i   ,   j       ⁢         (     Δ   ⁢           ⁢     jd   z       )     2     2       +         (         ∂   2     ⁢   T         ∂   t     ⁢     ∂   z         )       i   ,   j       ⁢         (     Δ   ⁢           ⁢   i   ⁢           ⁢   Δ   ⁢           ⁢     jd   t     ⁢     d   z       )     2     2       +   …             Eq   .           ⁢     (   3   )                 
where d t  and d z  are respectively the distances along the temporal axis and the spatial axis between two neighboring sensing points within the measurement space, as shown in  FIG. 3 . Eq. (3) defines a multiple term decomposition of the DTS data, wherein the decomposition includes a Taylor series decomposition having terms of selected orders, e.g. first order terms, second order terms, etc. Each term of the Taylor series decomposition generally has an associated physical meaning and provides a different level of resolution to the raw temperature measurement data. The present disclosure employs a non-orthogonal transform of the Taylor series decomposition of Eq. (3) limited to a selected number of these representations. In one embodiment, terms of the Taylor series composition up to the second order are used and terms that are of orders higher than two are not considered. Equation (3) may thus be rewritten as:
 
 T   i+Δi,j+Δj   =     i+Δi,j+Δj   ·     i,j =Σ k=0   5   h   Δi,Δj   k     i,j   k   Eq. (4)
 
where    i,j  denotes a non-orthogonal transformation vector, and    i,j  denotes a vector containing the terms that are to be determined for the giving point (i,j). A linear reconstruction of the measurement T i,j  in the subspace R i,j  may be obtained by maximizing the energy compaction for the given transformation vector or, equivalently, by minimizing an expectation value of a linear estimator function:
 
Σ k=0   5 Ε[∥Γ i,j   k −{circumflex over (Γ)} i,j   k ∥ 2 ]  Eq. (5)
 
where, {circumflex over (Γ)} i,j   k  is the mean value of Γ i,j,   k  is a collection of the k th  term of the decomposition of the temperature measurements in subspace R ij . In particular, Γ i,j   k  are the elements of vector    i,j,   k as illustrated with respect to Eq. (8) below. Referring back to Eq. (5),
 
Γ i,j   k =Γ i,j    k−1 {circumflex over (Γ)} i,j   k−1   Eq. (6)
 
where Γ i,j   0  ={circumflex over (Γ)} i,j  is the actual raw temperature measurement (T i,j , in the measurement subspace and which may be a function of time and depth. Eq. (6) defines a generally time-consuming approach to the non-orthogonal transform problem, in which a k th  representation is progressively obtained using the (k−1) th  representation. However, the present disclosure speeds this process by using a single step approach in which the expectation of the linear estimator function (Eq. (5)) is rewritten as:
 
Σ Δi=−n     t     n     t   Σ Δj=−n     z     n     z   (T i+Δi,j+Δj −Σ k=0   5 h ΔiΔj   k     i,j   k ) 2   Eq. (7)
 
where    i,j  is a vector containing the following physical quantities:
 
                       𝒯   ⇀       i   ,   j       =       (       T     i   ,   j       ,       (       ∂   T       ∂   t       )       i   ,   j       ,       (       ∂   T       ∂   z       )       i   ,   j       ,       (         ∂   2     ⁢   T       ∂     t   2         )       i   ,   j       ,       (         ∂   2     ⁢   T       ∂     z   2         )       i   ,   j       ,       (         ∂   2     ⁢   T         ∂   t     ⁢     ∂   z         )       i   ,   j         )     T             Eq   .           ⁢     (   8   )                 
By defining a linear transfer function:
 
 = H ( H   T   H   T   H ) −1   H   T   Eq.(9)
 
with
 
                     H   =     (           h       -     n   t       ,     -     n   z         0         …         h       -     n   t       ,     -     n   z         5             ⋮       ⋱       ⋮             h       n   t     ,     n   z       0         …         h       n   t     ,     n   z       5           )       ,           Eq   .           ⁢     (   10   )                 
we can obtain the following solution:
 
   i,j = Γ i,j   Eq.(11)
 
     This solution to the Taylor series decomposition may also be viewed as a 2-dimensional filter for digitally filtering the raw temperature measurement data. Since the higher-order terms (i.e., terms of order greater than 2)in the Taylor series decomposition are not considered,   in Eq. (9)is only an approximate transfer function in which the approximation error depends on the size of subspace R ij . Therefore, a window size suitable for obtaining selected filtration results may be selected. An iterative self-adaptive algorithm, as shown in  FIG. 4  achieves this filtration result to a selected approximation error. 
       FIG. 4  shows a schematic diagram  400  of an iterative self-adaptive filtering process of the present disclosure. The iterative filtering process may be used to provide an accuracy or resolution of temperature measurements to within a selected approximation error. The filtering process preserves transition information for the set of continuous temperature measurement data. 
     Temperature signal T(t,z)  410  represents a raw DTS temperature measurement obtained from a DTS system which is an input signal to the filter system  400 . Noise signal n(t,z)  412  indicates an unknown noise signal accompanying the temperature measurements  410  and which is also input to the filter system  400 . In general, the temperature signal  410  and the noise signal  412  are indistinguishable in DTS systems and thus are input to filter  402  as a single measurement. In addition, noise signal n(t,z)  412  is often not constant but changes with changes in environment. Therefore, both temperature signal T(t,z)  410  and noise signal n(t,z)  412  are dependent on time and depth of the measurement location in the DTS system. Output signal  414  is a filtered output signal and may include multiple terms of the decomposition of Eq. (3), such as for 
               T     i   ,   j       ,       (       ∂   T       ∂   t       )       i   ,   j       ,       (       ∂   T       ∂   z       )       i   ,   j       ,         
etc.
 
     In one embodiment, the exemplary filter  402  is a self-adaptive filter using a dynamic window (such as data window  304  in  FIG. 3 ) that may be adjusted to reduce noise in the temperature measurements. The temperature signal  410  and noise signal  412  are fed to filter  402  which provides an approximation to the temperature measurements using the methods disclosed above with respect to Equations (1)-(12). In various embodiments, the approximation may provide values for one or more of terms 
               T     i   ,   j       ,       (       ∂   T       ∂   t       )       i   ,   j       ,         (       ∂   T       ∂   z       )       i   ,   j       .           
A criterion  404  may then be applied to the terms output from the filter  402  to determine an effectiveness of the filter  420 . In one embodiment, the selected criterion may be a selected resolution of the temperature measurements or a selected resolution for a selected term of the decomposition. If the filtered terms are found to be within the selected resolution, the filtered terms may be accepted as output signals  414 . Otherwise, the filter  402  may be updated at updating stage  406 . Updating may include, for example, changing the dimensions of the measurements subspace R ij . In various embodiments, this decomposition process represents DTS measurement data as a Taylor series decomposition that includes terms having various levels of temperature resolution. The first order terms have a resolution that is greater than zero-order terms, the second order terms have a resolution greater than the first order terms, etc. The first order terms, which are thermal derivatives in depth or time and the second order derivatives (i.e., variance with respect to depth, variance with respect to time and variance with respect to depth and time) may reach temperature resolutions up to several hundredths of a degree.
 
     Although the methods are discussed with respect to temperature measurements, the present disclosure may also be applied to any suitable signal that is a continuous function measured in a two-dimensional measurement space. While the method is described with respect to a Taylor series decomposition (Eq. (3)), other numerical decompositions may be also used in various alternate embodiments. 
     The methods disclosed herein may be applied to both single and double ended DTS measurements. For the latter application, a correction of the asymmetry of temperature measurements may be performed. As shown in  FIG. 2 , the raw temperature data are obtained for both forward and backward propagation directions of the laser light transmitting along the double-ended DTS cable  136 . In general, the data from the two legs ( 136   a  and  136   b ,  FIG. 2 ) are not symmetric predominantly due to attenuation loss of the laser light which makes the amplitude of light propagating, at a selected fiber position (e.g. point C), in the forward direction not the same as the amplitude of the light propagating in the backward direction. Correcting for this asymmetrical attenuation using the methods disclosed herein may increase resolution, especially for the first order terms and higher. 
       FIG. 5  shows a flowchart  500  illustrating an exemplary method of correcting bi-directional DTS temperature measurements for asymmetric data. In block  502 , a two-dimensional digital filtration process, such as discussed with respect to  FIG. 4 , is performed. In block  504 , temperature curves for left and right legs are obtained for one or more sections of the member. In block  506 , for a selected section, cross-correlation coefficients are calculated for temperature measurements in the left and right legs. In block  508 , a maximal correlation is found using the cross-correlation coefficients obtained in block  506 . In block  510 , calibration parameters are modified. In one aspect, some of the calibration parameters may be used to correct a depth misalignment between the two legs ( 136   a  and  136   b ,  FIG. 2 ). In another aspect, at least one of the calibration parameters may be used to offset the systematic temperature differences in the forward and backward propagating data measurements. In block  512 , a determination is made on whether the modified calibration parameters provide a stronger correlation. If a stronger correlation is not obtained with the modified calibration parameters, then the method returns to block  506  to calculate cross-correlation coefficients. If a stronger correlation is obtained, the method proceeds to block  514  in which DTS data is updated using the calibration parameters that provide the stronger correlation. After block  514 , in block  516  the updated DTS data is mapped to a fixed depth position of the member. 
       FIG. 6  shows a flowchart  600  illustrating an exemplary method of reducing system level noises in the DTS data. The system level noise may include a systemic fluctuation of DTS data from one scan to another, or an oscillation of the temperature thermal gradient (TTG). In block  602 , temporal thermal gradient (TTG) data is calculated. The TTG data may be a representation output from the filtering process shown in  FIG. 4 , and specifically the partial derivative with respect to time, ∂T(t,z)/∂t. In block  604 , a two-dimensional wavelet transformation is performed on the TTG data. In block  606 , a featured noise profile is obtained. In block  608 , data filtration is conducted in the transformed space. In block  610 , a reverse two-dimensional wavelet transformation is performed to obtain filtered TTG data. In block  612 , the filtered TTG data may be used to obtain DTS temperature data with reduced noise. 
       FIG. 7  shows various thermal gradient data sets obtained using a DTS measurement. The data set  702  shows temporal thermal gradient data obtained from raw temperature data over a selected depth interval (along the y-axis) and over a selected time interval (along the x-axis). In one embodiment, the data set  702  may be obtained using a three-point central difference formula after taking five-point moving average of raw temperature data. The data set  702  may be color coded to indicate a cooling or a heating of the wellbore or formation. For example, a red color at a selected time and depth indicates that temperature is increasing at the selected time and depth. A blue color at a selected time and depth indicates that temperature is decreasing at the selected time and depth. A green color indicates that temperature is constant. The data set  704  is a temporal thermal gradient profile obtained using the same DTS data as in temperature data set  702  and the methods disclosed herein. While data set  702  shows a strong noise background that covers the actual temperature signal, data set  704  displays a strong temperature signal. In data set  704 , the temperature at substantially all depths is decreasing (cooling) during time intervals  710  and  712 , and is increasing (heating) during time interval  714 . Between these time intervals  710 ,  712  and  714 , the temperature remains constant, as indicted by the green color. The decrease in temperature in time intervals  710  and  712  may be related to the occurrence of two consecutive liquid injections, in one embodiment. 
     Temperature data set  706  shows a color map of a spatial thermal gradient (STG) obtained over a depth interval for a selected time period or time interval. The data set  706  is obtained using the same three-point central difference formula used with respect to data set  702 . Temperature data set  708  is the STG color map obtained using the same data set  706  and the methods disclosed herein. Data sets  704  and  708  provide evidences that the disclosed method is capable of retrieving clear signals on temperature gradient with respect to depth from a generally noisy raw DTS data set. While very little in the way of a distinguishable temperature signal may be found in data set  706 , distinctive signals at depths  720 ,  722 ,  724 ,  726  and  728  (in data set  708 ) are displayed. Any of the signals at depths  720 ,  722 ,  724 ,  726  and  728  may be related, in various embodiments, to a change in a size of a tubular used for water injection, in a change in fluid flow direction such as a crossover, a liquid entrance to the formation, an acid reaction with carbonate formation, etc. 
     Therefore in one aspect, the present disclosure provides a method of obtaining a temperature profile of a wellbore, the method including: obtaining raw temperature data from the wellbore using a distributed temperature sensor system, the raw temperature data including noise; performing a numerical decomposition of the raw temperature data within a dynamic window in measurement space of the raw temperature data to obtain decomposition terms of first order and higher; applying an adaptive filter to the decomposition terms of first order and higher within the dynamic window to reduce noise from the decomposition terms of first order and higher; and using the filtered decomposition terms of first order and higher to obtain a temperature profile of the wellbore. In one embodiment, the method further includes calibrating a measurement depth using a heat source at a known depth. 
     In another aspect, the present disclosure provides a system for obtaining a temperature profile at a downhole location, the system including: a distributed temperature system configured to obtain raw temperature data from the downhole location, wherein the raw temperature data includes noise; and a processor configured to: perform a numerical decomposition of the raw temperature data within a dynamic window in measurement space of the raw temperature data to obtain decomposition terms of first order and higher; apply an adaptive filter to the decomposition terms of first order and higher within the dynamic window to reduce noise from the decomposition terms of first order and higher; and use the filtered decomposition terms of first order and higher to obtain a temperature profile of the wellbore. 
     In yet another aspect, the present disclosure provides a computer-readable medium having instructions stored thereon that are accessible to a processor and enable the process to perform a method for obtaining a temperature profile at a downhole location, the method including: obtaining raw temperature data from the downhole location from a distributed temperature sensor system; performing a numerical decomposition of the raw temperature data within a dynamic window in measurement space of the raw temperature data to obtain decomposition terms of first order and higher; applying an adaptive filter within the dynamic window to reduce noise on the decomposition terms of first order and higher; and using the filtered decomposition terms of first order and higher to obtain a temperature profile of the wellbore. 
     While the foregoing disclosure is directed to the preferred embodiments of the disclosure, various modifications will be apparent to those skilled in the art. It is intended that all variations within the scope and spirit of the appended claims be embraced by the foregoing disclosure.