Document:

<PAGE>

                                                                     Exhibit 4.3

THE SECURITIES REPRESENTED BY THIS WARRANT AND THE COMMON STOCK ISSUABLE THEREBY
HAVE NOT BEEN AND WILL NOT BE, EXCEPT AS PROVIDED IN THE REGISTRATION RIGHTS
AGREEMENT REFERRED TO BELOW, REGISTERED UNDER THE SECURITIES ACT OF 1933, AS
AMENDED (THE "SECURITIES ACT"), OR ANY OTHER APPLICABLE SECURITIES LAW AND,
ACCORDINGLY, THE SECURITIES REPRESENTED BY THIS CERTIFICATE MAY NOT BE RESOLD,
PLEDGED, OR OTHERWISE TRANSFERRED, EXCEPT PURSUANT TO AN EFFECTIVE REGISTRATION
STATEMENT UNDER, OR IN A TRANSACTION EXEMPT FROM REGISTRATION UNDER, THE
SECURITIES ACT AND IN ACCORDANCE WITH ANY OTHER APPLICABLE SECURITIES LAWS.

                                    WARRANT

                to Purchase 9,000,000 Shares of Common Stock of

                          MILLER EXPLORATION COMPANY

     This Common Stock Purchase Warrant (the "Warrant") certifies that for value
                                              -------
received, Guardian Energy Management Corp., a Michigan corporation ("Guardian"),
                                                                     --------
or its assigns, is entitled to subscribe for and purchase from Miller
Exploration Company, a Delaware corporation (the "Company"), in whole or in
                                                  -------
part, 9,000,000 shares (the "Warrant Shares") of Common Stock, par value $0.01
                             --------------
per share, of the Company ("Common Stock") at the Exercise Price (as hereinafter
                            ------------
defined), subject, however, to the provisions and upon the terms and conditions
hereinafter set forth This Warrant and all rights hereunder shall expire at 5:00
p.m., Traverse City, Michigan time, on the fourth  anniversary of the approval
of the Conversion Shares by the stockholders of the Company (the "Termination
Date").

                                   ARTICLE I

                             Exercise of Warrants
                             --------------------

     1.1  Term; Exercise Price.  The Warrant represented hereby may be exercised
          --------------------
by the holder hereof, in whole or in part, at any time and from time to time
after the date hereof until 5:00 p.m., Traverse City, Michigan time, on the
Termination Date (the "Term").  Subject to adjustment as provided in Article IV
                       ----                                          ----------
hereof, the exercise price for shares of Common Stock under this Warrant shall
be $3.00 per share of Common Stock (the "Exercise Price").
                                         --------------

     1.2  Method of Exercise.  To exercise this Warrant, the holder hereof shall
          ------------------
deliver to the Company, at the Warrant Office designated in Section 2.1 hereof,
                                                            -----------
(i) a written notice in the form of the Subscription Notice attached as Exhibit
                                                                        -------
A hereto, stating therein the election of such holder to exercise the Warrants
-
in the manner provided in the Subscription Notice; and (ii) payment in full of
the Exercise Price in cash or by wire transfer or check for all Warrant Shares
purchased hereunder.
<PAGE>

This Warrant shall be deemed to be exercised on the date of receipt by the
Company of the Subscription Notice, accompanied by payment for the Warrant
Shares and surrender of this Warrant, as aforesaid, and such date is referred to
herein as the "Exercise Date". Upon such exercise, the Company shall, as
               -------------
promptly as practicable and in any event within five business days, issue and
deliver to such holder a certificate or certificates for the full number of the
Warrant Shares purchased by such holder hereunder. As permitted by applicable
law, the person in whose name the certificates for Common Stock are to be issued
shall be deemed to have become a holder of record of such Common Stock on the
Exercise Date and shall be entitled to all of the benefits of such holder on the
Exercise Date, including without limitation the right to receive dividends and
other distributions for which the record date falls on or after the Exercise
Date and to exercise voting rights.

     1.3  Expense and Taxes.  The Company shall pay all expenses and taxes
          -----------------
(including, without limitation, all documentary, stamp, transfer or other
transactional taxes) attributable to the preparation, issuance or delivery of
the Warrant and of the shares of Common Stock issuable upon exercise of the
Warrant.

     1.4  Reservation of Shares.  The Company shall reserve at all times so long
          ---------------------
as the Warrant remain outstanding, free from preemptive rights, out of its
treasury Common Stock or its authorized but unissued shares of Common Stock, or
both, solely for the purpose of effecting the exercise of the Warrant, a
sufficient number of shares of Company Stock to provide for the exercise of the
Warrant.

     1.5  Valid Issuance.  All shares of Common Stock that may be issued upon
          --------------
exercise of the Warrant will, upon issuance by the Company, be duly and validly
issued, fully paid and nonassessable and free from all taxes, liens and charges
with respect to the issuance thereof and, without limiting the generality of the
foregoing, the Company shall take no action or fail to take any action which
will cause a contrary result.

     1.6  No Fractional Share.  The Company shall not be required to issue
          -------------------
fractional shares of Common Stock on the exercise of this Warrant.  If more than
one Warrant shall be presented for exercise at the same time by the same holder,
the number of full shares of Common Stock which shall be issuable upon such
exercise shall be computed on the basis of the aggregate number of whole shares
of Common Stock purchasable on exercise of the Warrant so presented.  If any
fraction of a share of Common Stock would, except for the provisions of this

Section 1.6, be issuable on the exercise of this Warrant, the Company shall pay
-----------
an amount in cash calculated by it to be equal to the price of one share of
Common Stock as quoted on the NASDAQ National Market System at the close of
business on the date of such exercise multiplied by such fraction computed to
the nearest whole cent.

                                      -2-
<PAGE>

                                  ARTICLE II

                                   Transfer
                                   --------

     2.1  Warrant Office.  The Company shall maintain an office for certain
          --------------
purposes specified herein (the "Warrant Office"), which office shall initially
                                --------------
be the Company's offices at 3104 Logan Valley Road, P.O. Box 348, Traverse City,
Michigan 49685-0348, and may subsequently be such other office of the Company or
of any transfer agent of the Common Stock in the continental United States as to
which written notice has previously been given to the holder hereof.  The
Company shall maintain, at the Warrant Office, a register for the Warrants in
which the Company shall record the name and address of the person in whose name
this Warrant has been issued, as well as the name and address of each permitted
assignee of the rights of the registered owner hereof.

     2.2  Limited Right of Transfer.  This Warrant may not be directly or
          -------------------------
indirectly sold, assigned, pledged, hypothecated, encumbered, transferred or
otherwise disposed of without the prior written consent of the Company;
provided, however, that Guardian may assign its interest in this Warrant to one
or more of the entities or individuals listed on Schedule 2.2 hereto upon
receipt by the Company of (a) evidence, in form and substance satisfactory to
the Company, that each such assignee is an "Accredited Investor" as defined
under Rule 501 of the Securities Act, or (b) an opinion of the Company's counsel
that such assignment can be made without registration under the Securities Act.

     2.3  Transfer of Warrant.  The Company agrees to maintain at the Warrant
          -------------------
Office books for the registration and transfer of the Warrant.  Subject to the
provisions of Section 2.2, the Company, from time to time, shall register the
transfer of the Warrant in such books upon surrender of this Warrant at the
Warrant Office properly endorsed or accompanied by appropriate instruments of
transfer and written instructions for transfer satisfactory to the Company.
Upon any such transfer and upon payment by the holder or its transferee of any
applicable transfer taxes, new Warrants shall be issued to the transferee and
the transferor (as their respective interests may appear) and the surrendered
Warrant shall be canceled by the Company.  The Company shall pay all taxes
(other than securities transfer taxes) and all other expenses and charges
payable in connection with the transfer of the Warrant hereunder.

                                  ARTICLE III

                                 Anti-Dilution
                                 -------------

     3.1  Reclassification, Consolidation, Merger or Sale of Assets.  In the
          ---------------------------------------------------------
event that the Company shall be a party to any transaction (including without
limitation (i) any recapitalization or reclassification of the Common Stock
(other than a change in par value, or from par value to no par value, or from no
par value to par value), (ii) any combination or subdivision of the Common Stock
into a lesser or greater number of shares of Common Stock, (iii) any
consolidation of the Company with, or merger of the Company into, any other
person, (iv) any merger of another person into the

                                      -3-
<PAGE>

Company (other than a merger which does not result in a reclassification,
conversion, exchange or cancellation of outstanding shares of Common Stock of
the Company), or (v) any sale or transfer of all or a material amount of all of
the assets of the Company or any compulsory share exchange) pursuant to which
the Common Stock is converted into the right to receive other securities, cash
or other property, then lawful provision shall be made as part of the terms of
such transaction whereby the holder of this Warrant shall receive upon the
exercise hereof the kind and amount of securities, cash and other property
receivable upon such transaction by a holder of the number of shares of Common
Stock for which the Warrant was exercisable immediately prior to such
transaction. The entity formed by such consolidation or resulting from such
merger or which acquires such assets or which acquires the Company's shares, as
the case may be, shall make provision in its certificate or articles of
incorporation or other constituent document to establish such right. The above
provisions shall similarly apply to successive transactions of the foregoing
type.

     3.2  Common Stock Dilutions.  In the event that prior to exercise of this
          ----------------------
Warrant the Company issues or agrees to issue additional shares of Common Stock
pursuant to a stock split or stock dividend, or dividend or distribution of
rights to purchase stock, then the number of shares to be issued pursuant to the
exercise of this Warrant and the Exercise Price therefore shall be adjusted
proportionately.  In the event that prior to the exercise of this Warrant the
Company issues or agrees to issue cash or property pursuant to a dividend or
distribution, this Warrant shall represent the right to acquire, in addition to
the number of shares of Common Stock receivable upon exercise of this Warrant,
and without payment of any additional consideration therefor, the amount of such
cash or property which the holder hereof would have been entitled to receive
pursuant to such dividend or distribution had it been the holder of record of
the Common Stock receivable upon exercise of this Warrant on the record date
therefor.  The Company agrees and covenants that it will not, prior to the
exercise of this Warrant, in connection with any plan for the benefit of
employees or directors (i) issue a number of shares of Common Stock, securities
exercisable for Common Stock or other Common Stock equivalents (the "Plan
Securities") which in the aggregate exceeds fifteen percent (15%) of the total
shares of Common Stock outstanding as of the date thereof, and (ii) of the Plan
Securities issued pursuant to clause (i) of this Section 3.2, issue Plan
Securities which in the aggregate are equal to, convertible into or exercisable
for more than 300,000 shares of Common Stock for consideration, including any
applicable exercise or conversion price, which is below the price quoted for the
Common Stock at the close of business on the day prior to issuance of such Plan
Securities.

     3.3  Prior Notice of Certain Events.  In the event:
          ------------------------------

          (i)    the Company establishes a record date to determine the holders
of Common Stock who are entitled to (A) receive any dividend or other
distribution (other than a dividend payable in shares of Common Stock), (B)
participate in a redemption or repurchase of the then-outstanding shares of
Common Stock, or (C) vote in connection with any of the transactions identified
in clauses (ii),(iii), (iv) or (v) below; or

                                      -4-
<PAGE>

          (ii)   the Company shall (A) declare any dividend (or any other
distribution) on its Common Stock, other than a dividend payable in shares of
Common Stock or (B) declare or authorize a redemption or repurchase of the then-
outstanding shares of Common Stock; or

          (iii)  the Company shall authorize the granting to all holders of
Common Stock of rights or warrants to subscribe for or purchase any shares of
stock of any class or series or of any other rights or warrants; or

          (iv)   of any reclassification of Common Stock (other than a change in
par value, or from par value to no par value, or from no par value to par
value), or of any other capital reorganization or of any consolidation or merger
to which the Company is a party, or of the sale or transfer of all or
substantially all of the assets of the Company or of any compulsory share
exchange whereby the Common Stock is converted into other securities, cash or
other property; or

          (v)    of the voluntary or involuntary dissolution, liquidation or
winding up of the Company or such event is likely to occur.

then the Company shall cause to be mailed to the holder of this Warrant, at its
address as reflected in the Warrant Office transfer records, a notice stating
(A) the date on which a record (if any) is to be taken with respect to clause
(i) above, (B) the date on which any transaction or event contemplated in any of
clauses (i) through and including clause (v) above is expected to become
effective or the date of the anticipated closing of the public offering, and (C)
the time, if any is fixed, as to when the holders of record of Common Stock
shall be entitled to exchange their shares of Common Stock for securities or
other property deliverable upon any such reclassification, reorganization,
consolidation, merger, sale, transfer, share exchange, dissolution, liquidation
or winding up.  Each notice of an event specified in clause (i) above shall be
sent at least five (5) days prior to the record date specified therein and each
notice of an event specified in clauses (ii) through and including (v) above
shall be delivered at least fifteen (15) days prior to the proposed closing of
such transaction.

                                  ARTICLE IV

                                 Miscellaneous
                                 -------------

     4.1  Entire Agreement.  This Warrant, together with that certain
          ----------------
registration rights agreement of even date herewith by and between the Company
and Guardian (the "Registration Rights Agreement"), contain the entire agreement
between the holder hereof and the Company with respect to the Warrant Shares
purchasable upon exercise hereof and the related transactions and supersedes all
prior arrangements or understandings with respect thereto.

     4.2  Governing Law; Venue.  THIS WARRANT SHALL BE GOVERNED BY AND CONSTRUED
          --------------------
IN ACCORDANCE WITH THE APPLICABLE LAWS OF THE STATE OF

                                      -5-
<PAGE>

DELAWARE AND THE UNITED STATES OF AMERICA FROM TIME TO TIME IN EFFECT.

     4.3  Waiver and Amendment.  Any term or provision of this Warrant may be
          --------------------
waived at any time by the party which is entitled to the benefits thereof and
any term or provision of this Warrant may be amended or supplemented at any time
by agreement of the holder hereof and the Company, except that any waiver of any
term or condition, or any amendment or supplementation, of this Warrant shall be
in writing.  A waiver of any breach or failure to enforce any of the terms or
conditions of this Warrant shall not in any way effect, limit or waiver a
party's rights hereunder at any time to enforce strict compliance thereafter
with every term or condition of this Warrant.

     4.4  Illegality.  In the event that any one or more of the provisions
          ----------
contained in this Warrant shall be determined to be invalid, illegal or
unenforceable in any respect for any reason, the validity, legality and
enforceability of any such provision in any other respect and the remaining
provisions of this Warrant shall not, at the election of the party for whom the
benefit of the provision exists, be in any way impaired.

     4.5  Copy of Warrant.  A copy of this Warrant shall be filed among the
          ---------------
records of the Company.

     4.6  Notice.  Any notice or other document required or permitted to be
          ------
given or delivered to the holder hereof shall be in writing and delivered at, or
sent by telecopy or certified or registered mail to such holder at, the last
address shown on the books of the Company maintained at the Warrant Office for
the registration of this Warrant or at any more recent address of which the
holder hereof shall have notified the Company in writing.  Any notice or other
document required or permitted to be given or delivered to the Company shall be
delivered at, or sent by facsimile transmission or certified or registered mail
to, the Warrant Office, or such other address within the continental United
States of America as shall have been furnished by the Company to the holder of
this Warrant.

     4.7  Limitation of Liability; Not Stockholders.  No provision of this
          -----------------------------------------
Warrant shall be construed as conferring upon the holder hereof the right to
vote, consent, receive dividends or receive notices (other than as herein
expressly provided) in respect of meetings of stockholders for the election of
directors of the Company or any other matter whatsoever as a stockholder of the
Company.  No provision hereof, in the absence of affirmative action by the
holder hereof to purchase shares of Common Stock, and no mere enumeration herein
of the rights or privileges of the holder hereof, shall give rise to any
liability of such holder for the purchase price of any shares of Common Stock or
as a stockholder of the Company, whether such liability is asserted by the
Company or by creditors of the Company.

     4.8  Loss, Destruction, etc. of Warrant.  Upon receipt of evidence
          ----------------------------------
satisfactory to the Company of the loss, theft, mutilation or destruction of
this Warrant, and in the case of any such loss, theft or destruction upon
delivery of a bond of indemnity or such other security in such form and

                                      -6-
<PAGE>

amount as shall be reasonably satisfactory to the Company, or in the event of
such mutilation upon surrender and cancellation of this Warrant, the Company
will make and deliver a new Warrant of like tenor, in lieu of such lost, stolen,
destroyed or mutilated Warrant. Any Warrant issued under the provisions of this
Section 4.8 in lieu of any Warrant alleged to be lost, destroyed or stolen, or
-----------
in lieu of any mutilated Warrant, shall constitute an original contractual
obligation on the part of the Company. This Warrant shall be promptly canceled
by the Company upon the surrender hereof in connection with any exchange or
replacement. The Company shall pay all taxes (other than securities transfer
taxes) and all other expenses and charges payable in connection with the
preparation, execution and delivery of Warrants pursuant to this Section 4.8.
                                                                 -----------

     4.9  Compliance with Securities Act and Applicable Law.  The Company shall
          -------------------------------------------------
not be required to transfer this Warrant or to sell or issue Warrant Shares if
such transfer or issuance would constitute a violation by the Company of any
provisions of any law or regulation of any governmental authority.  While the
Company is required to register the resale of the Warrant Shares by holder(s)
pursuant to the Registration Rights Agreement, in the event the Warrant Shares
issuable on exercise of this Warrant are not then registered under the Act, the
Company may imprint the following legend or any other legend which counsel for
the Company considers necessary or advisable to comply with the Securities Act
of 1933:

          "The shares of stock represented by this certificate have not been
          registered under the Securities Act of 1933 or under the securities
          laws of any State and may not be sold, transferred or otherwise
          disposed of except upon such registration or upon receipt by the
          issuer, in form and substance satisfactory to the issuer, of an
          opinion of the issuer's counsel that registration is not required for
          such disposition."

     4.10 Registration Rights.  The Warrants Shares shall be entitled to such
          -------------------
registration rights under the Securities Act and under applicable state
securities laws as are specified in the Registration Rights Agreement.

     4.11 Headings.  The Article and Section and other headings herein are for
          --------
convenience only and are not a part of this Warrant and shall not affect the
interpretation thereof.

               [REMAINDER OF THIS PAGE INTENTIONALLY LEFT BLANK]

                                      -7-
<PAGE>

     IN WITNESS WHEREOF, the Company has caused this Warrant to be signed in its
name.

Dated:    July 11, 2000

                              MILLER EXPLORATION COMPANY

                              By: /s/ Kelly E. Miller
                                 -----------------------------------------------
                              Name: Kelly E. Miller
                                   ---------------------------------------------
                              Title: President
                                    --------------------------------------------

                                      -8-
<PAGE>

                                   EXHIBIT A
                                   ---------

                              SUBSCRIPTION NOTICE
                              -------------------

     The undersigned, the holder of the foregoing Warrant, hereby elects to
exercise purchase rights represented thereby for, and to purchase thereunder,
__________ shares of the Common Stock covered by such Warrant, and herewith
makes payment in full for such shares pursuant to Section 1.2 of such Warrant,
                                                  -----------
and requests that certificates for such shares (and any other securities or
other property issuable upon such exercise) be issued in the name of, and
delivered to ______________________________.

                                        ________________________________________

Dated:  ____________, _____.
<PAGE>

                                 SCHEDULE 2.2

                          LIST OF APPROVED ASSIGNEES
                          --------------------------

                        Jordan Exploration Company LLC
                               Martin G. Lagina
                                Robert M. Boeve
                               Wayne Sterenberg
                                 Craig Tester

          Any Persons (as defined in the Securities Purchase Agreement dated as
          of July 11, 2000 and entered into between Guardian and the Company)
          who control, are controlled by or are under common control with any of
          the above Persons.<PAGE>

                                                                     Exhibit 4.4

                FIRST AMENDMENT TO PROMISSORY NOTE, WARRANT AND
                         REGISTRATION RIGHTS AGREEMENT

     THIS FIRST AMENDMENT TO PROMISSORY NOTE AND WARRANT (the "Amendment") dated
as of July 18, 2000, is by and between Miller Exploration Company, a Delaware
corporation (the "Company"), and Veritas DGC Land, Inc., a Delaware corporation
("Veritas").

                              W I T N E S S E T H:

     WHEREAS, on April 15, 1999, the Company issued a promissory note (the
"Note") to Veritas in the principal amount of four million six hundred ninety-
six thousand forty and 60/100 dollars ($4,696,040.60), representing the
outstanding balance due Veritas for services provided to the Company in 1998 and
1999;

     WHEREAS, also on April 15, 1999, the Company issued a warrant (the
"Warrant") to Veritas that enables Veritas to purchase shares of the Company's
common stock, par value $0.01 per share (the "Common Stock"), in lieu of
receiving cash payments for the interest obligations under the Note, according
to the terms set forth in the Warrant;

     WHEREAS, also on April 15, 1999, the Company and Veritas entered into a
Registration Rights Agreement ("Registration Rights Agreement"), under which the
Company agreed to register shares of Common Stock issued upon exercise of the
Warrant as set forth therein;

     WHEREAS, as of the date hereof, the Warrant represents the right to
purchase 1,100,498 shares of Common Stock, which constitutes interest under the
Note from April 15, 1999 through October 14, 2000; and

     WHEREAS, the parties hereto desire to amend certain terms of the Note and
the Warrant.

     NOW, THEREFORE, in consideration of the premises and for good and valuable
consideration, the adequacy, receipt and sufficiency of which are hereby
acknowledged, the parties hereto hereby agree as follows:

     1.   Amendment to Note.
          -----------------

          (i) The Note is amended by deleting the introductory paragraph thereto
in its entirety and substituting in lieu thereof the following:

          FOR VALUE RECEIVED, MILLER EXPLORATION COMPANY, a Delaware
     corporation, promises to pay to the order of VERITAS DGC LAND, INC., a
     Delaware corporation, at Suite 932, 3701 Kirby Drive, Houston, Texas 77098-
     3982 (or such other place as the holder hereof may hereafter designate in
     writing), the principal sum of FOUR MILLION SIX HUNDRED NINETY-SIX THOUSAND
<PAGE>

     FORTY AND 60/100 DOLLARS ($4,696,040.60) (or the unpaid balance of all
     principal advanced against this note, if that amount is less), together
     with interest as follows: (a) provided that all obligations outstanding
     under this Note are paid in full by December 31, 2001 (the "Target Date")
     and subject to the provisions of Section 4 herein, at the Stated Rate, and
     (b) in the event that all obligations outstanding under this Note are not
     paid in full by the Target Date, at the Past Due Rate, subject to an
     additional adjustment as provided in Section 4 hereof; provided, that for
     the full term of this note the interest rate produced by the aggregate of
     all sums paid or agreed to be paid to the holder of this note for the use,
     forbearance or detention of the debt evidenced hereby (including, but not
     limited to, all interest on this note at the Stated Rate plus the
     Additional Interest) shall not exceed the Ceiling Rate.

          (ii) Section 1 of the Note is amended by deleting the following
definitions in their entirety and substituting in lieu thereof the following:

          "Credit Documents" means any and all papers now or hereafter
     governing, evidencing, guaranteeing, securing or otherwise relating to all
     or any part of the indebtedness evidenced by this note, including without
     limitation this note, the Warrant, the Registration Rights Agreement, and
     any and all existing and future amendments to such documents.

          "Maturity Date" means July 21, 2003.

          "Past Due Rate" means a rate per annum equal to the lesser of (i) the
     Ceiling Rate or (ii) thirteen and three-quarter percent (13 3/4%).

          "Stated Rate" means nine and three-quarters percent (9 3/4%).

          (iii)  Section 1 of the Note is also amended by adding the following
new definitions thereto:

          "Affiliate" means, as to any corporation, partnership, limited
     liability company, joint venture, association, joint-stock company, trust
     or unincorporated organization (an "Entity"), (a) any Entity controlling,
     controlled by or under common control with such Entity, (b) any director or
     officer of such Entity, (c) any Entity  that beneficially owns or holds
     five percent (5%) or more of the voting interest of such Entity, or (d) any
     natural person who, individually or with his immediate family members,
     beneficially owns or holds five percent (5%) or more of the voting interest
     of such Entity.

          "Amendment" means this Amendment to the Note, dated July 18, 2000.

          "Amended and Restated Credit Agreement" means the credit agreement
     entered into on or about July 18, 2000, by and between Maker and Bank One
     Texas, N.A, a national banking association, as such may be amended from
     time to time, a copy of which has been provided to Veritas.

                                       2
<PAGE>

          "Borrowing Base Compliance" means the compliance by Maker with the
     borrowing base requirements under the terms of the Amended and Restated
     Credit Agreement, as determined from time to time by the lender thereunder.

          "Change in Control" means (a) the sale, lease or other transfer of all
     or substantially all of the assets of Maker to any person or group (as such
     term is used in Section 13(d)(3) of the Securities Exchange Act of 1934, as
     amended); (b) the merger or consolidation of Maker with or into any other
     entity or the merger of another entity into Maker or any subsidiary thereof
     with the effect that immediately after such transaction the stockholders of
     Maker immediately prior to such transaction hold less than fifty percent
     (50%) of the total voting power of all securities generally entitled to
     vote in the election of directors, managers or trustees of the entity
     surviving such merger or consolidation; (c) the acquisition by any person
     or group, pursuant to a single transaction or series of related
     transactions, of more than fifty percent (50%) of the voting power of all
     securities of Maker generally entitled to vote in the election of directors
     of Maker (provided that this shall not be triggered by transactions with
     Guardian, Maker or Affiliates thereof); (d) a majority of the Board of
     Directors of Maker (the "Board") is composed of members who (i) have served
     less than twelve (12) months and (ii) were not approved by a majority of
     the Board at the time of their election or appointment; or (e) any
     transaction or series of related transactions, the result of which Maker is
     no longer subject to the reporting requirements of Section 13 of the
     Securities Exchange Act of 1934 or listed on Nasdaq.

          "Common Stock" means the Common Stock, par value $0.01 per share, of
     Maker.

          "Guardian" means Guardian Energy Management Corp., a Michigan
     corporation.

          "Guardian Transactions" means the transactions evidenced by that
     certain Securities Purchase Agreement, dated as of July 11, 2000, by and
     between Maker and Guardian, and the other documents related thereto, copies
     of which have been provided to Veritas.

          "Net Borrowing Base" means the engineered borrowing base, minus
     projected payments to Amerada Hess, plus projected cash flow after all
     payments (provided that the calculation of cash flow is satisfactory to the
     lender), minus capital expenditures.

          "Offer Term" means the period beginning on the date of this Amendment
     and ending on the first anniversary of the earlier of (i) the Maturity
     Date, or (ii) the date the obligations under the Note are paid in full, if
     prior to the Maturity Date.

          "Target Date" means December 31, 2001.

                                       3
<PAGE>

          "Warrant Shares" mean shares of Common Stock that are issuable to
     Payee upon the exercise of the Warrant, which represent interest under the
     terms of this Note pursuant to Section 3 hereof.  The number of Warrant
     Shares, as may be increased from time to time, shall be determined in
     accordance with the terms of the Warrant.

          (iv) Section 3 of the Note is amended by reordering the paragraphs
thereof such that the current paragraph (c) becomes paragraph (d), and by adding
the following new paragraph (c) immediately preceding such paragraph (d) as
follows:

          (c) From and after October 15, 2000, interest hereunder shall continue
     to accrue, and shall be payable on April 15, 2001 and continuing on each
     October 15 and April 15 thereafter until the principal amount of the Note
     has been paid in full.

          (v) Section 3 of the Note is also amended by adding the following new
paragraphs (e), (f) and (g) at the end thereof:

          (e) Anything in this Note to the contrary notwithstanding and subject
     to paragraph (f) of this Section 3, payments of interest hereunder, if any,
     shall be paid exclusively by an increase in the number of Warrant Shares
     issuable upon the exercise of the Warrant, the number of which shall be
     determined in accordance with the terms of the Warrant; provided, however,
     that interest hereunder shall be paid exclusively in Warrant Shares only
     until such time as Maker is in Borrowing Base Compliance.

          (f) At such time as Maker has achieved Borrowing Base Compliance
     payments of interest shall be paid in cash on the date due; provided,
     however, that payments of interest shall be made only to the extent that
     such payments do not cause Maker's projected loan balance to exceed 80% of
     the Net Borrowing Base. In the event that Maker is unable to pay the full
     amount of any requisite interest payment in cash due to the restrictions
     set forth in the previous sentence, Maker shall pay as much of the interest
     payment in cash as possible, with the remainder of the interest to be paid
     in the form of additional Warrant Shares.  In connection herewith, Maker
     shall use commercially reasonable efforts to ensure that it achieves
     Borrowing Base Compliance and that it is able to pay interest due under
     this Note in cash as soon as practicable.  In furtherance of the foregoing,
     Maker shall promptly provide Payee with copies of all financial information
     and other reports that it is required to provide to its lender under the
     terms of the Amended and Restated Credit Agreement.  If Maker does not make
     a cash payment of interest on the date due, the interest shall be
     automatically made by an increase in Warrant Shares pursuant to Section
     3(e).

          (g) Maker may at any time pay the full amount or any part of this Note
     without payment of any premium or fee.

                                       4
<PAGE>

     (vi) The Note is further amended by deleting the current Section 4 therein
and substituting in lieu thereof the following new Section 4:

          4.  Other Payments.  In addition to the other provisions of this Note
     regarding payment of the outstanding principal amounts hereunder, Maker
     covenants and agrees that:

          (a) Maker shall make a payment of five hundred thousand dollars
     ($500,000) to Payee on the effective date of the Amendment to be applied
     toward repayment of the outstanding principal under the Note.

          (b) Maker shall make an additional payment (the "Additional Principal
     Payment") of five hundred thousand dollars ($500,000) to Payee as set forth
     in this Section 4(b), which will be applied toward repayment of the
     outstanding principal under the Note. In order to fund this payment, Maker
     plans to sell approximately two million five hundred thousand dollars
     ($2,500,000) worth of Common Stock or Common Stock equivalents to Eagle
     Investments, Inc., in exchange for an equivalent amount of cash and
     property (the "Eagle Transaction").  The Board of Directors of Maker (i)
     has approved the Eagle Transaction and has recommended that the
     stockholders of Maker approve such transaction, (ii) will not rescind or
     revoke such recommendation, and (iii) will use its commercially reasonable
     efforts to ensure that the Eagle Transaction is consummated on or before
     October 31, 2000.  The Additional Principal Payment shall be paid to Payee
     within five (5) days after the Company obtains stockholder approval of the
     Eagle Transaction, provided, that if the stockholders fail to approve the
                        --------
     Eagle Transaction on or before October 31, 2000, then the Additional
     Principal Payment shall be paid on the earlier of (x) ninety (90) days
     following the date on which the Company fails to obtain such stockholder
     approval for, or the stockholders reject, the Eagle Transaction, or (y)
     December 31, 2000.

          (c) In the event that the payment in paragraph (b) is not made on the
     date or in the amount provided herein, the number of Warrant Shares
     issuable under the Warrant shall automatically be increased by an amount
     equal to Seven Hundred Fifty Thousand Dollars ($750,000), divided by the
     Market Value (as defined in the Warrant) of the Common Stock as of such
     date.  This increase in the number of Warrant Shares is a penalty, and is
     not in lieu of interest or principal due under the Note.

          (d) All proceeds derived from the exercise of the warrants issued to
     Guardian pursuant to the Guardian Transactions, and all proceeds of any
     kind whatsoever received by Maker pursuant to any transaction involving the
     issuance of equity or debt (other than borrowings under the Amended and
     Restated Credit Facility) subsequent to the Guardian Transactions, shall be
     paid only to Veritas until all amounts outstanding hereunder are paid in
     full, and shall be applied, to the extent consistent with the terms of this
     Note, first to the payment of interest and then to the payment of
     principal.

                                       5
<PAGE>

          (e) In the event that Maker does not pay Payee the $73,317.25 pursuant
     to Invoice No. 0700-LS-02 on or before August 4, 2000, the prevailing
     interest rate hereunder shall be increased by 400 basis points and shall
     apply retroactively to the date of the Amendment (subject to additional
     increases according to the terms hereof).

     (vii)  Section 6 of the Note is amended by adding the following new
paragraph (i) at the end thereof:

          (i)  A Change in Control of Maker.

     (viii)  The Note is further amended by adding the following new Sections 20
and 21 at the end thereof:

          Section 20.  Subordination.   Subject to the following sentence, all
     indebtedness and obligations of Maker to Payee evidenced by this Note,
     including without limitation, all obligations for the payment of principal,
     interest, penalties, fees, costs, expenses and indemnities, shall be and
     hereby are subordinated in right of payment and claim in favor of the
     indebtedness and obligations of Maker, not to exceed fourteen million
     United States dollars (US $14,000,000) in principal amount, under that
     certain credit agreement dated as of February 9, 1998, as amended from time
     to time, by and between Maker and Bank of Montreal, Houston Agency and any
     successor or replacement credit facilities or borrowings of Maker
     (including, without limitation, the Bank One Credit Agreement) that do not
     by their terms provide that they are subordinate to or pari passu with this
     Note (the "Senior Obligations").  Until the Maturity Date, Payee agrees
     that it will not request, demand, accept, or receive (by set-off or other
     manner), and Maker agrees that it will not pay or deliver to Veritas, any
     payment amount, credit, or reduction of all or any part of the amounts
     owing under this Note, provided, however, that nothing in this Section 20
     shall be interpreted to prohibit Maker from making, or Payee from receiving
     or otherwise exercising its respective rights to, the payments required
     pursuant to Sections 3, 4 and 21.

          Section 21.  Right of First Offer.  During the Offer Term, Maker
     shall, or shall cause its subsidiaries or contractors to, offer Veritas the
     right to perform any and all geophysical services with respect to
     properties of which Maker (or its subsidiaries) serve as the operator and
     has control over the selection of the geophysical services provider
     ("Geophysical Services").  In connection with this right of first offer,
     Maker shall notify Veritas in writing of any Geophysical Services to be
     performed during the Offer Term, and Veritas shall have fifteen (15) days
     within which to submit a bid in connection therewith before Maker contacts
     any third party providers (each, a "Third Party Provider"). In the event
     that Payee submits a bid to perform Geophysical Services, Maker (or its
     affiliate) shall negotiate in good faith with Payee with respect thereto.
     If, following such good faith negotiations, Maker (or its Affiliate) does
     not accept Veritas' bid and enters into a contract or

                                       6
<PAGE>

     subcontract for Geophysical Services with a Third Party Provider, Maker
     shall promptly deliver a copy of such contract or subcontract to Veritas to
     demonstrate that the terms thereof are materially superior to Veritas' bid.
     If, however, Veritas does not submit a bid to perform Geophysical Services
     as provided herein, Maker shall be free to enter into a contract or
     subcontract with any Third Party Provider without further notice to
     Veritas; provided, however, that this shall not limit or impair Veritas'
     rights with respect to any subsequent Geophysical Services to be performed
     during the Offer Term.

          Section 22.  Amendment to Credit Agreement.  Maker hereby agrees and
     covenants that it will not, without the prior written consent of Payee,
     agree to amend the Bank One Credit Agreement in a manner that could
     reasonably be expected to (i) adversely affect the rights of Payee under
     this Note, or (ii) extend the timing for payment of any amounts thereunder
     beyond the Target Date.

     2.   Amendment to Warrant.
          --------------------

          (i) The title of the Warrant is amended by deleting the caption
"Expiring on April 15, 2002," and replacing it with "Expiring on June 21, 2004,
subject to extension."  The introductory paragraph of the Warrant is amended by
deleting the final sentence thereof in its entirety and replacing it with the
following:

     This Warrant and all rights hereunder shall expire at 5:00 p.m., Houston,
     Texas time, on July 21, 2004, subject to extension as provided in Section
     2.1 hereof.

          (ii) Section 1 of the Warrant is amended by deleting the definitions
for the following terms in their entirety and substituting in lieu thereof the
following new definitions:

          "Note" shall mean that certain promissory note of the Company in the
           ----
     principal amount of $4,696,040.60, payable to Veritas and executed
     concurrently herewith, as the same shall be amended from time to time.

          "Warrant Shares" shall initially be an amount equal to nine percent
           --------------
     (9%) of the outstanding principal amount of the Note on April 15, 1999,
     divided by the Market Price.  On the six-month anniversary thereof, the
     number of Warrant Shares shall be increased by an amount equal to nine
     percent (9%) of the outstanding principal amount of the Note as of such
     date, divided by the Market Price.  In addition, on the one-year
     anniversary of the date thereof, the number of Warrant Shares shall be
     increased by an amount equal to nine percent (9%) of the outstanding
     principal amount of the Note as of such date, divided by the Market Value
     of the Common Stock as of such date. Beginning on April 15, 2001, and
     continuing on each April 15 and October 15 thereafter until all principal
     and accrued and unpaid interest due under the Note is paid in full, the
     number of Warrant Shares shall be increased by an amount equal to (A) the
     product of one-half of the prevailing rate of interest under the Note,
     multiplied by the principal amount outstanding thereunder, and (B) divided
     by the Market Value of the Common Stock as of such date; provided, that

                                       7
<PAGE>

     if the Company pays all or a portion of the Note during the interim, the
     number of Warrant Shares shall be increased in accordance with the formula
     set forth above, but shall be pro-rated to the date of payment.
     Notwithstanding the foregoing sentence, the Company shall have the right to
     reduce or eliminate all or any part of its obligation to increase the
     number of Warrant Shares on such respective dates by delivering to Veritas
     and/or its assigns all or a portion of the interest payment due such date
     in cash.

          (ii) Section 1 of the Warrant is amended by adding the following new
definition:

          "Amendment" means the amendment to the Warrant dated July 18, 2000.

          (iii)  Section 2.1 of the Warrant is hereby amended by deleting the
current Section 2.1 and substituting in lieu thereof the following Section 2.1

          2.1  Term; Exercise Price.  The Warrants represented hereby may be
               --------------------
     exercised by the holder hereof, in whole or in part, at any time and from
     time to time after the date hereof until 5:00 p.m., Houston, Texas time, on
     the earlier of (i) one year from the date the Note is paid in full, or (ii)
     July 21, 2004 ("Term"); provided, however, that to the extent that the Note
     is not paid in full by the Company on or before June 21, 2003, the Term
     shall be automatically extended on a daily basis so that the Term shall
     continue until the first anniversary of the date on which the Company has
     satisfied all of its obligations under the Note.  Subject to adjustment as
     provided in Article IV hereof, the exercise price for shares of Common
     Stock acquired under the Warrant shall be $0.01 per share ("Exercise
     Price").

     3.  Amendment to Registration Rights Agreement.
         ------------------------------------------

          (i) Section 2.1 of the Registration Rights Agreement is hereby amended
by deleting the current Section 2.1 and substituting in lieu thereof the
following Section 2.1:

          2.1  Registration.  Promptly after the date hereof, the Company shall
               ------------
     file a Form S-3 Registration Statement with the Securities and Exchange
     Commission to register shares of Common Stock for issuance upon exercise of
     the Warrant in an amount equal to the initial Warrant Shares (as defined in
     the Warrant), and shall use its reasonable best efforts to ensure that such
     Registration Statement is declared effective within 180 days of the date
     hereof.  The Company shall use its reasonable best efforts to file an
     additional such registration statement to cover any increase in the number
     of Registrable Securities underlying the Warrant, which registration
     statement shall be effective as promptly as practicable following the date
     of such increase. The Company shall also use its reasonable best efforts to
     keep such registration statements effective until at least six months after
     the expiration of the Warrant.  Each such registration statement referred
     to herein as a "Registration Statement."  The registration rights granted
     under this Section 2.1 are in addition to those provided in Section 2.2
     below, and the Company's obligations hereunder shall not terminate or be
     limited by the fact that piggy-back registration rights are available.

                                       8
<PAGE>

     4.   Counterparts.  This Amendment may be executed in any number of
          ------------
counterparts, each of which shall be considered an original for all purposes,
and all of which when taken together shall constitute a single counterpart
instrument.  Executed signature pages to any counterpart instrument may be
detached and affixed to a single counterpart, with such single counterpart with
multiple executed signature pages affixed thereto constituting the original
counterpart instrument.  All of those counterpart pages shall be read as though
one, and they shall have the same force and effect as if all the signers had
executed a single signature page.

     5.   Notice of Amendment; Indemnification.  Veritas hereby covenants and
          ------------------------------------
agrees that it will attach this Amendment to the Note and otherwise notify any
potential transferee of the Note of the existence of this Amendment.  Veritas
also covenants and agrees to indemnify the Company of any and all damages,
losses, claims, liabilities, demands, charges, suits, penalties, costs and
expenses (including court costs and reasonable legal fees and expenses incurred
in investigating and preparing for any litigation or proceeding) incurred by the
Company in the event that Veritas transfers the Note without giving the notice
required in the previous sentence.

     6.   Reaffirmation.  Except for the amendments to the Note and the Warrant
          -------------
expressly set forth herein, the Note, the Warrant and all other documents
executed in connection therewith shall remain in full force and effect, and the
Company and Veritas hereby restate and reaffirm all of the terms and provisions
of the Note, the Warrant and such other documents.  This Amendment shall not
affect the number of Warrant Shares currently issuable upon the exercise of the
Warrant or the exercise price in connection therewith.

     7.   Effective Date.  This Amendment shall be effective when executed by
          --------------
each party hereto as of the date first written above.

               [REMAINDER OF THIS PAGE INTENTIONALLY LEFT BLANK]

                                       9
<PAGE>

     IN WITNESS WHERE, the parties hereto have executed this Amendment as of the
date first written above.

                                    MILLER EXPLORATION COMPANY

                                    By:    /s/ Kelly E. Miller
                                        ----------------------------
                                        Name:  Kelly E. Miller
                                        Title:  President

                                    VERITAS DGC LAND, INC.

                                    By:   /s/ Deanna Goodwin
                                        ----------------------------
                                        Name:  Deanna Goodwin
                                        Title:  Vice President

                                       10

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